Download Warm Up New Concepts

LES SON
Understanding Advanced Factoring
61
Warm Up
1. Vocabulary A polynomial of degree one with two terms is a
(11)
.
2. True or False: The GCF of -2x4y5 - xy2 + 2 is 2xy2.
(23)
3. Divide 4x2 - 2x + 5 by x - 1 using synthetic substitution.
(51)
New Concepts
Recall that synthetic division is used to divide a polynomial P(x) by a linear
binomial of the form x - a and that the last number in the bottom row is
the remainder and also the value of P(a). If the remainder is 0, the linear
binomial is a factor of the polynomial. This is known as the Factor Theorem.
Factor Theorem
For polynomial P(x), (x - a) is a factor of P(x) if and only if P(a) = 0.
For example, x + 2 is a factor of x2 + 10x + 16 and
Hint
x + 2 = x - (-2), so the
value of a is -2.
P(-2) = (-2)2 + 10(-2) + 16
= 4 - 20 + 16
= 0.
Example 1 Determining Whether a Linear Binomial is a Factor
Determine whether the linear binomial is a factor of the polynomial.
a. P(x) = 2x3 - x2 - 43x + 60, x - 4
SOLUTION Use synthetic division with a = 4.
4
2 -1 -43
60
8
28 -60
2
7 -15
0
Because P(4) = 0, x - 4 is a factor of P(x) = 2x3 - x2 - 43x + 60.
b. P(x) = x3 + x2 - 24x + 36, x + 3
SOLUTION Use synthetic division with a = -3.
-3 1
1 -24 36
6 54
-3
1 -2 -18 90
Because P(-3) ≠ 0, x + 3 is not a factor of P(x) = x3 + x2 - 24x + 36.
Online Connection
www.SaxonMathResources.com
436
Saxon Algebra 2
When the remainder is zero, the quotient of P(x) and (x - a) is another
factor of the polynomial. This factor may be able to be further factored by
other factoring methods. For instance, in Example 1a, the quotient
2x2 + 7x - 15 factors into (2x - 3)(x + 5). Therefore, P(x) factors into
(x - 4)(2x - 3)(x + 5).
If the remaining quotient is a cubic, it may be factorable by one of the
following factoring methods.
Sum and Difference of Cubes
3
Sum of Two Cubes: a + b3 = (a + b)(a2 - ab + b2)
Difference of Two Cubes: a3 - b3 = (a - b)(a2 + ab + b2)
Example 2
Math Reasoning
Justify Tell why 3x - 1 is
not a difference of cubes.
Factoring the Sum and Difference of Two Cubes
Factor each expression.
a. d 3 + 125
SOLUTION The expression is a sum of two cubes, where a = d and b = 5.
d 3 + 125
(d + 5)(d 2 - 5d + 25)
b. 27x6 + y9
SOLUTION The expression is a sum of two cubes, where a = 3x2 and b = y3.
27x6 + y9 = (3x2)3 + (y3)3
(3x2 + y3)(9x4 - 3x2y3 + y6)
c. 3m5 - 24m2
SOLUTION First factor out the GCF, 3m2. Then factor the difference of the
two cubes, where a = m and b = 2.
3m5 - 24m2
3m2(m3 - 8)
3m2(m - 2)(m2 + 2m + 4)
Another method of factoring is factoring by grouping. To factor by
grouping, group the terms evenly so that each group has a common factor.
Factor out the GCF from each group. Then factor out the common
polynomial factor.
Example 3
Factoring by Grouping
Factor each expression.
a. x3 + 5x2 + 3x + 15
SOLUTION The first two terms have a common factor as do the last two terms.
(x3 + 5x2) + (3x + 15)
x2(x + 5) + 3(x + 5)
Factor x2 from the first group and
3 from the second.
(x + 5)(x2 + 3)
Factor out the GCF of (x + 5).
Neither factor can be factored further.
Lesson 61
437
b. x4 + 4x3 - x - 4
Math Reasoning
Analyze Can a
polynomial with five
terms be factored by
using the grouping
method? Explain.
SOLUTION Factor x3 from the first two terms and -1 from the last two terms.
x4 + 4x3 - x - 4
x3(x + 4) - 1(x + 4)
(x + 4)(x3 - 1)
Factor out the GCF of (x + 4).
(x + 4)(x - 1)(x2 + x + 1)
Factor the difference of cubes.
c. x6 + 3x5 + 2x4 + x2 + 3x + 2
SOLUTION Use two groups of three terms each.
(x6 + 3x5 + 2x4) + (x2 + 3x + 2)
x4(x2 + 3x + 2) + 1(x2 + 3x + 2)
(x2 + 3x + 2)(x4 + 1)
Factor out the GCF of (x2 + 3x + 2).
(x + 1)(x + 2)(x4 + 1)
Factor the trinomial.
Example 4
Application: Three-Dimensional Figures
The volume of a rectangular solid with a length of x - 5 feet has a volume
of x3 - 23x2 + 170x - 400 cubic feet. Factor the expression for the volume
completely.
SOLUTION Since volume is the product of the length, width, and height, the
expression for the length, x - 5, is a factor of the polynomial. Use synthetic
division to find the quotient of the polynomial and x - 5.
5 1 -23 170 -400
5 -90
400
1 -18
80
0
The expression for the volume factors into (x - 5)(x2 - 18x + 80). But the
trinomial can still be factored: (x - 10)(x - 8).
The expression for the volume factors into (x - 5)(x - 10)(x - 8).
Lesson Practice
Determine whether the linear binomial is a factor of the polynomial.
a. P(x) = 3x3 + 19x2 + 32x + 16, x + 3
(Ex 1)
b. P(x) = 4x3 - 33x2 + 56x - 12, x - 6
(Ex 1)
Factor each expression.
c. 64 - h3
(Ex 2)
e.
(Ex 2)
8 7
_
x + x4
27
g. 4m2n + 12m2 - n - 3
(Ex 3)
d. s6t12 - 125r3
(Ex 2)
f. xy - 9y + 5x - 45
(Ex 3)
h. x5 + 12x4 + 36x3 + 5x2 + 60x + 180
(Ex 3)
i. The volume of a rectangular solid with a length of x - 15 feet has a
volume of x3 + 5x2 - 600x + 4500 cubic feet. Factor the expression for
the volume completely.
(Ex 4)
438
Saxon Algebra 2
Practice
Distributed and Integrated
Graph.
1. x = -5
(13)
2. 2x - y = 4
3. 7y + 7x = 21
(13)
(13)
4. Find the intercepts of the equation 5x - 4y + 7z = 21.
(Inv 3)
Simplify.
5. -16
1
-_
2
(59)
6. 27
1
-_
3
_3
7. 9 2
(59)
(59)
Find an inverse for the equations.
2
8. y = _x - 3
3
9. y = 5x + 15
(50)
(50)
10. Let K(u) = 17 and L(u) = -13. Find K(L(u)) and L(K(u)).
(53)
x≤4
*11. Minimize P = 4x + 3y for the constraints y ≤ 6
(54)
x+y≥7
*12. Mary deposited $980 at 7.0% compounded continuously. How much money will
(57)
she have after 9 years?
13. Meteorology The number of tornadoes in the United States in 2006 is shown in
(55)
the table.
Season
Number of Tornadoes
Jan-Feb
Mar-Apr
May-Jun
Jul-Aug
Sep-Oct
Nov-Dec
59
395
258
151
160
82
What is the experimental probability that a tornado occurs in the first
six months of the year?
*14. Write Tell how factoring a sum of cubes is similar to factoring a difference of cubes.
(61)
15. Probability A student graphed a set of data points and found that the correlation
(45)
coefficient was -0.73. What is the probability that the slope of the line of best fit
1
is greater than _2 ? Explain.
*16. Geometry The probability that a randomly selected point lies within a
(60)
region can be found using the ratio of the area the point would lie in
to the total area. Find the probability that a random point within the
region lies within one of the smaller circles.
6 in.
10 in.
2 in.
Lesson 61
439
17. Buildings One building is 310 feet tall and a nearby building is 374 feet tall. From
(46)
the top of the taller building, you can spot the top of the shorter building at an
angle of depression of 30°. How far apart are the two buildings? Round to the
nearest foot.
18. Walking Shadows While walking the dog one night, you notice how the length of
(53)
your shadow changes as you approach each lamppost. The length of the shadow
1
seems to be _5 of your distance to the lamppost. Your dog is dragging you along
at about 3 feet per second, aiming for a lamppost. Show how to write a composite
function that gives the length of your shadow in terms of time as you approach
the lamppost.
*19. Error Analysis Is the work below correct? If not, find and explain the error, and
(59)
give the correct solution.
_3
x8 = 3
(x _) _ = (3) _
3
8
3
8
3
8
_3
x1 = 3 8
8
x = √
27
20. Analyze Suppose you spin the spinner shown 8 times. What is the probability
(49)
that you will spin black 5 times?
21. Verify Use the form x2 + 2cx + c2 for a quadratic perfect square trinomial to verify
(58)
the method of completing the square.
*22. Graphing Calculator Describe the feasible region for the inequalities:
(54)
x ≥ 0, y ≥ 0, y ≤ -0.75x + 8
23. Multiple Choice Which value of r, the correlation coefficient, describes the set of
(45)
points that are closest to forming a line?
A r = -0.75
B r=0
C r = 0.6
D r = 0.65
*24. Commuting to Work According to data collected in the 2005 American Community
(60)
Survey, approximately 10.67% of workers carpool to work, 4.66% use public
transportation, and 2.47% walk. Find the probability that a randomly selected
worker carpools or uses public transportation.
25. Write A few different equations of the form y = bx are graphed to the
(47)
right. Describe in your own words how the shape of the graph depends
on the value of b.
6
4
y = .5 x
-4
-2
Saxon Algebra 2
y = 2.5 x
2
O
-2
440
y
2
y = 1.25 x
x
4
_3
*26. Multi-Step
(59)
_3
55 · 55
Use _.
_4
55
a. Which property of rational exponents should be applied first to simplify
the expression? Explain.
b. Apply the property from part a.
c. Which property of rational exponents should be applied next?
d. Apply the property from part c.
e. If possible, finish simplifying the expression from part d.
27. Multiple Choice In the diagram to the right, find the length of the
(52)
hypotenuse of triangle LMN.
A 7.5
B √
2
C 15
L
N
15 cm
_
√2
3
D 15 √
M
*28. Multi-Step The area of a twin size mattress is x2 - 275 - 11x + 25x square inches.
(61)
a. Factor the polynomial by grouping.
b. Find the area given that x = 50.
29. Amusement Parks The Giant Swing at Silver Dollar City in Branson, Missouri, rotates
(56)
through an angle measuring 230°. Find the reference angle for this angle.
*30. Verify Show that there are two different ways to group the terms in
(61)
3d + 6f + d2 + 2df and that both lead to the same factorization.
Lesson 61
441
LES SON
Using Complex Numbers
62
Warm Up
1. Vocabulary Any point on the number line is a
(1)
number.
2. Simplify (5 + 3x) - (6 - 2x).
(2)
3. Solve 5x2 - 125 = 0.
(58)
4. Simplify √
150 .
(40)
New Concepts
The equation x = ± √
-16 has no real solutions because there are no
real numbers that, when squared, equal -16. But the equation does have
solutions when a new set of numbers, outside of the real numbers, is
introduced. These numbers are the imaginary numbers.
Imaginary Numbers
An imaginary number is written in the form bi where b is a real number
and i is the imaginary unit.
i is a solution of x2 = -1.
Math Reasoning
-1 .
Because i2 = -1, i = √
Generalize For any
positive real n,
√-n
.
=
The square root of a negative number is an imaginary number.
√
-16 = √
16 · -1 = √
16 √
-1 = 4i
Example 1
Simplifying Square Roots of Negative Numbers
a. Simplify -2 √
-49 .
SOLUTION
-2 √
- 49
49 · -1
-2 √
Write -49 as the product of 49 and -1.
-2 √
49 √
-1
The square root of a product is the product of the
square roots.
-2 · 7 · i
√
49 = 7 and √
-1 = i.
-14i
Multiply the real numbers.
-117 .
b. Simplify √
SOLUTION
√-117
Online Connection
www.SaxonMathResources.com
442
Saxon Algebra 2
√9
· 13 · -1
Write -117 as a product with a perfect square.
√9
√13
√-1
The square root of a product is the product of the
square roots.
(i)
3 √13
√9
= 3 and √
-1 = i.
3i √13
Commutative Property of Multiplication
Example 2 Solving a Quadratic Equation with Imaginary
Numbers
Solve x2 + 25 = 0. Write the solutions in terms of i. Check the answers.
SOLUTION x2 + 25 = 0
x2 = -25
Math Language
Complex solutions of
quadratic equations
always occur in
conjugate pairs of the
form a + bi and a - bi.
Subtract 25 from each side.
x = ± √
-25
Take the square root of each side.
x = ±5i
Simplify the square root.
The solutions are the conjugate pairs 5i and -5i.
x2 + 25 = 0
x2 + 25 = 0
(5i)2 + 25 0
(-5i)2 + 25 0
25i2 + 25 0
25i2 + 25 0
25(-1) + 25 0
25(-1) + 25 0
Check
0=0 ✓
0=0 ✓
An imaginary number bi is part of a complex number.
Complex Numbers
A complex number is a number that can be written in the form a + bi,
where a and b are real numbers.
Math Reasoning
Analyze Are real
numbers a subset of
complex numbers or
are complex numbers a
subset of real numbers?
In a complex number, a is called the real part and bi is called the
imaginary part. If b = 0, then the imaginary part is 0 and the number is a
real number. If a = 0 and b ≠ 0, then the real part is 0 and the number is
an imaginary number.
Complex
Real
Imaginary
2 + 3i
2 + 0i = 2
0 + 3i = 3i
To add or subtract complex numbers, add or subtract the real parts and then
add or subtract the imaginary parts.
Add or Subtract Complex Numbers
(a + bi) + (c + di) = (a + c) + (b + d )i
(a + bi) - (c + di) = (a - c) + (b - d )i
Example 3 Simplifying Expressions with Complex Numbers
Add or subtract. Write the answer in the form a + bi.
a. (7 + 4i) + (-2 - 5i)
SOLUTION (7 + (-2)) + (4 + (-5))i = 5 + (-1)i = 5 - i
b. (5 - 3i) - (5 - 11i)
SOLUTION (5 - 5) + (-3 - (-11))i = 0 + 8i = 8i
Lesson 62
443
Example 4 Solving Quadratic Equations
Solve each equation. Write the solutions in the form a + bi.
a. (x - 3)2 = -4
SOLUTION (x - 3)2 = -4
-4
x - 3 = ± √
Take the square root of each side.
x - 3 = ±2i
Simplify the square root.
x - 3 = 2i
or x - 3 = -2i
x = 3 + 2i
Solve both equations.
x = 3 - 2i
or
The solutions are 3 + 2i and 3 - 2i.
b. x2 + 8x + 18 = 0
SOLUTION x2 + 8x = -18
2
x + 8x + 16 = -18 + 16
2
(x + 4) = -2
Subtract 18 from each side.
Complete the square.
Write the left side as a perfect square.
x + 4 = ± √
-2
Take the square root of each side.
x + 4 = ±i √
2
Simplify the square root.
x = -4 ± i √
2
Subtract 4 from each side.
The solutions are -4 + i √2 and -4 - i √2.
Lesson Practice
a. Simplify _12 √
-100 .
(Ex 1)
b. Simplify 3 √
-450 .
(Ex 1)
c. Solve 0 = 196 + x2. Write the solutions in terms of i. Check the answer.
(Ex 2)
d. Solve -6x2 = 216. Write the solutions in terms of i.
(Ex 2)
Add or subtract. Write the answer in the form a + bi.
(Ex 3)
e. (-2 + i) + (14 + 4i)
f. (10 - 6i) - (-4 + 3i)
Solve each equation. Write the solutions in the form a + bi.
(Ex 4)
g. (x + 9)2 = -9
h. x2 - 2x + 2 = 0
Practice
Distributed and Integrated
Factor.
*1. 125x4 + 375x3 - x - 3
(61)
444
Saxon Algebra 2
*2. 64x12 - 125y9
(61)
Add or subtract.
a
3. a + _
(37)
b
ax2 - c + _
2
4. _
m
(37) m2p
gh2
g2h2
_
5. -_
+
xh
+
(37)
x3
gx3
Factor the greatest common factor.
6. 35x7y5m - 7x5m2y2 + 14y7x4m2
7. 6x2ym5 - 2x2ym + 4xym
(23)
(23)
Simplify.
8.
(59)
1
_
81
-3
_
4
9. (-27)
-2
_
10.
3
(59)
(59)
1
_
-3-2
11. School It is estimated that 15% of students walk to school. What is the probability
(49)
that exactly 2 out of 5 randomly selected students will walk to school?
12. Find an equation for the inverse of y = _15 x3 + 6.
(50)
13. What are the odds in favor of rolling a 3 on a number cube?
(55)
*14. Multiple Choice Which is a real number?
(62)
A 15i
B 0 + 15i
C 5 + 0i
D 15 + 15i
1
*15. Verify Show that the expression _
is equivalent to _
1
x+2 .
_
x+1
(37)
1+
x+1
*16. Electrical Engineering A circuit has a current of (4 - 2i) amps, and a second circuit
(62)
has a current of (6 - 6i) amps. Find the sum of the currents.
17. Analyze Explain why the odds in favor of an event with a probability of 50% are 1 to 1.
(55)
18. Projectile Motion The Leaning Tower of Pisa measures 180.4 feet. If an object is
(58)
thrown into the air, with an initial velocity of 80 feet per second, from the top of
the lower side, the height, h, of the object in feet after t seconds can be described
by the equation h = -16t2 + 80t + 180.4. Find the time at which the object will
reach a height of 200 feet. Round your answer to the tenths place.
*19. A number cube is rolled once. Are the events a number greater than 3 or a multiple
(60)
of 2 mutually exclusive or inclusive? Explain why.
*20. Transportation The approximate volume of a moving truck, in cubic feet, can be
(61)
given by x3 - 6x2 + 8x + 5x2 - 30x + 40. Factor this expression by grouping.
Then find the volume when x = 10.
21. Multiple Choice Which of the following angles is not coterminal with 225º?
(56)
A 585º
B -135º
C 935º
D -495º
22. Multi-Step The population of an insect colony is growing at a rate of 1.5% per week.
(57)
a. Write an expression for the function N(P) that gives the population one week
after the population is P.
b. What composite function gives the population three weeks after the population
is P?
Lesson 62
445
*23. Graphing Calculator Factor 2x5 - x4 - 2x + 1. Then enter the original expression for
(61)
Y1 and the factored expression for Y2 on a graphing calculator. Access the Table
function and study the table of values for the expressions. What is true? What does
it mean?
24. Write Explain why radical expressions with even indices and positive radicands
(59)
6
have two real roots. Use √
64 to demonstrate your explanation.
*25. Geometry An expression for the volume of the rectangular
(51)
prism is V = 2x3 + 7x2 - 14x - 40. Find an expression for
the missing dimension.
x+2
?
26. Analyze Suppose the graph of y = bx + k passes through the
(57)
points (3, -3) and (5, 0). Determine whether the equation
models exponential growth or decay.
x+4
27. Error Analysis A student made an error while studying the relation
(52)
between the two triangles shown. Explain the error.
Find the missing angles.
Y
30°
60°
m∠X = 90° m∠Y = 60° m∠Z = 30°
m∠P = 90° m∠Q = 60° m∠R = 30°
The corresponding angles are congruent. So, the triangles are congruent.
R
28. Probability A sales clerk has a weekly quota of 15 sales. The probability, p, that the
(58)
sales clerk will meet this quota after working d days, in a 5-day work week, can
be approximated by p = -0.04d2 + 0.4d. How many days a week should the sales
clerk work for the probability of making 15 sales to be 0.8?
*29. Verify Show that 11i and -11i are solutions of x2 = -121.
(62)
30. Cell Phones Mobile-U has two cell phone plans, the Metropolitan Plan and the
(54)
Continental Plan. The cost for each plan is shown in the table. A company is
comparing the two plans. The company needs at least 12 cell phones, one for each
sales representative. Each representative will use at least 500 minutes per month.
What options does this company have?
446
Saxon Algebra 2
P
X
m∠Y = 90° - 30° = 60° m∠R = 90° - 60° = 30°
Start-up Fee
Free minutes
Per minute
rate above free
minutes
Monthly
charge
Z
Continental Plan
$50
2000
Metropolitan Plan
$100
1200
10 cents
7 cents
$20 × # of cell phones
$25 × # of cell phones
Q
LESSON
63
Understanding the Unit Circle and
Radian Measures
Warm Up
1. Vocabulary Half the diameter of a circle is the
(SB)
of the circle.
2. If both legs of a right triangle have length 1, then the length of the
.
hypotenuse is
(41)
3. If the shortest leg of a 30°–60°–90° triangle has length 1, then the lengths
of the other two sides of the triangle are
and
.
(52)
New Concepts
A unit circle is a circle with a radius of 1 unit.
Exploration
Exploring the Unit Circle
Use trigonometric ratios to determine the coordinates of points on a unit
circle that is centered at the origin.
1. The figure shows a unit circle and a 60° angle in
standard position. What is the length of the
hypotenuse of the 30°-60°-90° triangle? How do
you know?
y
P(x, y)
60°
y
x
x
1
2. Use what you know about special right triangles
to find the lengths of the legs of the 30°-60°-90°
triangle.
3. What are the coordinates of point P?
4. What are the exact values of cos 60° and sin 60°?
5. How are the values of cos 60° and sin 60° related to the coordinates of
point P?
y
6. Describe how you can use a similar method to
find the coordinates of point Q.
7. Explain how the values of cos 45° and sin 45°
are related to the coordinates of point Q.
Q(x, y)
45°
y
x
x
1
Reading Math
Remember the symbol θ
is read “theta.”
Online Connection
www.SaxonMathResources.com
Angles can be measured in degrees or in radians.
The measure of an angle in radians is based on arc length,
the distance between two points on a circle. In a circle with
radius r, the measure of a central angle θ is one radian
when it intercepts an arc that has a length equal to the
radius.
r
r
θ = 1 radian
Lesson 63
447
y
There are 360°or 2π radians in
a circle. Thus the conversion
2π
factor _
360° , which is equivalent to 1,
can be used to convert measures
between degrees and radians.
θ = -210° = -
θ = 360° = 2π radians
x
7π
6
All angles are measured clockwise or
counterclockwise from the positive
x-axis. Measured clockwise, they are
negative angles. In the figure, the
-210° angle is equivalent to an angle measure of 150°.
Converting Angle Measures
Degrees to Radians
Radians to Degrees
Multiply the number of degrees
π radians
.
by _
Multiply the number of radians
180°
by _
.
(
Example 1
Math Language
Angle measures in
radians often appear
without the label radians.
π
For example, - _
radian
6
is often written simply
π
as -_
.
6
180°
)
( π radians )
Converting Between Degrees and Radians
Convert each measure from degrees to radians or from radians to degrees.
a. -30°
SOLUTION
π radians = -_
π radians
-30° _
180°
6
(
b.
)
π radians .
Multiply by _
180°
)
(
3π
_
radians
4
SOLUTION
180°
3π radians _
= 135°
(
)
(_
π radians )
4
For every point P(x, y) on
the unit circle, the value of
r is 1. Therefore, if θ is an
angle in standard position
whose terminal side passes
through point (x, y) on the
unit circle, then:
y _
y
sin θ = _
r = 1 =y
x =_
x =x
cos θ = _
r
1
y
tan θ = _
x
So, the coordinates of P can
be written as
(cos θ, sin θ).
Multiply by
180°
.
(_
π radians )
The Unit Circle
( 2 2)
2π
( _22 , _22 ) _
3π 3
_
120°
4
( _23 , _12 ) _
135°
5π
√3
1, _
_
√
√
√
6 150°
II
III
(-1, 0) π 180°
(
√3
_
,
2
(
)
1
_
2
7π
_
6
225°
5π
240°
_
4 4π
)
2
( _12 ,
√2
√2
_
, _
2
210°
_
3
√3
_
2
)
)
(
(
√3
1, _
_
(0, 1)
2
2
π
_
√2 _
√2
π
_
_
2
,
90° 3 _
2
2
π
60°
4
√
3 _
_
π
45° _
,1
2
2
6
30°
)
(
I
IV
)
(1, 0)
0° 0
330°
11π
_
315°
6
√3
1
_
7π
300°
_
, _
2
2
4
270° _
5π
3
√
√
3π
2
2
_
_
, _
2
2
2
√3
1, _
(0, -1) _
2
2
)
(
(
(
)
)
The diagram shows the equivalent degree and radian measures of special
angles, and the corresponding x- and y-coordinates of points on the unit circle.
448
Saxon Algebra 2
Example 2 Using the Unit Circle to Evaluate Trigonometric
Functions
Use the unit circle to find each trigonometric function value. Find exact
values.
a. sin 120°
SOLUTION
(
√3
)
The terminal side of 120° passes through the point -_12 , _
on the unit
2
circle.
Hint
The exact value of
sin 120° is irrational. An
approximate value is
0.866.
√
3
sin 120° = _
≈ 0.866
2
exact approximate
sin θ = y
√3
sin 120° = _
2
11π
_
b. tan 6
SOLUTION
(
)
√3
11π
The terminal side of _
passes through the point _
, -_12 on the unit
2
6
circle.
y
tan θ = _
x
1
-_
11π _
_
tan
= 2
6
√
3
_
2
1 ·_
2
= -_
2 √
3
To divide by a fraction, multiply by its reciprocal.
1
= -_
√
3
√
3
1 ·_
= -_
√
3 √
3
Rationalize the denominator.
√
3
= -_
3
You can use reference angles and the portion of the unit circle in Quadrant I
to determine trigonometric function values.
Evaluate Trigonometric Functions using Reference Angles
To find the sine, cosine, or tangent of θ:
Step 1: Determine the measure of the reference angle of θ.
Step 2: Use the portion of the unit circle in Quadrant I to find the sine,
cosine, or tangent of the reference angle.
Step 3: Use the quadrant of the terminal side of θ in standard position to
determine the sign of the sine, cosine, or tangent.
Lesson 63
449
The diagram shows how the signs of the
trigonometric functions are determined by the
quadrant that contains the terminal side of θ in
standard position.
QII
sinθ
cosθ
tanθ
QIII
sinθ
cosθ
tanθ
+
+
sinθ
cosθ
tanθ
+
+
+
QI
sinθ
cosθ
tanθ
+
QIV
Example 3 Using Reference Angles to Evaluate Trigonometric
Functions
Find the sine, cosine, and tangent of 135°. Find exact values.
y
SOLUTION
Step 1: Find the measure of the reference angle.
The measure of the reference angle is 45°.
45°
135°
x
Step 2: Find the sine, cosine, and tangent of the
reference angle.
Hint
Recall that a reference
angle is an acute angle.
So for θ < 360°, the
measure of θ is:
QII - 180° - θ
QIII - θ - 180°
QIV - 360° - θ
√
2
sin 45° = _
2
Use sin θ = y.
√
2
cos 45° = _
Use cos θ = x.
2
tan 45° = 1
y
(_22 , _22 )
√
Use tan θ =
_y
x
45°
√
x
.
Step 3: Determine the sign.
√
2
sin 135° = _
In Quadrant II, sin θ is positive.
2
Math Language
An arc is an unbroken
part of a circle consisting
of two points on the
circle, called endpoints,
and all the points on the
circle between them.
This arc is named RS .
R
S
√
2
cos 135° = -_
2
In Quadrant II, cos θ is negative.
tan 135° = -1
In Quadrant II, tan θ is negative.
The arc length of a circle intercepted by the central angle is related to the
central angle.
Arc Length Formula
For a circle of radius r, the arc length s intercepted by
a central angle θ (measured in radians) is given by the
following formula.
s
θ
r
s = rθ
The arc length s of a circle is a portion of the circumference of a circle. This
portion is the ratio between the measure of the central angle and the measure
of the entire circle multiplied by the circumference of the circle. Thus,
θ
s=_
2π · 2πr or s = θr.
450
Saxon Algebra 2
Example 4 Finding Arc Lengths
a. Find the length of arc s1. Approximate to the nearest tenth.
SOLUTION
s = rθ
Write the formula.
2π
s1 = 5 · _
3
s1
2π
_
2π
Substitute s1 for s, 5 for r, and _
for θ.
3
3
5 cm
10π
=_
3
Simplify.
≈ 10.5 cm
Use a calculator to approximate.
b. Find the length of arc s2. Approximate to the nearest tenth.
SOLUTION
Step 1: Convert 225° to radians.
π radians = _
5π radians
225° _
4
180°
s2
)
(
225°
2.5 ft
Step 2: Use the formula for arc length.
s = rθ
s2 = 2.5
Write the formula.
5π
(_
4 )
5π
Substitue s2 for s, 2.5 for r, and _
for θ.
4
= 3.125π
Simplify.
≈ 9.8 ft
Use a calculator to approximate.
Example 5
Application: Planet Rotation
Earth makes one complete
rotation in 24 hours, and its
radius is approximately
3955 miles. If an object is fixed
on Earth’s equator, how far
does it travel in 1 hour due to
Earth’s rotation?
SOLUTION
1
_
of a rotation in 1 hour
24
r = 3955 mi
The central angle that corresponds to a complete rotation is 2π radians. In 1
π
1
1
hour, Earth makes _
of a complete rotation. So, θ = _
· 2π = _
.
24
24
12
s = rθ
π
= 3955 · _
12
≈ 1035
The object travels about 1035 miles in 1 hour.
Check Find the approximate circumference of Earth: 2πr ≈ 2π · 3955 ≈
24,850 miles.
1
_
of 24,850 is slightly greater than 1000, so 1035 miles is reasonable.
24
Lesson 63
451
Lesson Practice
a. Convert 150° to radians.
4π
b. Convert -_
radians to degrees.
(Ex 1)
3
(Ex 1)
c. Use the unit circle to find the exact value of cos 315°.
(Ex 2)
3π
d. Use the unit circle to find the exact value of tan _
.
4
(Ex 2)
e. Use a reference angle to find the sine, cosine, and tangent of 210°. Find
exact values.
(Ex 3)
f. Find the length of arc s1. Approximate to the nearest
tenth.
(Ex 4)
s1
4π
_
3
8.5 m
g. Find the length of arc s2. Approximate to the nearest
tenth.
s2
(Ex 4)
135°
8 in.
h. The minute hand of a certain clock is 15 centimeters long. How far
does the tip of the minute hand travel in 25 minutes, to the nearest
centimeter?
(Ex 5)
Practice
Distributed and Integrated
Simplify.
1.
(48)
_1 + _4
x
y
_
1
3+_
xy
2.
(48)
1
x+_
x2
3. _
(48)
2
x2 - _
x2
_4 - 3
x
_
_7 + 2
x
Determine if the following probability experiments are binomial experiments. If not,
explain why.
4. Rolling a number cube 18 times and recording the results
(Inv 5)
5. Flipping 2 coins 20 times to test if they match half of the time
(Inv 5)
Simplify.
45 - 2 √
75 + 2 √
108
6. 5 √
(40)
7. 3 √
12 (4 √
3 - 3 √
3)
(40)
Expand.
8. (x + 5)3
(19)
452
Saxon Algebra 2
9. (x + 4)3
(19)
Solve.
10. 36x2 - 36 = 0
(35)
11. 24x = -11x2 - x3
(35)
12. Error Analysis Julian is finding the inverse of the function y = 2x3. Which step is
(50)
incorrect?
x = y3
Step Two: x = 2y3
Step Three: _
Step One: y = 2x3
2
x
_
Step Four: ± =y
2
-4 ) + (10 + √
-1 ).
*13. Write the expression in terms of i and then add: (15 - √
√
(62)
C
14. Geometry In the triangle to the right, find x.
(44)
90°
xm
xm
45°
45°
B
9m
A
*15. Planet Rotation Jupiter is the largest planet. Its radius is approximately 44,365
(63)
miles. Jupiter rotates faster than Earth. It takes approximately 9.8 hours for Jupiter
to make one complete rotation. If an object is fixed on Jupiter’s equator, how far
does it travel in 1 hour due to Jupiter’s rotation?
16. Multiple Choice Which of the following functions could be used to model
(57)
exponential growth?
2.4
1
2
A y=_
C y=_
D y=_
B y = 2e-x
ex
e2x
e-2x
17. Surveying A surveyor stands on the roof of a building that is 180 feet above the
(46)
ground and spots the top of a taller building at an angle of elevation of 34°. The
two buildings are 480 feet apart. Find the height of the taller building. Round to
the nearest foot.
*18. Graphing Calculator Describe the feasible region for the set of inequalities x ≥ 0,
(54)
y ≥ 0, y ≤ -1.5x + 9.5.
*19. Formulate A fair coin is flipped. Write the equation that can be used to solve for
(60)
the probability that three flips result in heads, then four flips, and seven flips. Write
a formula using exponents that can be used to find the probability that the same
independent event will occur n times.
1
_
+ _1
20. Multiple Choice Which of the following expressions is equivalent to _
?
1
_
(48)
x
y
1-
x
2(1 - x)
x2 - xy - x - y
x+y
1-x
A _
B __
D _
C _
2
xy
y
x+y
x+y
xy
*21. Write How is adding and subtracting complex numbers similar to adding and
(62)
subtracting polynomials?
*22. Model Sketch a graph of five points whose correlation coefficient
(45)
is about 0.1.
Lesson 63
453
180°(n - 2)
23. Multi-Step The formula θ = _
gives the measure of each interior angle of
n
(56)
an n-sided regular polygon.
a. Use the formula to find the measure of an interior angle of a regular
decagon.
b. Find the reference angle for this angle.
24. Estimate Use what you know about the unit circle and special angles to estimate
(63)
cos(0.3π). Do not use a calculator. Explain your method.
25. Jupiter Gravity on the planet Jupiter is almost three times stronger than on Earth.
(58)
If an object is thrown into the air, with an initial velocity of 100 feet per second,
from the surface of Jupiter, the height, h, of the object in feet after t seconds can be
approximated by the equation h = -42t2 + 100t. Find the time at which the object
will first reach 50 feet. Round your answer to the tenths place.
26. Analyze Let f(x) = ax + b and g(x) = cx + d. Under which conditions will the
(53)
composite functions f(g(x)) and g(f(x)) be equal?
27. Construction The volume of a typical American-made brick in cubic millimeters
(61)
can be represented by the polynomial x3 - 20x2 - 4500x + 126,000 where the
length is given by x + 70. Fully factor the expression for the volume of a brick.
3
28. Coordinate Geometry Sketch a graph of the equation y = √
x , where y is the side
(59)
length of a cube and x is the volume of the cube. Use the graph to estimate the
volume of a cube with side lengths of 2.4 in. Use the graph to estimate the side
length of a cube with a volume of 6.9 in3.
*29. Error Analysis A student incorrectly found s, the length of the
(63)
indicated arc, as shown below.
s = rθ
s
60°
10 m
= 10 · 60
= 600 m
What is the error? Find the correct arc length.
*30. Pass codes A computer generates temporary pass codes for users. Each pass code
(55)
is exactly 3 letters long with no repeating letters. What is the probability of getting
a pass code with 3 vowels (A, E, I, O, or U)?
454
Saxon Algebra 2
LAB
10
Using the Log Keys
Graphing Calculator Lab (Use with Lesson # 64, 72, 81, 87, 93, 102, and 110)
Evaluating Logarithmic Expressions
1. Calculate the common logarithmic expression
4 log (5).
a. To enter the value 4 log (5), press
.
.
b. To calculate this value, press
2. Calculate the natural logarithmic expression 4 ln (5).
a. To enter the value 4 ln (5), press
.
b. To calculate this value, press
Hint
Use the Change of Base
Law when calculating a
log that does not have a
base of 10 or e.
.
3. Use the Change of Base Law to calculate the
logarithmic expression log6 (15).
a. To enter the value log6 (15), press
.
b. Press
to calculate this value.
Graphing Logarithmic Functions
Graphing
Calculator Tip
When graphing
logarithmic expressions,
use the standard zoom
window. Adjust the
window to view specific
parts of the graph by
zooming in or out.
1. Graph the common logarithmic function
log (3x + 2).
a. Press
to access the screen to enter
the logarithmic function.
b. To enter the function log (3x + 2), press
.
, and then press
c. Press
the ZStandard window.
d. Then press
to select
to view the graph.
Online Connection
www.SaxonMathResources.com
Lab 10
455
2. Graph the logarithmic function log6 (13x)
whose base is not natural or common.
a. Press
to access the screen to enter
the logarithmic function.
b. To enter the function log6 (13x), press
.
, and then press
c. Press
the ZStandard window.
d. Then press
to view the graph.
Lab Practice
1. Calculate 6 log (8).
2. Calculate 3 ln (1).
3. Calculate log7 (9).
4. Graph y = log (15x + 8).
5. Graph y = log5 (x + 6).
456
Saxon Algebra 2
to select
LESSON
Using Logarithms
64
Warm Up
1. Vocabulary In the expression 45, the base is 4 and the
is 5.
(3)
2. What is the value of 23 ?
(3)
3. What value of x makes the equation 10x = 100 true?
(47)
New Concepts
A logarithm is the exponent that is applied to a specified base to obtain a given
value. Every exponential equation has a logarithmic form and vice versa.
Exponential Equation
Logarithmic Equation
bx = a
logb a = x
b > 0, b ≠ 1
The logarithmic equation logb a = x is read “the log base b of a equals x.”
Notice that x is both the exponent and the logarithm in the equations above.
Example 1
Converting from Exponential to Logarithmic Form
Write each exponential equation in logarithmic form.
a. 25 = 32
Reading Math
Read the symbol as
“if and only if.” When
used between two
equations, it means the
equations are equivalent.
SOLUTION
25 = 32 log2 32 = 5
The base is the same in both the exponential
equation and the logarithmic equation.
The logarithmic form is log2 32 = 5.
b. 51 = 5
SOLUTION
51 = 5 log5 5 = 1
The exponent is the logarithm.
The logarithmic form is log5 5 = 1.
c. 80 = 1
SOLUTION
Online Connection
www.SaxonMathResources.com
80 = 1 log8 1 = 0
Any nonzero base to the zero power is 1.
The logarithmic form is log8 1 = 0.
Lesson 64
457
Reading Math
The equations in
Example 1d can be
1
written 5-2 = _
and
25
1
log5 _
=
-2.
25
d. 5-2 = 0.04
SOLUTION
5-2 = 0.04 log5 0.04 = -2
An exponent (or log) can be negative.
The logarithmic form is log5 0.04 = -2.
Example 2 Converting from Logarithmic to Exponential Form
Write each logarithmic equation in exponential form.
a. log10 1000 = 3
SOLUTION
log10 1000 = 3 103 = 1000
The base is the same in both the logarithmic
equation and the exponential equation.
The exponential form is 103 = 1000.
b. log7 7 = 1
SOLUTION
log7 7 = 1 71 = 7
The logarithm is the exponent.
The exponential form is 71 = 7.
c. log3 81 = x
SOLUTION
log3 81 = x 3x = 81
The logarithm (and the exponent) can be a
variable.
The exponential form is 3x = 81.
A logarithm with base 10 is called a common logarithm. If no base is written
for a logarithm, the base is assumed to be 10. For example, log 5 = log10 5.
Example 3
Application: Acidity of Rainwater
Because of the phenomenon of acid rain, the acidity of rainwater is
important to environmental scientists. The acidity of a liquid is measured
in pH, given by the function pH = -log[H+], where [H+] represents the
concentration of hydrogen ions in moles per liter. In 1999, the hydrogen
ion concentration of rainwater in western Nevada was found to be
approximately 0.0000032 moles per liter. What was the pH of the rainwater?
SOLUTION
pH = -log[H+]
pH = -log(0.0000032)
Substitute the known
value in the function.
Use a calculator to find the value of the
logarithm in base 10. Press the
key.
The rainwater had a pH of about 5.5.
458
Saxon Algebra 2
Check
Hint
The display
3.16227766E - 6 means
3.16227766 × 10-6,
which is approximately
3.2 × 10-6, or 0.0000032.
-log (0.0000032) ≈ 5.5
Write the result found in the Solution.
log (0.0000032) ≈ -5.5
Multiply both sides by –1.
10-5.5 ≈ 0.0000032
Write the logarithmic equation in
exponential form.
Use a calculator to verify that the value of
10-5.5 is approximately 0.0000032.
Lesson Practice
Write each exponential equation in logarithmic form.
(Ex 1)
a. 32 = 9
b. 41 = 4
c. 90 = 1
d. 8-1 = 0.125
Write each logarithmic equation in exponential form.
(Ex 2)
e. log10 100 = 2
f. log8 8 = 1
g. log5 125 = x
h. The acidity of a liquid is measured in pH, given by the function pH =
-log[H+], where [H+] represents the concentration of hydrogen ions in
moles per liter. In 1999, the hydrogen ion concentration of rainwater
in northern Maine was found to be approximately 0.0000200 moles per
liter. What was the pH of the rainwater, to the nearest tenth?
(Ex 3)
Practice
Distributed and Integrated
Use synthetic substitution.
1. Find f(6) for f(x) = x4 - 4x3 + 2x2 + 4x - 18.
(51)
2. Find f(-7) for f(x) = x4 + 5x3 - 12x2 - 4x + 8.
(51)
3. Find the equation of the line that passes through (-2, 5) and (-6, -3).
(26)
4. The sides of a triangle measure 7 in., 6 in., and 8 in. Is the triangle a right triangle?
(41)
Find the zeros of each quadratic function.
5. f(x) = 2x2 + x - 15
(35)
6. f(x) = -32x2 - 28x
(35)
7. f(x) = x2 - 9
(35)
Determine if (0, -3) is a solution of the inequalities.
8. 5x + 2y < -4
(39)
9. y - 2x > 6
(39)
10. 16y - 2 ≥ x
(39)
11. Formulate A parasail is attached to a boat with a rope 300 feet long. The angle of
(52)
elevation from the boat to the parasail is 48 degrees. Write a formula that would
help you to estimate the parasail’s height above the boat.
Lesson 64
459
*12. Coordinate Geometry Find the area of triangle OQR. Give the answer in
(63)
exact form and to the nearest tenth of a square unit.
2
y
(_23 , _12 )
P
O(0, 0)
√
Q
x
O
2
3
R(3, 0)
13. Find an equation for the inverse of y = 5x - 8. Identify the domain and range of the
(50)
inverse function.
*14. Write Explain why log0 9 and log1 9 do not exist.
(64)
15. Population Currently the U.S. population is growing at a rate of approximately
(53)
0.6 percent per year. According to the 2000 census, the population was about
281 million. Use composite functions to predict the U.S. population in the year
2015, and justify your prediction.
*16. Multi-Step Use
(59)
_9
_1
_9
_1
(7) (7) .
4
4
a. Which property of rational exponents should be applied first to simplify the
expression? Explain.
b. Apply the property from part a.
c. Which property of rational exponents should be applied next?
d. Apply the property from part c.
e. If possible, finish simplifying the expression from part d.
*17. Transportation The approximate volume of a moving truck, in cubic feet, that can
(61)
move four to five household rooms can be given by x3 - 15x2 + 56x + 6x2 - 90x + 336.
Factor this expression by grouping. Then find the dimensions when x = 15.
18. Geometry The figures below show how to create Sierpinski’s Triangle. Start with a
(57)
solid black equilateral triangle (Iteration 0); at every iteration, join the midpoints
of the sides of each black triangle to get a smaller triangle, and color the interior
of that triangle white.
Iteration 0
Iteration 1
Iteration 2
Iteration 3
Iteration 4
Formulate an exponential function for the number of black triangles N(x) in
the xth iteration.
3 _
1
19. Find the slope of the line that passes through the points A _
x , 3 and
(31)
1 5
B _5 , _
x , where x is any non-zero real number.
(
)
(
*20. Windshield Wipers A rear windshield wiper moves through
(63)
an angle of 135° on each swipe. To the nearest inch, how much
greater is the length of the arc traced by the top end of the
wiper blade than the length of the arc traced by the bottom
end of the wiper blade?
460
Saxon Algebra 2
)
Top end
14 in.
Bottom end
9 in.
21. Multiple Choice Which of the following expressions is equivalent to _
?
√
3
5- 2
(44)
A
3 - 15 √
2
_
21
B
15 + 3 √
2
_
23
C
15 - 2 √
3
_
21
D
21 - 3 √
2
_
15
22. Multi-Step What number needs to be added to both sides of the equation to make
x2 + 12x + 42 = 0 a perfect square? Solve the equation.
(Inv 6)
*23. Rainwater The acidity of a liquid is measured in pH, given by the function
(64)
pH = -log[H+], where [H+] represents the concentration of hydrogen ions in moles
per liter. In 1999, the hydrogen ion concentration of rainwater in the Chesapeake
Bay region of Maryland was found to be approximately 0.0000316 moles per liter.
What was the pH of the rainwater, to the nearest tenth? Show how to check your
answer by writing a logarithmic equation and its equivalent exponential equation.
(Hint: You will need to use the ≈ symbol instead of the = symbol.)
*24. Write Two number cubes are tossed once. Explain why the events, one cube
(60)
shows a number less than 5 and the sum of the two cubes is a multiple of 2, are
dependent.
25. Justify Give an example of an event that has a probability of 0. Explain why the
(55)
probability is 0.
*26. Graphing Calculator Set the mode on your graphing calculator to a + bi by pressing
(62)
the Mode key and using the arrow keys. Then find √
-1296 ÷ √
-324 .
27. Error Analysis Rizwan tried to find the value of cosθ, where θ is an angle in standard
(56)
position with the point Q(-6, 8) on its terminal side. His work is shown below.
What was Rizwan’s error?
x2 + y2
√
(-6)2 + 82
r = √
r=
100
r = √
r = 10
x
cosθ = _
r
6 = 0.6
cosθ = _
10
*28. Multiple Choice Which equation is equivalent to log2 16 = x?
(64)
A 2x = 16
B x2 = 16
C 162 = x
D 216 = x
29. Analyze Compare and contrast correlation coefficient values of -0.45 and 0.45.
(45)
*30. Falling Objects The height, in feet, of an object that is falling or is projected into
(58)
the air can be described by h = -16t2 + v0t + h0, where h is the height in feet after
t seconds, v0 is the initial velocity of the object in feet per second, and h0 is the
initial height of the object in feet per second. A coin is tossed from the top of
a 100 foot tall building, with an initial velocity of 92 feet per second. Write the
equation that models the height of the coin. Find the time when the coin will
reach the ground. Round your answer to the tenths place.
Lesson 64
461
Using the Quadratic Formula
LES SON
65
Warm Up
1. Vocabulary The square root of a negative number is an
number.
(62)
2. Solve by completing the square x2 - 26x = 13.
(58)
3. Simplify 3 √
-25 .
(62)
4. Evaluate b2 - 4ac when a = 2, b = 7, and c = 5.
(2)
New Concepts
Math Reasoning
Analyze
-c
b2
_
Explain how _
a + 4a2
b2
4ac
becomes -_
+_
.
4a2
4a2
In general, a quadratic equation in standard form is ax2 + bx + c = 0, with
a ≠ 0. The general equation can be solved for x by completing the square.
ax2 + bx = -c
Subtract c from each side.
b x = -_
c
x2 + _
a
a
Divide both sides by a.
bx + _
b2 = -_
c +_
b2
x2 + _
2
a
a
4a
4a2
Complete the square.
b
x+_
2a
2
b
( ) = -_ac + _
4a
b = -_
4ac + _
b
(x + _
2a )
4a
4a
2
Factor the left side.
2
2
2
2
Make common denominators on
the right.
2
b - 4ac
b =_
)
(x + _
4a
2a
b - 4ac
b = ± _
x+_
√
4a
2a
b - 4ac
b ± _
x = -_
√
4a
2a
2
2
2
Combine the fractions on the right.
2
2
Apply the Square Root Property.
2
2
√
b2 - 4ac
b ±_
x = -_
2a
2a
2
-b ± √b - 4ac
x = __
2a
Isolate x.
Simplify the radical.
Combine the fractions.
The last line is the quadratic formula.
Quadratic Formula
The solutions of the quadratic equation ax2 + bx + c = 0 (a ≠ 0) are
given by
-b ± √
b2 - 4ac
x = __ .
2a
Online Connection
www.SaxonMathResources.com
462
Saxon Algebra 2
The quadratic formula can be used to find the solutions of any quadratic
equation.
Example 1 Solving Quadratic Equations with Real Zeros
Solve each equation.
Math Language
The solutions of a
quadratic equation are
the zeros of the related
function.
a. 5x2 + 34x - 7 = 0
SOLUTION Use the quadratic formula.
2
- 4ac
-b ± √b
x = __
2a
2
- 4(5)(-7)
-34 ± √34
___
=
2(5)
-34 ± √1296
= __
10
-34 ± 36
1 and -7
=_ =_
5
10
1
The solutions are _ and -7.
Substitute 5 for a, 34 for b, and -7 for c.
Simplify the radicand and denominator.
Evaluate the square root and simplify.
5
b. 9x2 + 6x + 1 = 0
2
-b ± √b
- 4ac
2a
SOLUTION Use the quadratic formula x = __.
2
-6 ± √6
- 4(9)(1)
__
x=
2(9)
-6 ± √0
=_
18
Substitute 9 for a, 6 for b, and 1 for c.
Simplify.
-6 ± 0
6 = -_
1
= _ = -_
3
18
18
1
The solution is -_.
3
Example 2 Solving Quadratic Equations with Complex Zeros
Hint
The standard form of a
complex number is
a + bi.
a. Solve the equation 3x2 + 5x + 4 = 0. Write the solutions as complex
numbers in standard form.
SOLUTION Use the quadratic formula.
2
- 4ac
-b ± √b
x = __
2a
2
-5 ± √5
- 4(3)(4)
= __
2(3)
-5 ± √-23
= __
6
-5 ± i √23
= __
6
√23
5 ±_
= -_
i
6
6
Substitute 3 for a, 5 for b, and 4 for c.
Simplify the radicand and denominator.
Write the negative root as an imaginary
number.
Write the solutions in standard form.
√23
√23
The solutions are -_56 + _
i and -_56 - _
i.
6
6
Lesson 65
463
_1
1
b. Solve the equation x2 - 2 x + _8 = 0. Write the solutions as complex
numbers in standard form.
SOLUTION To make the calculations simpler, clear the fractions before using
the quadratic formula.
8x2 - 4x + 1 = 0
Multiply both sides of the equation by 8.
2
-b ± √b
- 4ac
Use the quadratic formula: x = __
.
2a
2
- 4(8)(1)
-(-4) ± √(-4)
= ___
2(8)
Substitute 8 for a, -4 for b, and 1 for c.
4 ± √-16
=_
16
Simplify the radicand and denominator.
4 ± 4i
=_
16
Write the negative root as an imaginary
number.
1±i
=_
4
Factor 4 out of the numerator and
denominator.
1 ±_
1i
=_
4
4
Write the solutions in standard form.
The solutions are _14 + _14 i and _14 - _14 i.
Because the solutions of a quadratic equation are the zeros, or x-intercepts
of the equation, a graphing calculator can be used to find the solutions of a
quadratic equation when the solutions are real.
Example 3
Solving Quadratic Equations using a Graphing
Calculator
Solve 2x2 + x - 15 = 0 by using a graphing calculator.
SOLUTION Graph the related function, y = 2x2 + x - 15.
Math Reasoning
Verify Graph y = 3x2 + 5x
+ 4. How does the graph
confirm that there are no
real solutions to
3x2 + 5x + 4 = 0?
If needed, adjust the window so that the x-intercepts of the graph are
displayed. Access the Calc function (2nd + Trace) and choose zero.
When asked for left and right bounds, choose points that are on either side
of one of the x-intercepts. Repeat for the second intercept.
The solutions are -3 and 2.5.
464
Saxon Algebra 2
Example 4
Application: Travel
Train A leaves the station at 1 p.m. and travels due north at 55 miles per hour.
Train B leaves the same station at 4 p.m. and travels due east at 35 miles
per hour. About how long will it take for the trains to be 500 miles apart?
SOLUTION
Write each distance as rate × time.
Train B is 3 hours behind Train A.
Train A: d = rt = 55t
Train B: d = rt = 35(t - 3)
55t
a2 + b2 = c2
2
2
35(t - 3)
2
(55t) + (35(t - 3)) = 500
3025t2 + (35t - 105) 2 = 250,000
2
2
3025t + 1225t - 7350t + 11,025 = 250,000
2
4250t - 7350t - 238,975 = 0
500
Use the Pythagorean
Theorem.
Simplify.
Square the binomial.
Combine like terms.
Now use the quadratic formula.
2
- 4(4250)(-238,975)
-(-7350) ± √(-7350)
t = _____
2(4250)
7350 ± √4,116,597,500
= ___
8500
t ≈ 8.41 and t ≈ -6.68
Disregard the negative time. It will take about 8.41 hours for the trains to be
500 miles apart.
Check Find the distance each train traveled after 8.41 hours.
Train A: 55(8.41) = 462.55 miles, Train B: 35(8.41 - 3) = 189.35 miles
2
+ 189.352 = 499.8 ≈ 500.
Use the Pythagorean Theorem: √462.55
Lesson Practice
Solve each equation.
(Ex 1)
a. 3x2 - 7x + 2 = 0
b. 9x2 + 12x + 4 = 0
Solve each equation. Write the solutions as complex numbers in standard form.
c. x2 + 5x = - 9
(Ex 2)
1x - _
1 =0
d. -x2 + _
3
9
(Ex 2)
e. Solve 8x2 + 7x - 100 = 0 by using a graphing calculator.
(Ex 3)
f. A hiker leaves a campground at 8 a.m. and heads due south at a rate of
4 miles per hour. Another hiker leaves the same campground at 9 a.m.
and heads due east at a rate of 2 miles per hour. Approximately how
long will it take for the hikers to be 15 miles apart?
(Ex 4)
Lesson 65
465
Practice
Distributed and Integrated
1. Secret ingredient X has a half-life of 23 hours. What portion of 360 kg will
remain after 7 hours?
(57)
Use long division.
2. Divide (3x3 + 12x2 - 15x + 15) by (3x - 9)
(38)
3. Divide (2x3 + 14x2 - 4x - 48) by (2x + 4)
(38)
Solve.
*4. x2 = 7 + 3x
*5. x2 = -x + 1
(58)
(58)
Simplify.
3 (5 √
3 - 2 √
6)
6. 2 √
(40)
7. 4 √
63 - 3 √
28
8. 3 √
2 · 2 √
6 · 3 √
6
(40)
(40)
9. Jordon spins a spinner with six equal-sized sections numbered 1–6. In one spin,
what is the likelihood that the spinner will stop on a 1 or a 5?
(55)
*10. Error Analysis Find and correct the error a student made in solving 4x2 - 37x + 9 = 0.
(65)
b2 - 4ac
-b ± √
x = __
2a
2
- 4(4)(9)
-37 ± √(-37)
___
=
8
1369 - 144
-37 ± √
= __
8
-37 ± √1225
-37 ± 35
1 and -9
= __ = _ = -_
4
8
8
11. List the coefficients in the 6th row of Pascal’s Triangle.
(49)
12. Probability A history test has 20 questions, 7 of which are about the Civil War.
(55)
Students are asked to answer 5 of the 20 questions. What is the probability that a
student randomly selecting questions will select 5 Civil War questions?
*13. Write Describe how to determine sin _
2 by using the unit circle.
3π
(63)
14. Find an equation of a line perpendicular to y = _3 x - 4.
1
(36)
15. Analyze In what type of right triangles will the sine and cosine values of the acute
(46)
angles be equal?
Find x and y.
17.
16.
(52)
(52)
y
x
y
5
30°
3
466
Saxon Algebra 2
30°
x
*18. Multi-Step Complete parts a–d to graph a logarithmic function and use the graph
(64)
to estimate a logarithm.
1
a. Write the equation y = log2 _4 in exponential form. What value of y makes both
equations true?
b. Writing exponential equations
if needed, complete the table
of values for the function
y = log2x.
x
_1
_1
4
2
1
2
4
8
y = log2 x
c. Plot the points and sketch the graph of the function.
d. Use your graph to estimate the value of log26. Check your estimate by
evaluating the appropriate power of 2 on a calculator.
19. Multiple Choice Choose the letter that best represents the value of x in the equation
(58)
x2 + 2x + 7 = 8.
A x = -1 ± √
2
B x = -2, 0
C x = 2, 0
D x = 1 ± √
2
3
4
20. Measurement Given that 1 strip of red ribbon = 3 inches and 1 strip of blue ribbon
(39)
= 4 inches, write and graph the inequality that represents that the total number of
inches must be no less than 60.
21. Multi-Step A car is sitting outside in a driveway. Consider the following statement,
“If it starts raining, the car will get wet.”
a. Is the converse of the statement true? Explain.
(Inv 1)
b. Is the contrapositive true? Explain.
22. Geometry The base of a triangle is 5 √
2 feet, and its height is _
feet. Find the
√
3
(44)
area of the triangle.
1
*23. Physics A fireworks technician wants a firework to explode 70 feet below the
(62)
top of the Gateway Arch in St. Louis. The Gateway Arch has a height of 630 feet.
The technician has figured the height of the firework to be modeled by
-16t2 + 160t, where t is the time in seconds after it is launched. The solutions
of -16t2 + 160t = 560 give the number of seconds it will take for the firework to
reach the desired height. Solve the equation. Will the firework reach the desired
height? If so, when?
24. Justify Let f(x) = 3x - 7 and g(x) = _13 x + _73 .
(53)
a. Justify the claim that f (g(x)) = x.
b. Does g(f(x)) = x also? Explain.
25. Error Analysis Find and correct the error a student made in factoring the expression
(61)
below.
9x4 - 63x3 - 8x + 56
9x3(x - 7) - 8(x - 7)
(x - 7)(9x3 - 8)
(x - 7)(3x - 2)(6x2 + 6x + 4)
Lesson 65
467
*26. Physical Science The North Falls waterfalls in Silver Falls State Park in Oregon are
(65)
136 feet high. The equation -16t2 + 136 = 0 gives the number of seconds it takes
for a drop of water at the top of the falls to reach the pool of water below. Solve
the equation by using the quadratic formula. Round to the nearest hundredth of
a second.
27. Formulate Let θ be an angle in standard position and let n be any integer. You can
(56)
use the expression θ + n(360°) to indicate all the angles coterminal with θ. Write
an expression for all the angles coterminal with 15°, and find 3 different ones.
*28. Graphing Calculator Solve 5x2 - 2x - 16 = 0 by using a graphing calculator.
(65)
29. Velocity of an Electron The velocity at which an electron orbits the nucleus of
(59)
an atom is known as the Fermi-Thomas velocity and can be found using the
_2
Z3
equation _ c, where Z is the number of protons in the nucleus and c is the speed
137
of light. What is the Fermi-Thomas velocity for platinum, which has 78 protons,
in terms of c? Round to the nearest thousandths place.
*30. Analyze Write the following equations in exponential form. What property of
(64)
exponents is illustrated?
log2 1 = 0, log3 1 = 0, log15 1 = 0
468
Saxon Algebra 2
LESSON
Solving Polynomial Equations
66
Warm Up
-b ± √
b2 - 4ac
1. Vocabulary The formula x = __
is called the
2a
.
(65)
3
2. Factor 27cd - 12cd.
(23)
3. Solve x(x - 4) = 0.
(35)
4. True or False: x + 7 is a factor of x3 + 8x2 + x - 42.
(61)
5. Solve x2 + x + 1 = 0 by using the quadratic formula.
(65)
New Concepts
Sometimes a polynomial equation can be solved just by factoring and using
the Zero Product Property.
Example 1 Using Factoring to Solve Polynomial Equations
Solve each equation.
Hint
Recall that the Zero
Product Property states
that if ab = 0, then a = 0
or b = 0.
a. 2x4 - 10x3 - 72x2 = 0
SOLUTION Since one side is already equal to 0, begin by factoring the
polynomial.
2x2(x2 - 5x - 36) = 0
The GCF is 2x2.
2x2(x + 4)(x - 9) = 0
Factor the trinomial.
2x2 = 0 or x + 4 = 0 or x - 9 = 0
Use the Zero Product Property.
x = 0 or
x = -4 or
x=9
Solve each equation.
The solutions are 0, -4, and 9.
b. 4x3 + x = 4x2
SOLUTION First write the equation in standard form.
4x3 - 4x2 + x = 0
Subtract 4x2 from each side.
x(4x2 - 4x + 1) = 0
The GCF is x.
x(2x - 1)(2x - 1) = 0
Factor the perfect square trinomial.
x = 0 or 2x - 1 = 0 or 2x - 1 = 0
x = 0 or
x = _12 or
x = _12
The solutions are 0 and _1 .
Use the Zero Product Property.
Solve each equation.
2
Online Connection
www.SaxonMathResources.com
1
Notice in Example 1b that the solution, or root, of _2 appears twice. This is
called a double root. It has multiplicity 2.
Lesson 66
469
Multiplicity of Roots
The multiplicity of a root r of a polynomial equation is the number of
times x - r is a factor of the polynomial.
Example 2
Identifying Multiplicity
Find the roots of each equation. Give the multiplicity of each root.
a. x5 + 8x4 + 16x3 = 0
SOLUTION Factor the polynomial to use the Zero Product Property.
x3(x2 + 8x + 16) = 0
The GCF is x3.
x3(x + 4)(x + 4) = 0
Factor the perfect square trinomial.
3
x = 0 or x + 4 = 0 or x + 4 = 0
Use the Zero Product Property.
x = 0 or
Solve each equation.
x = -4 or
x = -4
The roots are 0 and -4. Because x3 = x · x · x, x is a factor 3 times, so the root
of 0 has a multiplicity 3. Because x + 4 is a factor 2 times, the root of -4
has multiplicity 2.
b. x4 = 2x2 - 1
SOLUTION First write the equation in standard form: x4 - 2x2 + 1 = 0.
(x2 - 1)(x2 - 1) = 0
(x + 1)(x - 1)(x + 1)(x - 1) = 0
Factor the perfect square trinomial.
Factor the differences of squares.
x + 1 = 0 or x - 1 = 0 or x + 1 = 0 or x - 1 = 0
x = -1 or x = 1 or x = -1 or x = 1
The roots are -1 and 1. Because both (x + 1) and (x - 1) occur twice, each
root has multiplicity 2.
A polynomial equation like x3 - 2x2 - 11x + 12 = 0, where the degree of
the polynomial is greater than 2, can be factored by using synthetic division
if one of the factors is a linear binomial and is already known. When none
of the factors are already given, begin factoring the polynomial by using the
Rational Root Theorem.
Math Reasoning
Analyze What can you
do to use the Rational
Root Theorem to find the
possible rational roots
of _15 x3 + 2x2 - x + 2 = 0?
The Rational Root Theorem
If a polynomial P(x) has integer coefficients, then every rational root of
p
P(x) = 0 can be written in the form _q , where p is a factor of the constant
term and q is a factor of the leading coefficient of P(x).
The Irrational Root Theorem states that irrational roots come in pairs.
The Irrational Root Theorem
If a polynomial P(x) has rational coefficients, and a + b √c is a root of
P(x) = 0, where a and b are rational and √c is irrational, then a - b √c is
also a root of P(x) = 0.
470
Saxon Algebra 2
Example 3 Identifying All Real Roots of a Polynomial Equation
a. Identify all the real roots of x3 - 2x2 - 11x + 12 = 0.
SOLUTION By the Rational Root Theorem the possible rational roots
Hint
Remember that x - r is a
factor if P(r) = 0.
±1, ±2, ±3, ±4, ±6, ±12
are: __
= ±1, ±2, ±3, ±4, ±6, ±12. Use synthetic
±1
division to test roots, r, until a factor, x - r, is found.
1 1 -2 -11 12
1 -1 -12 .
1 -1 -12
0
The first try works. By the Factor Theorem P(1) = 0,
so x - 1 is a factor.
The quotient is also a factor, so the polynomial can be written as
(x - 1)(x2 - x - 12), which factors into (x - 1)(x - 4)(x + 3).
The equation can be written as (x - 1)(x + 3)(x - 4) = 0.
Using the Zero Product Property, the roots are 1, -3, and 4.
b. Identify all the real roots of 3x4 - 5x3 - 29x2 + 3x + 4 = 0.
SOLUTION By the Rational Root Theorem the possible rational roots are:
±1, ±2, ±4
_
= ±1, ±_1 , ±2, ±_2 , ±4, ±_4 . Test roots. Begin with integers.
±1, ±3
3
3
3
Organize the work in a synthetic substitution table.
The remainder is 0 for P(4), so
x - 4 is a factor.
3x3 + 7x2 - x - 1 is the other
factor.
Possible root 3 -5 -29
3
4
1
3 -2 -31 -28 -24
2
3
1
-27 -51 -98
4
3
7
-1
-1
0
±1
The possible rational roots for 3x3 + 7x2 - x - 1 = 0 are: _
= ±1, ±_13 .
±1, ±3
- _3 3
1
7 -1 -1
-1 -2
3 6 -3
1
0
x + _13 is a factor. 3x2 + 6x - 3 is the other factor.
The trinomial factors into 3(x2 + 2x - 1). Solve
x2 + 2x - 1 = 0 by using the quadratic formula.
2
-2 ± √2
-4(1)(-1)
-2 ± √8
-2 ± 2 √2
x = __ = _ = _ = -1 ± √2
2
2
2(1)
The polynomial equation is
1 (x - (-1 + √2 ))(x - (-1 - √2 )) = 0.
(x - 4) x + _
3
The real roots are 4, -_1 , -1 + √2, and -1 - √2.
(
)
3
The x-intercepts of a polynomial function are real roots of the related
equation. Therefore, a graph can be used to help find the roots of a
polynomial equation.
Lesson 66
471
Example 4 Using a Calculator to Solve Polynomial Equations
Solve x3 + 5x2 - 9x - 10 = 0 by using a graphing calculator.
SOLUTION Graph the related function, y = x3 + 5x2 - 9x - 10. Use the
Calc command to find the x-intercepts.
Math Reasoning
Since 2 is a rational root, use synthetic division with r = 2.
Verify Show that the
Rational Root Theorem
gives 2 as a possible root.
2 1
1
5 -9 -10
2
-7 ± √7
- 4(1)(5)
-7 ± √29
2 14 10 Solve x2 + 7x + 5 = 0: x = __ = _
.
2
2(1)
7 5
0
-7 + √29
-7 - √29
, and _
. Notice
The roots are 2, _
2
2
that the x-intercepts given by the graphing
calculator above are estimates of the exact
values of the irrational roots.
Example 5
Application: Packaging
A box is to be designed so that the height is 6 inches greater than the length
and the width is 3 inches less than the length. What must the length be if
the volume has to be 40 cubic inches?
SOLUTION
1. Understand The volume of a box, or rectangular prism, is the product of
the length, width, and height: V = lwh.
2. Plan Write expressions for each dimension. If x represents the length,
then x + 6 represents the height and x - 3 represents the width.
40 = x(x - 3)(x + 6)
3. Solve 40 = x3 + 3x2 - 18x
Hint
Use the Rational Root
Theorem to determine
all the possible rational
roots.
0 = x3 + 3x2 - 18x - 40
4 1 3 -18 -40
4 28 40
1 7 10
0
The polynomial factors into(x - 4)(x2 + 7x + 10), which factors into
(x - 4)(x + 2)(x + 5). The roots are 4, -2, and -5. Since the length
must be positive, disregard the negative roots.
The length is 4 inches.
4. Check If the length is 4 inches, the height is 10 inches, and the width is
1 inch. Find the volume: V = lwh = 4(1)(10) = 40.
472
Saxon Algebra 2
Lesson Practice
Solve each equation.
(Ex 1)
a. x4 - 3x3 - 28x2 = 0
b. 18x3 - 18x = 0
Find the roots of each equation. Give the multiplicity of each root.
(Ex 2)
c. 2x3 - 32x2 + 128x = 0
d. x6 + 12x5 + 27x4 = 0
e. Identify all the real roots of x3 - 5x2 - x + 5 = 0.
(Ex 3)
f. Identify all the real roots of x3 + 4x2 - 5x - 14 = 0.
(Ex 3)
g. Solve x3 + 2x2 - 45x - 18 = 0 by using a graphing calculator.
(Ex 4)
h. A box is to be designed so that the height is 3 inches greater than the
length and the width is 2 inches less than the length. What must the
length be if the volume has to be 18 cubic inches?
(Ex 5)
Practice
Distributed and Integrated
Simplify each expression.
_2
1.
(59)
4.
(37)
81
_
4
2.
_3
(37)
81 4
3+x
x + __
_
x+2
2
x + 4x + 4
p - 4px
_
p
3.
(37)
_1
a+2
4a + _
_
a+4
_1
5. (4 6) 3
2a
_1
6. 343 3 · 343 3
(59)
(59)
*7. There are 7 cherries and 3 grapes in a bag. What is the probability of selecting
(60)
2 grapes at random without replacement?
8. Find the equation of the line that passes through (2, -3) and is parallel to
y = -_38 x + 2.
(36)
*9. Packaging A new food product is to be sold in cylindrical cans. The designer wants
(66)
the height of the can to be 6 inches greater than the radius. The manager wants
the can to have a volume of about 125 in3.
a. Write a polynomial equation in standard form that can be used to find the
radius. Leave in terms of π. Hint: The formula for the volume of a cylinder is
V = πr2h.
b. Find the approximate radius that should be used for the can by using a
graphing calculator. Round the radius to the nearest tenth.
10. Find an equation for the inverse of y = 7x2 + 21 for x ≥ 0. Identify the domain
(50)
and range of the inverse function.
11. Multiple Choice f(x) = x - 5 is an example of a:
(22)
A continuous function
B discrete function
C discontinuous function
D both B and C
Lesson 66
473
b 2
12. Write Explain why it is necessary to add the constant term _2 to both sides of
(58)
the equation when using the method of completing the square to solve a quadratic
equation.
()
*13. Graphing Calculator Calculate the binomial probability, to the nearest hundredth of
(Inv 5)
a percent, of 8 successes out of 50 trials, if the probability of success for each trial
is 0.17.
14. Geometry Consider the circle with radius 5 units, centered at the origin. A 270°
(56)
angle in standard position cuts the circle into a larger and a smaller sector. Find
the area of the larger sector.
*15. Error Analysis A student converted the logarithmic equation log2 64 = x to
(64)
exponential form by writing x2 = 64. What is the error? Write the correct
exponential equation.
*16. Multi-Step A 24 in. by 36 in. photo is framed and the total area of the photo and
(65)
the frame is 1200 in2.
a. Write a quadratic expression in standard form to represent the situation.
b. Find the width of the frame to the nearest hundredth of an inch.
17. Formulate Write the equation of an exponential function whose graph passes
(57)
through the points (0, 5) and (-2, 125). Determine whether the function is
increasing or decreasing.
*18. Analyze If using the quadratic formula to solve 16 = -3x2, what would be the
(65)
values of a, b, and c?
*19. Tires The diameter of a certain tire is 24 inches. Through how many degrees does
(63)
the tire rotate when the automobile travels 5 feet? Give your answer to the nearest
degree.
20. Population In 2006, the U.S. Census Bureau estimated there were 36,457,549
(55)
people living in California and 299,398,484 living in the United States.
a. Estimate the probability that a person randomly selected in the United States
lives in California.
b. Estimate the odds in favor that a person randomly selected in the United States
lives in California.
21. Probability A student taking a test has enough time to solve 2 out of the
(62)
5 equations shown.
x2 + 1 = 0,
x2 + 4 = 0,
x2 - 25 = 0,
x2 - 16 = 0,
x2 + 9 = 0
If she chooses equations randomly, what is the probability that both equations
have imaginary solutions?
Use trigonometric ratios to solve for each of the variables.
y
√
3
x
1 and sin 30° = _
22. cos 30° = _ and cos 30° = _
23. sin 30° = _
(52)
(52)
2
2
12
12
474
Saxon Algebra 2
24. Multi-Step A student creates a chart classifying
(60)
his books by subject and cover.
a. Are the events math textbook or paperback
textbook mutually exclusive or inclusive?
Explain why.
b. Find the probability that four books, selected
randomly with replacement, are math or
paperback. Round to the thousandths
place.
Cover
Subject
Hardcover
Paperback
Math
7
3
Sciences
23
14
Social Sciences
6
16
History
1
4
25. Music Frets are small metal bars placed across the neck of a guitar so that the
(59)
guitar can produce notes of a specific scale. To find the distance a fret should be
n
_
placed from the bridge, multiply the length of the string by 2 - 12 , where n is the
number of notes higher than the string’s root note. Where should a fret be placed
on the low E string to produce a note that is 5 notes higher, if the length of the
string is 64 cm, rounded to the nearest hundredths place?
*26. Generalize Tell how to factor by grouping.
(61)
27. Probability Find the probability of getting at least 1 tail if a coin is tossed 4 times.
(49)
*28. Multiple Choice Which is not a possible rational root of 2x3 + 2x2 - 21x - 27 = 0?
(66)
9
2
A -3
B -1
C _
D _
3
2
29. Let f(x) = 8x - 6 and g(x) = 4x2 + 1. Evaluate the composition (f ◦ g)(-2) using
(53)
two different methods.
*30. Write Explain why log0 5 and log1 5 do not exist.
(64)
Lesson 66
475
LES SON
Finding Inverse Trigonometric Functions
67
Warm Up
1. Vocabulary A circle with a radius of 1 unit is called a
.
(63)
2. What is sin 30°?
π
3. What is cos _
?
(63)
4
(63)
New Concepts
Evaluation of trigonometric functions for given angle measures was
explained in Lesson 46. Angle measures that have given trigonometric
function values can be found by using inverse trigonometric relations.
If a function value is:
sin θ = a
cos θ = a
tan θ = a
Reading Math
The symbol sin -1
represents the inverse
sine relation, NOT a
1
reciprocal. sin -1 a ≠ _
.
sin a
Then one of the inverse relation values is:
sin -1 a = θ
cos -1 a = θ
tan -1 a = θ
The inverses of the trigonometric functions
are not functions themselves because there are
many values of θ for a particular value of a.
√
2
For example, consider cos -1 _
2 . Angles with
7π
π
measures of _ and _ radians have the cosine
√
2
_
2 .
4
(_22 , _22 )
√
y
π
_
√
x
4
7π
_
4
4
So do all angles that are coterminal
value
with these angles.
(_22 ,
√
√2
_
2
)
Unit Circle
Example 1
Finding Trigonometric Inverses
√3
Describe all the values of sin -1 _
2 .
Hint
Substitute positive and
negative integers for n in
π
+ 2πn
the expressions _
3
2π
and _
+
2πn
.
Evaluate
3
the sine of each result.
Step 1: Find the values of θ between 0 and 2π
√3
radians for which sin θ is equal to _.
(
√3
1, _
_
2
)
2
2
√3
π √3
2π
_
= sin _, _ = sin _
2
3
2
3
y
(_12, _23 )
√
2π
_
3
π
_
3
x
Use y-coordinates of
points on the unit circle.
Step 2: Describe the measures of all angles
π
2π
_
that are coterminal with _
3 and 3 .
√3
All the values of sin -1 _
2 are given
π
2π
+ (2π)n and _
+ (2π)n,
by _
3
Add integer multiples
of 2π radians.
3
where n is an integer.
Online Connection
www.SaxonMathResources.com
476
Saxon Algebra 2
To define inverse trigonometric functions, it is necessary to restrict the
domains of the trigonometric functions. The trigonometric functions that
a calculator uses have restricted domains and will give only one value in the
range that you will need to generalize to find all the possible values.
π ≤θ≤_
π⎬
Sin θ = sin θ for ⎨⎩θ -_
2
2⎭
⎧
⎫
2
y-axis, the positive
x-axis, or in Quadrant
I or IV.
Cos θ = cos θ for {θ 0 ≤ θ ≤ π}
θ can terminate on the
x-axis, the positive
y-axis, or in Quadrant
I or II.
π <θ<_
π ⎬ θ can terminate on the
Tan θ = tan θ for ⎨⎩θ - _
2
2⎭
⎧
π
_
θ can terminate on the
x
π
-_
2 _
π
2
x
⎫
positive x-axis, or in
Quadrant I or IV.
Sin θ, Cos θ, and Tan θ can be used to define the inverse
trigonometric functions. For each value of a in the domain
of any inverse trigonometric function, there is one and only
one value of θ in the range.
π
-_
2 _
π
2
x
π
-_
2
Inverse Trigonometric Functions
Reading Math
The inverse
trigonometric functions
are also called the
arcsine, arccosine, and
arctangent functions.
Words
The inverse sine
function is Sin-1a = θ,
where Sin θ = a.
The inverse cosine
function is Cos-1a = θ,
where Cos θ = a.
The inverse tangent
function is Tan-1a = θ,
where Tan θ = a.
Example 2
Symbol
Domain
Sin-1a
{a-1 ≤ a ≤ 1}
Cos-1a
{a-1 ≤ a ≤ 1}
Tan-1a
Range
⎧
π
π ⎫
⎨θ - _
≤θ≤_
⎬
2
2 ⎭
⎩
{θ -90° ≤ θ ≤ 90°}
{θ0 ≤ θ ≤ π}
{θ0° ≤ θ ≤ 180°}
⎧
π
π ⎫
⎨θ - _
<θ<_
⎬
2
2 ⎭
⎩
{a-∞ < a < ∞}
{θ -90° < θ < 90°}
Evaluating Inverse Trigonometric Functions
Find each inverse trigonometric function value that exists. Give answer in
y
both radians and degrees.
a. Tan-1 (-1)
x
SOLUTION
Hint
π , or
_
4
Use the unit circle.
y
Tan θ = _x
√
2
_
π
2
Tan(-_
= -_
= -1
√
4)
2
_
2
π
-1 = Tan -_
4
( )
Tan
-1
(-1) =
π
-_
is the angle measure between
4
π
π
_
- and _ whose tangent is -1.
π
-_
2
45°
(_22 ,
√
√2
_
2
)
2
( 4 ), or Tan
-1
(-1) = -45°
-1
b. Cos 2
SOLUTION
The domain of the inverse cosine function is {a -1 ≤ a ≤ 1}. Because 2 is
not in this domain, Cos -1 2 does not exist.
Lesson 67
477
Example 3 Solving Trigonometric Equations
a. Solve 2 cos θ = √
3 , for 0° ≤ θ ≤ 360°.
Caution
SOLUTION
The inverse cosine key
on a calculator will only
give the solution in the
domain of the inverse
cosine function.
2 cos θ = √
3
√
3
2 cos θ _
_
=
2
2
√3
Cos -1 _
= 30°
2
y
(_23 , _12 )
√
cos θ is unknown.
30° x
Divide each side by
2 to isolate cos θ.
330°
√
3
cos θ =_
2
(_23 , _12 )
√
θ = 30° or 330°
Use the unit circle.
The solutions are 30° and 330°.
b. Solve 5 sin θ = 5, for 0° ≤ θ ≤ 360°.
(0, 1)
y
SOLUTION
5 sin θ = 5
90°
sin θ is unknown.
sin θ = 1
x
Divide each side by 5.
θ = 90°
Use the unit circle.
The solution is 90°.
c. Solve 6 sin θ + 3 = 0, where θ is any real number of radians.
SOLUTION
D
2
y
6 sin θ + 3 = 0
sin θ is unknown.
6 sin θ = -3
7π
_
6
Subtract 3 from each side.
1
sin θ = -_
Divide each side by 6.
2
7π
11π
θ=_
or _
or any coterminal angle
6
6
x
11π
_
(
√3
1
_
, _
2
)
2
6
(_23 , _12 )
√
7π
11π
The solutions are _
+ 2πn and _
+ 2πn, where n is an integer.
6
6
Example 4 Solving Trigonometric Equations Using a Calculator
Solve each equation to the nearest tenth. Use the given restrictions.
Caution
If the answer on your
calculator screen is
0.927295218 when you
enter cos -1 (0.6), your
calculator is set to radian
mode instead of degree
mode.
a. 10 Cos θ = 6, for 0° ≤ θ ≤ 180°
SOLUTION
10 Cos θ = 6
Cos θ is unknown.
Cos θ = 0.6
θ = Cos
Divide each side by 10.
-1
(0.6)
θ ≈ 53.1°
The solution is 53.1°.
478
Saxon Algebra 2
Use the inverse cosine key on your calculator.
b. 10 cos θ = 6, for 270° < θ < 360°
y
SOLUTION
Because of the restrictions, the terminal side
of θ is in Quadrant IV. Find the angle that
terminates in Quadrant IV that has the same
cosine value as 53.1°.
306.9°
x
53.1°
θ ≈ 360° - 53.1°
θ ≈ 306.9°
Example 5
θ has a reference angle of
53.1°, and 270° < θ < 360°
Application: Navigation
A pair of hikers plan to walk from a campground to a waterfall. The
waterfall is 2.5 miles east and 0.5 mile north of the campground. To the
nearest degree, in what direction should the hikers head?
Step 1: Draw a diagram.
The hikers’ direction should be based on θ, the measure of an acute angle
of a right triangle.
N
Waterfall
0.5
θ
2.5
Campground
Step 2: Find the value of θ.
opposite
Tan θ = _
adjacent
Use the tangent ratio.
0.5 = 0.2
Tan θ = _
2.5
Substitute 0.5 for
opposite and 2.5 for
adjacent. Then
simplify.
θ = Tan -1 0.2
θ ≈ 11°
The hikers should head 11° north of east.
Lesson Practice
√
3
a. Describe all the values of cos -1 _
.
2
(Ex 1)
-1
√
3
in both radians and degrees if it exists
b. Evaluate Cos -_
2
(Ex 2)
3
-1
c. Evaluate Sin -_ in both radians and degrees if it exists.
)
(
(Ex 2)
(
2
)
d. Solve 2 sin θ = -1, for 0° ≤ θ ≤ 360°.
(Ex 3)
e. Solve -cos θ = -1, for 0° ≤ θ ≤ 360°.
(Ex 3)
f. Solve 10 cos θ - 5 = 0, where θ is any real number of radians.
(Ex 3)
Lesson 67
479
g. Solve the equation to the nearest tenth. Use the given restrictions.
5 Sin θ = 2, for - 90° ≤ θ ≤ 90°
(Ex 4)
h. Solve the equation to the nearest tenth. Use the given restrictions.
5 Sin θ = 2, for 90° < θ < 180°
(Ex 4)
i. A group of hikers plan to walk from a parking lot to an overlook. The
overlook is 1.6 miles west and 0.6 mile north of the parking lot. To the
nearest degree, in what direction should the hikers head?
(Ex 5)
Practice
Distributed and Integrated
1. Given θ = -1787°, find the measure of the reference angle.
(56)
Convert each measure from degrees to radians.
3. 125°
2. 60°
(63)
4. -240°
(63)
(63)
5. Use synthetic substitution to find f(-3) for f(x) = 7x4 - 3x3 + x2 + 11.
(51)
6. Complete the square and factor the resulting perfect square trinomial: x2 + 6x.
(58)
Simplify.
7.
(48)
_a
b
_
a
+
_b
8.
(48)
s +_
x
_
a+b
b
_
a
_
s+x
b
*9. Error Analysis A student incorrectly solved the equation sinθ = -0.55, for
(67)
90° ≤ θ ≤ 270°, as shown below. What error did the student make? Find the
correct solution.
sin θ = -0.55
sin
-1
(-0.55) ≈ -33.4°
θ ≈ 180° - 33.4°
θ ≈ 146.6°
*10. Tennis A tennis player 6 feet from the net hits
(67)
an overhead shot, hitting the ball at a height of
8.5 feet. Assume that the path of the ball is a
straight line. To the nearest degree, what is the
maximum angle θ at which the ball can be hit
and land within the court?
θ
39 ft
45 ft
*11. Graphing Calculator a. Graph y = x4 + 2x3 - 12x2 + 14x - 5 on a graphing
(66)
calculator. What are the x-intercepts?
b. Find the multiplicity of each root of x4 + 2x3 - 12x2 + 14x - 5 = 0
by factoring.
480
Saxon Algebra 2
0≤x≤4
12. Minimize P = 4x + 6y for the constraints y ≥ 1
(54)
y ≥ -x + 4
*13. Multiple Choice Which is a solution of 5 + 2x2 = -3x?
(65)
√
31
5
3 +_
A 1
B _
C _
2
4
4
√
31
3 -_
D -_
i
4
4
*14. Contact Lenses The contact lens prescription for Asia’s left eye is stronger than the
(Inv 5)
prescription for her right eye. If Asia accidentally mixed up her contact lenses,
what is the probability that she will put the correct lens in either eye on the first
try? Explain.
*15. Multi-Step Complete parts a–d to determine the measure of the angle formed
(63)
by the hands of a clock when the time is 3:24. Find all measures in radians.
a. Find x, the angle formed at 3:00.
b. Find (x + y + z), the angle through which the minute hand moves in
24 minutes.
y
x
z
c. Find y, the angle through which the hour hand moves in 24 minutes.
d. Find z, the angle formed by the minute hand and hour hand when the time
is 3:24.
16. Model Make a Venn Diagram to illustrate how the complex numbers, real numbers,
(62)
and imaginary numbers are related.
17. Finance The value of a stock over the past five years where x is the number of
(61)
years is modeled by x3 - 2x2 + 5x - 10. Factor this expression.
18. Geometry The Community Center wants to build a playground area and enclose it
(54)
within a wooden, split rail fence. The cost of fencing is $20 per finished foot. The
cost for painting the fence is $12 per finished foot. The Center cannot spend more
than $2000 on fencing. The playground cannot be any longer than 30 ft and needs
to be at least 15 ft wide. What is the maximum area the Center can get for the size
of the playground? How much will this cost?
19. Error Analysis A cup contains 15 pink beads, 24 purple beads, and 18 gold beads. Three
(60)
beads are individually selected with replacement. The work below is used to find the
probability that one of each color bead is selected in the order pink, purple, gold.
Is the work correct? If not, explain the error and find the correct solution. If so,
explain each step of the work.
K = event of pink bead drawn
U = event of purple bead drawn
G = event of gold bead drawn
5
15 = _
P(K ) = _
57
19
P(K,U,G ) = P(K ) · P(U ) · P(G )
8 ·_
6 ≈ 0.035
5 ·_
=_
19 19 19
8
24 = _
P(U ) = _
57
19
20. Multiple Choice Choose the letter that best represents
(59)
3
3x4 √
4
3
3
A _
B 3x4 √
2
C 6x4 √
4
2
18 = _
6
P(G ) = _
57
19
216x
_
simplified.
√
4
3
12
3
D
3x4 √
2
_
2
Lesson 67
481
*21. Analyze When solving x3 + 7x2 + 7x - 6 = 0, why is it not necessary to use
(66)
synthetic substitution after one of the roots is found?
22. Write A ball is tossed with an initial velocity of 74 feet per second from the top of
(58)
a building that is 50 feet tall. The height, h, of the ball in feet after t seconds can
be approximated by the equation h = -16t2 + 74t + 50. The ball will be 20 feet
from the ground when t = -0.375 second and t = 5 seconds. Explain why you
would not use the solution t = -0.375 second.
*23. Coordinate Geometry Find m∠AOB to the nearest tenth of a degree, using these
(67)
points: A(8, 3), O(0, 0), and B(8, 0).
24. Verify Explain why the probability of choosing first, second, and third place
(55)
1
winners for a photo when randomly choosing from the top 10 athletes is _
.
C
10
3
*25. Storage The volume of a can of dog food is 450 cubic inches. The height of the
(66)
can is 5 inches greater than its radius. Write a polynomial in standard form, using
3.14 for π, that can be used to find the radius. (Round the constant to the nearest
integer.) Then find the approximate radius by using a graphing calculator.
26. Generalize Suppose θ is an angle in standard position and Q(x, y) is a point on the
(56)
terminal side of the angle. Explain how the signs (positive or negative) of sin θ,
cos θ, and tan θ depend on which quadrant Q lies in.
27. School A teacher has 3 red crayons, 7 yellow crayons, 5 green crayons, and 2 blue
(60)
crayons. A yellow crayon and then a blue crayon are chosen at random without
replacement. Are these events dependent or mutually exclusive?
28. The number of rabbits increases exponentially. At first there were 400. Three
(57)
years later there were 1600. How many rabbits would there be after 10 years?
y≥0
x≥0
29. Maximize P = 5x + 2y for the constraints
(54)
y ≤ -x + 10
y ≤ 2x + 1
*30. Physical Science The South Falls waterfalls in Silver Falls State Park in Oregon
(65)
are 177 feet high. The equation -16t2 + 177 = 0 gives the number of seconds it
takes for a drop of water at the top of the falls to reach the pool of water below.
Solve the equation by using the quadratic formula. Round to the nearest hundredth
of a second.
482
Saxon Algebra 2
LESSON
Finding Conditional Probability
68
Warm Up
1. Vocabulary If the occurrence of one event affects the probability of the
second event, the two events are
.
(55)
2. As a percent, what is the probability of getting heads on all three flips of a
coin?
(55)
3. A bag contains 8 red marbles, 16 white marbles, and 4 black marbles. What
is the probability of randomly selecting a white or black marble?
(55)
New Concepts
The probability of two independent events is the product of their individual
probabilities. However, finding the probability of two dependent events
requires finding a conditional probability. Recall that P(B|A) is a conditional
probability. It is read as “the probability of B given that A has occurred.”
Dependent Events
If A and B are dependent events, then P(A and B) = P(A) · P(B|A).
Example 1
Finding the Probability of Dependent Events
a. Suppose the two spinners shown
are each spun once. Find the
probability that the first spinner
lands on 2 and the sum of the
spinners is 6.
Math Reasoning
Analyze Name another
way of finding the
probability of the events
in Example 1a.
1
2
1
2
4
3
4
3
SOLUTION Assign event A to be the
first spinner landing on 2 and event B to be the sum of the spinners is 6.
Determine if the events are dependent. It helps to list all the outcomes
in the sample space. The first numbers are from the first spinner, and the
second numbers are from the second spinner. If event A does not occur,
3
P(B) = _
because out of 16 possible outcomes, 3 have a sum of 6. These
16
pairs are boxed. If event A does occur, P(B) = _14 because out of 4 possible
pairs only one has a sum of 6.
1, 1
2, 1
3, 1
4, 1
1, 1
2, 1
3, 1
4, 1
1, 2
2, 2
3, 2
4, 2
1, 2
2, 2
3, 2
4, 2
1, 3
2, 3
3, 3
4, 3
1, 3
2, 3
3, 3
4, 3
1, 4
2, 4
3, 4
4, 4
1, 4
2, 4
3, 4
4, 4
Since the occurrence of event A affects the probability of event B, the
events are dependent.
Online Connection
www.SaxonMathResources.com
1 =_
1
1 ·_
P(A and B) = P(A) · P(B|A) = _
4 4 16
Lesson 68
483
b. Suppose the two spinners shown are
each spun once. Find the probability
that the first spinner lands on an
odd number and the sum of the
spinners is greater than 4.
1
2
1
2
4
3
4
3
SOLUTION
Confirm that the events are dependent.
10
= _58 because out of 16 possible
If event A does not occur, P(B) = _
16
outcomes, 10 have a sum greater than 4.
If event A does occur, P(B) = _4 = _1 because out of 8 possible pairs (those
8
2
in the first and third columns), 4 have a sum greater than 4.
1, 1
2, 1
3, 1
4, 1
1, 1
2, 1
3, 1
4, 1
1, 2
2, 2
3, 2
4, 2
1, 2
2, 2
3, 2
4, 2
1, 3
2, 3
3, 3
4, 3
1, 3
2, 3
3, 3
4, 3
1, 4
2, 4
3, 4
4, 4
1, 4
2, 4
3, 4
4, 4
Since the occurrence of event A affects the probability of event B, the
events are dependent.
1 =_
1
1 ·_
P(A and B) = P(A) · P(B|A) = _
2 2
4
Example 2 Using a Table to Find Conditional Probability
The table shows the results of a
survey that asked students to state
the year and season of their birthday.
a. Find P(Winter|1993).
Hint
First circle or highlight
the row(s) or column(s)
with the given
condition(s), that is,
event A.
1992
1993
1994
Winter
10
16
14
Spring
6
9
11
Summer
12
11
8
SOLUTION P(Winter|1993) is the
Fall
14
9
12
probability of a birthday occurring
in the winter, given that it occurred in 1993. Since the condition is that it
must be in the year 1993, focus on the 1993 column.
winter : __
16
16 ≈ 35.6%
_
=_
all seasons 16 + 9 + 11 + 9
45
b. Find P(1992|Spring or Summer).
SOLUTION P(1992|Spring or Summer) is the probability of a birthday
occurring in 1992, given that it occurred in the spring or summer. Since the
condition is that it must be in the spring or summer, focus on those rows.
6 + 12
18 = _
1992
6 ≈ 31.6%
___
: ___ = _
spring or summer all years 6 + 9 + 11 + 12 + 11 + 8
57
19
When items are selected and not replaced, the events are dependent. For each
successive selection, the total number of outcomes, which are the values in
the denominators, decreases by one.
484
Saxon Algebra 2
Example 3 Using a Calculator to Find Conditional Probability
A raffle is being held at a school function. A container has the names of
the 587 juniors and 618 seniors who are participating on slips of paper.
Prizes are being given to the first, second, and third names pulled from the
container. Once a student wins a prize, his or her name is not returned to
the container. What is the probability that the first two prizes go to juniors
and the third prize goes to a senior?
Math Reasoning
Analyze Calculate
P(junior, senior, junior).
How does it compare to
Example 3?
SOLUTION After the first junior wins, there is one
less junior name in the container, which is also one
less name overall. After the second junior wins,
there is one less name overall again. There are
587 + 618 = 1205 names in all.
587 · _
586 · _
618 ≈ 0.122
P( junior, junior, senior) = _
1205 1204 1203
The probability is about 12.2%.
Example 4
Math Reasoning
Estimate Estimate the
probability that an
envelope was stamped on
Thursday given that it was
stamped upside down.
Application: Quality Control
At a mailing center, it was found that on Tuesday, 44 out of 662 envelopes
were stamped upside down, on Wednesday, 16 out of 251 envelopes were
stamped upside down, and on Thursday, 3 out of 598 envelopes were
stamped upside down. Find the probability that an envelope was stamped
on Wednesday given that it was stamped right-side up.
SOLUTION
1. Understand Make a table to
organize the information.
Subtract to find the number
of envelopes stamped
right-side up.
Tues.
Wed.
Thurs.
Right-side Up
618
235
595
Upside Down
44
16
3
2. Plan The condition is that the envelope was stamped right-side up.
Find the sum of the envelopes stamped right-side up; this will be the
denominator.
3. Solve Use a calculator.
235 ≈ 0.162
235
__
=_
618 + 235 + 595
1448
The probability is about 16.2%.
4. Check Use estimation to check for reasonableness. From the given
information, about 1500 envelopes were stamped and about 250 of
those were stamped on Wednesday. The majority of those stamped on
2
250
Wednesday were stamped right-side up, and _
= _16 = 16 _3 %.
1500
Lesson 68
485
Lesson Practice
Suppose the two spinners shown are each spun once.
(Ex 1)
a. Find the probability that the first
spinner lands on 4 and the product
of the spinners is less than 10.
b. Find the probability that the first
spinner lands on a number less than
4 and the sum of the numbers is less
than 6.
The table shows the results of a survey that
asked students from three grade levels to pick
their favorite subject from those listed.
(Ex 2)
c. Find P(Math|12).
d. Find P(10th|History).
1
2
1
2
4
3
4
3
Math
History
Science
English
10th
34
45
52
9
11th
56
19
45
12
12th
11
26
66
58
e. A raffle is being held at a mall. A box
has the names of the 237 men and 318 women who are participating on slips
of paper. Gift certificates are being given to the first five names pulled
from the box. Once a person wins a certificate, his or her name is not
returned to the box. What is the probability that the first three certificates
go to men and the last two certificates go to women?
(Ex 3)
f. At a factory, it was found that during the morning shift, 32 out of
the 2666 boxes of nails that were filled had unacceptable weights, and
during the evening shift, 11 out of the 1870 boxes of nails that were
filled had unacceptable weights. Find the probability that a box was
filled in the evening, given that its weight was found acceptable.
(Ex 4)
Practice
Distributed and Integrated
1. Solve s2 - 22 = -112. Write the solutions in terms of i.
(62)
Factor completely.
2. -3pax + pax2 + 2pa
(23)
3. 30 + 3x2 - 21x
(23)
4. Multi-Step Use x2 - 62x + 961 = 26.
a. Explain why the left side of the equation is a perfect square trinomial.
(58)
b. Factor x2 - 62x + 961.
c. Solve the equation for x.
486
Saxon Algebra 2
*5. Error Analysis Find and correct the error a student made in determining the roots
(66)
and their multiplicity in x3 - 4x2 + 4x = 0.
x3 - 4x2 + 4x = 0
x(x2 - 4x + 4) = 0
x(x - 2)(x - 2) = 0
The only root is 2 and its multiplicity is 2.
6. Analyze What does it mean for an event to have the first number in the odds in
favor ratio greater than the second number?
(55)
7. Geometry Write an expanded expression for the volume
of the cube shown.
(49)
2b + 5 cm
*8. Housing The table shows estimated housing
(68)
occupancy for three states for the year 2006.
Find the probability that a randomly selected
house from one of these states is in Nevada.
Then find the probability that a randomly
selected house from one of these states is in
Nevada given that it is vacant. Round each
percent to the nearest tenth.
Simplify.
_2
9. -64 - 3
(59)
10.
(59)
Vacant Houses
1,075,521
165,918
936,828
128,439
7,088,376
819,138
Mississippi
Nevada
New York
-3
_
-9
Occupied Houses
_1
11. (-8) 3
-_3
(59)
2
12. Find the arc length of a sector with a radius of 4 in. and central angle θ = _π6 .
(63)
*13. Probability A bag contains 30 red marbles and 10 white marbles. If three marbles
(68)
are randomly chosen without replacement, what is the probability of selecting all
red or all white marbles?
14. Surveying A surveyor is standing on top of a hill that is 600 meters from a bridge.
(46)
Her angle of elevation to the top of the bridge is 14° and her angle of depression
to the bottom of the bridge is 26°. To the nearest tenth of a meter, what is the
height of the bridge?
*15. Generalize Insert = or ≠ to make each statement true.
(68)
If A and B are independent events, then P(B|A)
If A and B are dependent events, then P(B|A)
P(B).
P(B).
16. Statistics How do probability and statistics compare and contrast?
(60)
17. Multiple Choice Which angle does not have a reference angle of 30º?
(56)
A -30º
B 120º
C 570º
D -150º
Lesson 68
487
*18. Estimate Use what you know about the unit circle and special angles to estimate
(67)
Cos-1(0.48) in degrees. Do not use a calculator. Explain your method.
19. Write Describe how to determine cos 150° by using its reference angle.
(63)
Simplify.
20. 3
(44)
√_52 - 2 √_25
21. 3 √
18 + 2 √
50 - √
98
(44)
22. Calculate the binomial probability P(12, 200, 0.09) to the nearest hundredth of a
N!
percent using the formula P(n, N, p) = _
pn(1 - p) N-n.
n!(N - n)!
(Inv 5)
*23. Graphing Calculator Describe the feasible region for the set of inequalities x ≥ 0,
(54)
x ≤ 4, y ≥ 0, 2y + x ≤ 25.
24. Error Analysis Find and correct the error a student made in subtracting complex
(62)
numbers.
(7 - 4i) - (2 + 3i)
5-i
*25. Multiple Choice Which describes all the values of sin-1_12 ?
(67)
A
5π
_
C
2π
_
6
3
B
_π
D
_π
6
3
3
√
6π2x
26. Estimate Sketch a graph of the equation y = _
, where y is the radius of
2π
(59)
a sphere and x is the volume of the sphere. Use the graph to estimate the
volume of a sphere with a radius of 2 cm. Use the graph to estimate the radius of
a sphere with a volume of 14 cm3.
27. Volume The volume of a California king size mattress is 12x3 - 36x2 - 432x - 384
(61)
cubic inches. Factor the polynomial given that x - 8 is a factor. Then find the
volume given that x = 20.
*28. Plant Nutrition The pH of a soil is given by the function pH = -log [H+], where
(64)
[H+] represents the concentration of hydrogen ions in moles per liter. Most plants
require soil with a hydrogen ion concentration between 0.0001 and 0.00000001
moles per liter. What pH values correspond to these hydrogen ion concentration
values?
*29. Travel A driver leaves a parking lot and heads due south at a rate of 40 miles per
(65)
hour. An hour later, another driver leaves the same parking lot and heads due west
at a rate of 60 miles per hour. Approximately how long will it take for the cars to
be 300 miles apart?
*30. Multi-Step Two bags each contain papers labeled with numbers 1 through 5. One
(68)
paper is selected from each.
a. Explain why selecting a 5 from the first bag and the sum of the numbers is
greater than 5 are dependent events.
b. Find the probability of both events.
488
Saxon Algebra 2
LESSON
Simplifying Complex Expressions
69
Warm Up
1. Vocabulary A number that can be written in the form a + bi where
a and b are real numbers is a
.
(62)
2. Multiply (3x - 5)(4x + 2).
(19)
5
3. Simplify _.
√
5
(40)
4. Add (3 + 2i) + (6 - 8i).
(62)
New Concepts
Math Language
The imaginary axis is
sometimes referred to as
the i-axis.
Complex numbers (a + bi) can be graphed on a complex plane, where the
horizontal axis represents the real part, a, and the vertical axis represents the
imaginary part, bi.
Example 1
Graphing Complex Numbers
Graph each number on the complex plane.
a. 3 + 4i
b. -2 - 3i
c. -4i
d. 1
imaginary
SOLUTION
4i
a. Move 3 units to the right and 4 units up.
3 + 4i
2i
b. Move 2 units to the left and 3 units down.
1
-4
c. Move 0 units right or left and 4 units down.
-2
2
4
real
-2i
d. Move 1 unit to the right and 0 units up or
down.
-2 - 3i -4i -4i
The absolute value of a complex number is its distance from the origin. Unless
the point is on the x- or i-axis, the distance is the length of a line segment
that forms the hypotenuse of a right triangle and can be found using the
Pythagorean Theorem.
Absolute Value of a Complex Number
imaginary
a + bi
4i
⎪a + bi⎥
-4
Online Connection
b
2i
2
+ b2
⎢a + bi = √a
-2
-2i
a
2
4
real
www.SaxonMathResources.com
Lesson 69
489
Example 2 Finding the Absolute Value of Complex Numbers
Find the absolute value of each number.
a. 3 + 4i
SOLUTION ⎢a + bi =
2
√a
+ b2
2
+ 42
⎢3 + 4i = √3
=5
+ 16 = √25
= √9
b. -2 - 3i
Math Reasoning
Verify Use the formula
for the absolute value
of a complex number to
show that ⎢-1 is 1.
SOLUTION ⎢a + bi =
⎢-2 - 3i =
2
√a
+ b2
2
+ (-3)2
√(-2)
+ 9 = √13
= √4
c. -5i
SOLUTION ⎢a + bi =
⎢-5i =
2
√a
+ b2
2
+ (-5)2
√0
=5
= √25
When multiplying complex numbers, remember that i2 = -1.
Example 3
Multiplying Complex Numbers
Multiply. Write answers in the form a + bi.
a. -4i(7 + 3i)
SOLUTION
-4i(7 + 3i)
-28i - 12i2
Distributive Property
-28i - 12(-1)
i2 = -1
-28i + 12
Simplify.
12 - 28i
Write in standard form.
b. (5 + 4i)(6 - 2i)
SOLUTION
(5 + 4i)(6 - 2i)
30 - 10i + 24i - 8i2
2
490
Saxon Algebra 2
Use the FOIL method.
30 + 14i - 8i
Simplify.
30 + 14i - 8(-1)
i2 = -1
30 + 14i + 8
Simplify.
38 + 14i
Write in standard form.
Study the powers of i below.
Powers of i
i5 = i · i4 = i(1) = i
1
i =i
i2 = -1
i6 = i · i5 = i(i) = i2 = -1
i3 = i · i2 = i(-1) = -i
i7 = i · i6 = i(-1) = -i
i4 = i · i3 = i(-i) = -i2 = -(-1) = 1
i8 = i · i7 = i(-i) = -i2 = -(-1) = 1
Notice that the four simplified values of i, -1, -i, and 1 repeat such that
i = i5, i2 = i6, i3 = i7, and i4 = i8. Any power of i can be found by dividing the
exponent of i by four and using the remainder.
When the remainder is …
The expression is equivalent to …
0
i4 = 1
1
i1 = i
2
i2 = -1
3
i3 = -i
Example 4 Evaluating Powers of i
Simplify each expression.
a. i30
b. 5i12
SOLUTION 30 ÷ 4 = 7 R 2,
30
2
SOLUTION 12 ÷ 4 = 3 R 0, so i12 = i4 = 1.
5i12 = 5(1) = 5
so i = i = -1.
c. -2i13
SOLUTION 13 ÷ 4 = 3 R 1, so i13= i1 = i.
-2i13 = -2(i) = -2i
Caution
A simplified expression
does not have a radical
in the denominator.
Because i = √
-1 , an
expression with i in
the denominator is not
simplified.
To eliminate a radical in a denominator, multiply both the numerator and
denominator of the fraction by the conjugate of the denominator. The
complex conjugate of a + bi is a - bi.
Example 5
Dividing Complex Numbers
a. Divide 3 by 6 + i √
5 . Write the answer in the form a + bi.
SOLUTION
3
_
(
)
6 + i √5
6 - i √5
3
_
_
6 + i √5 6 - i √5
Multiply the numerator and the denominator by
the complex conjugate of the denominator.
18 - 3i √5
_
Distribute and simplify.
36 - 5i2
18 - 3i √5 _
3 √5
_
= 18 - _i
41
41
41
Lesson 69
491
b. Divide 4 + i by 6i. Write the answer in the form a + bi.
SOLUTION
Math Reasoning
Analyze Why is -6i the
conjugate of 6i?
4+i
_
6i
4+i _
-6i
_
( -6i )
6i
Multiply the numerator and the denominator by
the complex conjugate of the denominator.
-24i - 6i2
_
Distribute.
-36i2
-24i - 6(-1)
__
Simplify.
-36(-1)
-24i + 6
6 = -_
24 i + _
2i + _
1 =_
1 -_
2i
_
= -_
36
36
3
36
6
6
3
Lesson Practice
Graph each number on the complex plane.
(Ex 1)
a. 1 - 3i
b. -2 + 4i
d. 2i
c. -3
Find the absolute value of each number.
(Ex 2)
f. -5 + i
e. 7 - 2i
g. 10i
Multiply. Write answers in the form a + bi.
(Ex 3)
h. 8i(-9 - 5i)
i. (2 - 3i)(5 + 9i)
Simplify each expression.
(Ex 4)
j. i18
k. -3i11
l.
_1 i21
2
m. Divide 3 + 3i by 3i. Write the answer in the form a + bi.
(Ex 5)
n. Divide -4 by 2 + i √
3 . Write the answer in the form a + bi.
(Ex 5)
Practice
Distributed and Integrated
Solve using the quadratic formula.
1. f(x) = x2 + 8x - 3
(65)
2. f(x) = 3x2 - 10x + 4
(65)
3. Probability A bag holds 53 yellow marbles, 17 red marbles, and 30 green marbles.
What is the probability of not choosing a red marble?
(55)
4. A savings account earns interest at an annual rate of 3.8%, compounded quarterly.
If the account begins with a principal amount of $2,700, what will its value be
after 14 years?
(57)
Solve for x.
_1
5. x 7 = 5
(59)
492
Saxon Algebra 2
_2
6. x 3 = 36
(59)
*7. Multiple Choice In which quadrant of the complex plane is 5 - i located?
(69)
A Quadrant I
B Quadrant II
C Quadrant III
D Quadrant IV
8. Write Explain how to use synthetic division to determine if x + 6 is a factor of
x3 + 3x2 - 28x - 60 and then perform the division.
(61)
Simplify.
9.
(62)
_1 √
-256 .
10.
4
(62)
_1 √
-2500 .
2
11. Given θ = -920°, find the measure of the reference angle.
(56)
*12. Generalize Show that the product of any two complex conjugates, a + bi and
(69)
a - bi, is a real number.
(
√
2
)
*13. Evaluate the inverse trigonometric function Sin-1 -_
. Give your answer in both
2
(67)
radians and degrees.
14. Why can’t the matrix below be used as an encoding matrix for a cipher?
(Inv 4)
⎡5
0
⎣0
⎢
-2
3⎤
0
7
0 13 ⎦
15. Amusement Parks You get on a ferris wheel ride, and the ferris wheel turns through
(56)
1370°. How much further must the wheel turn before you are at the bottom again?
*16. Error Analysis
(68)
Find and correct the error a student made in finding P(AM | Male).
16 = _
4 = 80%
P(AM | Male) = _
5
20
AM
PM
Male
16
8
Female
4
12
*17. Graphing Calculator a. Graph y = x3 + 6x2 - 32 on a graphing calculator. What are
(66)
the x-intercepts?
b. Find the multiplicity of each root of x3 + 6x2 - 32 = 0 by factoring.
*18. Formulate Give a rule for finding P(B|A). Hint: Consider the rule for P(A and B)
(68)
for dependent events.
19. Multiple Choice Which equation is equivalent to 10x = 0.1?
(64)
A log0.110 = x
B log0.1x = 10
C log100.1 = x
D log10x = 0.1
Lesson 69
493
20. Formulate Write a formula for P(A or B or C). Use the diagram to the
(60)
right to help you.
A
B
C
21. Multi-Step Let f(θ) = Sinθ and g(θ) = Sin-1θ, where θ is in radians.
(67)
1
a. Evaluate f g _ . Show your steps.
( ( 2 ))
π
b. Evaluate g f (_
6 ) . Show your steps.
(
)
c. Based on part a, what is the value of f(g(x))? Based on part b, what is the value
of g( f (x))?
22. Error Analysis Examine the work below. Find and explain the error. Then find the
(58)
correct solution.
x2 - 44x + 484 = 121
(x - 22)2 = 121
x - 22 = 11
x = 33
*23. Probability A student graphs y = 9x2 - 29x + 22 and uses a graphing calculator
(65)
to find the value of the zeros one at a time. What is the probability that the first
zero the student finds is negative? Why?
*24. Multi-Step Simplify i22 + i15. Then graph the number on the complex plane.
(69)
25. Planet Rotation Venus is only slightly smaller than Earth; its radius is
(63)
approximately 3757 miles. But it rotates much more slowly; it takes 243 Earth days
for Venus to make one complete rotation. If an object is fixed on Venus’s equator,
how far does it travel in 1 hour due to Venus’s rotation?
26. Analyze Describe the value(s) of bx when b = 1 and when b ≠ 1.
(64)
*27. Find the absolute value of ⎢3 - 4i.
(69)
*28. Find the absolute value of ⎢6 + 2i.
(69)
29. Sports A football player playing in a domed stadium kicks a football straight
(62)
up into the air. The height of the football after t seconds can be modeled
by -16t2 + 80t. The dome has a height of about 288 feet. The solutions of
-16t2 + 80t = 288 give the number of seconds it will take for the football to reach
the top of the dome. Solve the equation. Will the football reach the top of the
dome? If so, when?
30. Geometry Find the area of the triangle to the right.
(59)
7
(2 ) 9 cm
11
(2) 9 cm
494
Saxon Algebra 2
LESSON
Solving Radical Equations
70
Warm Up
1. Vocabulary In the expression
(40)
3
√x
,
3 is referred to as the
.
64 .
2. Simplify √
(40)
3
3. Simplify √
-125 .
(40)
4. Solve x2 + 5x - 30 = 6.
(35)
_1
5. Write (x + 6) 2 in radical form.
(40)
New Concepts
A radical equation is an equation that contains at least one radical whose
radicand contains a variable.
Radical Equations
√x
+3 +1=4
3
3
√
2x + 5 = √
x-1
Math Reasoning
Analyze Does cubing
each side of x = 5 result
in extraneous solutions?
Explain.
Not Radical Equations
x - √5 = 8
4
√
25 = x - 3
To solve a radical equation, first isolate the radical on one side of the equation.
Then raise both sides of the equation to the power of the index of the radical.
This will clear the radical.
It is critical that the solutions of a radical equation be checked because some of
the solutions obtained may be extraneous solutions. An extraneous solution
is a solution of a derived equation that is not a solution of the original equation.
For example, the equation x = 5 has one solution, 5. But when both sides
of the equation are squared, the resulting equation x2 = 25 has two solutions,
-5 and 5. The solution -5 is extraneous because it is not a solution of the
original equation.
Example 1 Solving Equations Containing One Square Root
Solve each equation.
a. √
x - 5 = 14
SOLUTION The radical is already isolated on the left side.
√x
- 5 = 14
2
( √x
- 5 ) = 142
x - 5 = 196
x = 201
Square both sides of the equation.
Simplify.
Add 5 to both sides.
Check √x
- 5 = 14
√201
- 5 14
Online Connection
www.SaxonMathResources.com
√196
14
14 = 14 ✓
Lesson 70
495
b. √
6x + 7 + 3 = 9
SOLUTION First isolate the radical on the left.
√6x
+7+3=9
√6x
+7=6
( √6x
+ 7)
2
Subtract 3 from both sides of the equation.
= 62
Square both sides.
6x + 7 = 36
Simplify.
6x = 29
Subtract 7 from both sides.
29
x=_
6
Divide both sides by 6.
Check √6x
+7+3=9
29
+7+39
(_
√6
6)
√36
+39
9=9 ✓
Example 2
Solving Equations Containing One Cube Root
Solve each equation.
3
a. √
4x + 4 - 6 = 0
SOLUTION First isolate the radical.
3
√
4x
+4-6=0
3
√
4x
+4=6
( √4x
+ 4)
3
3
Add 6 to each side.
3
=6
Cube both sides to clear the cube root.
4x + 4 = 216
4x = 212
x = 53
Simplify.
Subtract 4 from each side.
Divide both sides by 4.
3
Check √4x
+4-6=0
3
4(53)
+4-60
√
3
√
- 6 0
216
6-60
0=0 ✓
3
Math Reasoning
Verify Check that
-335 is a solution of
3
-2 √
x - 8 = 14.
b. -2 √
x - 8 = 14
SOLUTION Isolate the radical by dividing.
3
- 8 = 14
-2 √x
3
√
x
- 8 = -7
( √x
- 8)
3
3
Divide both sides by -2.
3
= (-7)
x - 8 = -343
x = -335
496
Saxon Algebra 2
Cube both sides.
Simplify.
Add 8 to each side.
Math Language
Indices is the plural of
index.
Sometimes an equation has two radicals. If their indices are the same and
each radical is on a different side of the equation, it is simplest to raise each
side of the equation to the power of the index.
If the indices on the radicals differ, the solution can be estimated by using
a graphing calculator. Graphing calculators are also very useful in checking
solutions.
Example 3 Solving Equations Containing Two Radicals
Solve each equation. Check the solutions graphically.
a. √8x
- 15 = √
4x + 5
SOLUTION √8x
- 15 = √4x
+5
2
2
( √8x
- 15 ) = ( √4x
+ 5)
Square both sides.
8x - 15 = 4x + 5
Simplify.
4x = 20
Isolate x.
x=5
Check Graph Y1 = √8x
- 15 and Y2 = √4x
+ 5 . The intersection point is
the solution. Use the intersect command located in the Calculate (2nd Trace)
menu. The curves intersect at x = 5.
3
3
b. √
3x = 2 √
x-1
3
3
= 2 √x
SOLUTION √3x
-1
3
3
3
3
(√
) = (2 √x
3x
- 1)
Cube both sides.
3x = 8(x - 1)
Simplify.
3x = 8x - 8
Distribute 8.
Isolate x.
-5x = -8
8 = 1.6
x=_
5
Graphing
Calculator Tips
To type the cube root
symbol, press Math and
choose it from the items
in the Math list.
3
3
and Y2 = 2 √x
Check Graph Y1 = √3x
- 1 . The curves intersect at
x = 1.6.
Lesson 70
497
3
- √
c. √2x
-5x - 6 = 0
SOLUTION Move one of the radicals to the right side. Then use a graphing
calculator.
3
√2x
- √-5x
- 6 = 0
3
√2x
- 6 = √-5x
3
Add √5x to each side.
3
- 6 and Y2 = √-5x
. The curves intersect at x ≈ 4.83.
Graph Y1 = √2x
Example 4 Solving Equations with Extraneous Solutions
Math Reasoning
Analyze How can you
see in the first step of 4a
that the equation has no
solutions?
Solve each equation.
a.
√
x - 4 = -8
SOLUTION √x
- 4 = -8
√x
= -4
2
Add 4 to each side to isolate the radical.
2
( √x
) = (-4)
Square both sides.
x = 16
Simplify.
Check √x
- 4 = -8
√16
- 4 -8
4 - 4 -8
0 ≠ -8
The solution of 16 is extraneous. The equation has no solutions.
b. x + 1 = √
2x + 10
SOLUTION x + 1 = √2x
+ 10
The radical is isolated on the right.
(x + 1)2 = ( √2x
+ 10 )
x2 + 2x + 1 = 2x + 10
2
x -9=0
2
Square both sides.
Simplify.
Solve the quadratic equation.
x2 = 9
x = ±3
Check x + 1 = √2x
+ 10
3+1
x + 1 = √2x
+ 10
+ 10 -3 + 1 √2(-3)
+ 10
√2(3)
4 √16
-2 √4
4=4 ✓
-2 ≠ 2
The solution of -3 is extraneous. The only solution is 3.
498
Saxon Algebra 2
Example 5
Application: Physics
The period of a pendulum, T, is the time it takes for the pendulum to make
L
_
, where L is the
one back-and-forth swing, and is given by T = 2π √ 9.8
length of the pendulum in meters. Find the length of a pendulum whose
period is 3 seconds.
SOLUTION Substitute 3 for T in the formula and solve for L.
L
_
√
9.8
3 = 2π
L
3 = _
_
√ 9.8
3
L
= ( √_ )
(_
2π )
9.8
Isolate the variable.
2π
2
2
9 =_
L
_
4π 2
Simplify.
9.8
9
=L
(_
4π )
9.8
Square both sides.
Isolate L.
2
2.23 ≈ L
The length of the pendulum is about 2.23 meters.
Lesson Practice
Solve each equation.
2x + 3 = 21
a. √
(Ex 1)
3
c. √
2x + 1 = 0
b. 11 = √
x-7-2
(Ex 1)
3
d. 4 √
3x - 4 = 16
(Ex 2)
(Ex 2)
e. √
9x = √
5x + 2
(Ex 3)
3
g. √
3x + √
x-4-5=0
(Ex 3)
h. √
-1 - 4x = √
x - 11
(Ex 4)
3
3
f. - √
2x = √4x
+7
(Ex 3)
i. x = √
6 - 5x
(Ex 4)
j. The period of a pendulum, T, is the time it takes for the pendulum to
L
_
, where L
make one back-and-forth swing, and is given by T = 2π (Ex 5)
√ 9.8
is the length of the pendulum in meters. Find the length of a pendulum
whose period is 5 seconds.
Practice
Distributed and Integrated
Solve the equations. Write the solutions in the form a + bi.
1. 3p2 + 7 = -9p2 + 4
(62)
2. 2x2 + 31 = 9
(62)
*3. Find the roots and give the multiplicity of each root: x3 - 7x2 + 11x - 5 = 0
(66)
4. Let f (x) = 3x + 1 and g(x) = x - 2. Find the composite function f (g(x)).
(53)
Lesson 70
499
5. Of the 220 people who came into the Italian deli on Friday, 104 bought pizza and 82
used a credit card. Half of the people who bought pizza used a credit card. What is
the probability that a customer bought pizza or used a credit card?
(60)
Solve for x by completing the square.
6. x2 = 9x - 7
(58)
7. -5x - 6 = -x2
(58)
*8. Lighthouses The number of miles m a person can see on a clear day is
(70)
approximated by m = 1.2116 √
h , where h is the person’s height in feet above sea
level. If a person standing at the top of the Barnegat Lighthouse in New Jersey on a
clear day can see a distance of 16 miles, what is the person’s height above sea level?
9. Error Analysis The odds in favor of an event are 9 to 2. To the nearest percent, what
is the probability of the event?
(55)
10. Multiple Choice What is P(-3) if x + 3 is a factor of 2x3 + 2x2 - 10x + 6?
(61)
A -3
B 0
C 3
D 6
11. Velocity of an Electron The velocity at which an electron orbits the nucleus of an atom
_
(59)
Z
is known as the Fermi-Thomas velocity and can be found using the expression _ c,
2
3
137
where Z is the number of protons in the nucleus and c is the speed of light. What
is the Fermi-Thomas velocity for cobalt, which has 27 protons, in terms of c?
Round to the nearest thousandths place.
*12. Write What is an extraneous solution? When can it occur?
(70)
13. Estimate Use what you know about the unit circle and special angles to estimate
(63)
tan 43°. Do not use a calculator. Explain your method.
*14. Graphing Calculator With your calculator in a + bi mode, find the following.
(69)
a. (64 - 82i)(26 + 31i)
b. -3.7i(2.25 + 4.5i)
15. Electrical Engineering A circuit has a current of 5 amps, and a second circuit has a
(62)
current of (14 - 9i) amps. Find the sum of the currents.
16. Temporary Employees A department store is planning to hire up to 24 temporary
(54)
employees for a tent sale. Experienced workers will be paid $20 per hour and
inexperienced workers will be paid $15 per hour. The company can pay up to
$400 per hour for the temporary employees. Experienced workers produce 1.5 times
more profit than inexperienced workers. How many of each type of worker should be
hired to maximize the company’s profits?
*17. Multi-Step The product of a number and its multiplicative inverse is 1.
(69)
a. State the multiplicative inverse of 2 + 3i.
b. Simplify the multiplicative inverse.
c. Verify Show that your answer in part b is the multiplicative inverse by
multiplying it by 2 + 3i.
500
Saxon Algebra 2
Factor each expression completely.
18. 8c3 + 343
19. -5z3 + 320
(61)
(61)
20. Generalize How does the Rational Root Theorem make factoring polynomials
(66)
easier? What must be true about a polynomial before it used?
*21. Wheelchair Accessibility A contractor designed the wheelchair ramp shown in
(67)
the diagram. The ramp must satisfy a building code that specifies 4.7° as the
maximum angle of elevation, labeled θ in the diagram. Does the design satisfy
the building code? Justify your answer.
θ
50 ft
4 ft
22. Find an equation for the inverse of y = 5x3 - 20. Determine whether each relation
(50)
is a function.
23. Verify A student used factoring to find that the solutions of 4x2 - 11x - 20 = 0
(65)
are -1.25 and 4. Show how to verify these solutions with the Quadratic Formula.
y≥0
x≥0
24. Maximize P = 4.5x + 2.4y for the constraints y ≥ x
(54)
y ≤ -2x + 9
1
25. Error Analysis A student converted the exponential equation 4 -2 = _
to logarithmic
16
(64)
1
_
form by writing log4(-2) = 16 . What is the error? Write the correct logarithmic
equation.
26. Geometry Each side of the regular hexagon shown is 7 cm. Find
(52)
−−
the length of the diagonal AE. Round your answer to the
nearest tenth. (The measure of one interior angle of an n-gon
(n - 2)180˚
is _
.)
n
A
B
F
C
E
D
27. Write Examine the formula for the probability of two inclusive events. Explain
(60)
why P(A and B) is subtracted from the formula.
P(A or B) = P(A) + P(B) - P(A and B)
28. Multi-Step A clothing store has a sale in which all the shirts are 20% off. On the
(53)
last day of the sale, all shirts have an additional 25% off. Find a function describing
how much a shirt with price p will cost on the last day. Solve the function.
*29. Probability A number cube is rolled twice. What is the probability that the number
(70)
facing up on each roll shows a solution of √
2x - 8 = x - 4?
*30. Multiple Choice A bag of grapes contains 9 green grapes and 15 red grapes. If 4 grapes
(68)
are chosen at random, without replacement, what is the approximate probability
that the first two grapes are green and the last two grapes are red?
A 4.6%
B 5.9%
C 7.1%
D 13%
Lesson 70
501
INVESTIGATION
7
Collecting Data
Exercise Suppose you want to find out what students at your school do for
exercise, and then make a graph to display the results.
1. You construct this survey question: “What do you do for exercise?”
Think about some possible responses. How many different responses
do you think you would get if you asked 100 students?
2. Is “dance” a response you thought of for question 1? Is “skateboarding”
or “martial arts” a response you thought of?
3. Is it possible that some students do more than one type of exercise?
Online Connection
www.SaxonMathResources.com
The answers to survey questions are data items, so collecting the answers is
a form of collecting data. When constructing a survey question, it is often
important to limit the number of possible responses so that you will be able
to organize, display, analyze, and interpret the data you collect. Provide
enough response choices so that there is a choice for everyone, and make sure
the choices don’t overlap. That is, construct them so that there is clearly only
one response. In the table below, the question about exercise is rephrased and
answer choices are supplied, along with numbers of responses.
“Which of the following best describes your main type of exercise?”
Math Reasoning
Response
Analyze Refer to the
table. Which choice is
needed so that everyone
can choose at least one
response? What word(s)
in the question allows
everyone to be able to
choose exactly one
response?
Frequency
A. Competitive sports (team and individual)
24
B. Run, jog, or walk (not competitive)
16
C. Swim or bicycle (not competitive)
D. Strength training (weights and exercise
equipment)
E. Other
5
15
20
A bar graph and a circle graph are both appropriate ways to display this
data. Copy and complete the bar graph and circle graph below.
4.
502
Saxon Algebra 2
Competitive
Sports
30%
%
20
15
10
Other
0
Strength
Training
5
Swim or
Bicycle
0.30 × 360° = 108°
25
Run, Jog,
or Walk
80
Exercise
30
Competitive
Sports
24 of 80 responses were
Competitive. To get the
central angle measure for
the Competitve sector in
the circle graph, multiply
360° by the appropriate
decimal.
24
_
= 0.30
5.
Exercise
Number of Students
Hint
%
%
6%
Analyze and Write A survey question is biased if there is a reasonable
chance that it could influence a response because of the way it is
constructed. Determine whether each question below is biased. Explain
your answer. If the question is biased, rewrite it so that it is not biased.
6. Do you agree with most high school students that parents give them
too little freedom to make their own decisions?
7. Do you support the unfair policy of requiring community service as a
high school requirement?
8. Do you think energy conservation should be a high priority for state
and national elected officials?
9. Write a biased survey question. Then rewrite it to remove the bias.
A population is a group being studied. Populations are often too large
to survey, so a subset of the population called a sample is surveyed. If
a sample is not a fair representation of the population, it is biased. If a
sample is a fair representation of the population, it is unbiased. When
properly chosen in an appropriate situation, any of the following can be an
unbiased sample.
• Random sample – Each member of the population has an equal chance
of being selected.
• Systematic sample – Members are chosen based on a rule.
• Stratified sample – The population is divided into non-overlapping
subgroups, and members are chosen at random from the subgroups.
• Self-selected sample – Members volunteer to be in the sample.
Analyze and Write Identify the type of sample and determine whether it
is biased. Explain your answer.
10. A manager asks every tenth customer who enters her store whether
they like the recent remodeling of the store.
11. A candidate mails a survey about an upcoming election along with a
request for campaign contributions, enclosing a return envelope for
responses.
12. For a survey about school policies, a student randomly chooses
25 students from each grade. (There are approximately 200 students
in each grade.)
13. There are 645 employees at a company. The president assigns the
integers 1 through 645, one to each employee, and then randomly
chooses 50 integers and surveys the corresponding employees.
14. Every voter in a town is registered as a Democrat, a Republican, or an
Independent. For a survey of registered voters, 50 are randomly chosen
from each group.
15. Multi-Step Complete parts a–d.
a. Construct a survey question with 4 or 5 response choices.
b. Identify a population. Describe 3 or 4 possible samples of different
types. Identify each type and state whether it is biased or unbiased.
Investigation 7
503
Math Reasoning
Analyze A bar graph
and circle graph both
show relative sizes of
categories. Which type
better shows how the
size of each category
compares to the whole?
c. Construct a frequency table. Show a reasonable distribution of
responses.
d. Construct a bar graph and circle graph to show the data.
It is useful to describe some data sets with measures of center such as
mode, median, and mean, and with measures of spread such as variance and
standard deviation.
A teacher chose 2 students at random from each of the 10 sophomore
homeroom classes in a school and asked “To the nearest half hour,
what is the average number of hours per school night that you spend on
homework?” The table below shows the data distribution.
Hours
Frequency
Hint
To find the median, order
the 20 data values from
least to greatest, and
then find the average of
the 10th and 11th values.
0
0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 or more
1
4
5
5
3
1
1
0
16. Identify the population and the type of sample. Determine whether the
sample is biased.
17. Find the mode(s) and median.
18. Supply the missing terms in the numerator, and then evaluate the
1(0.5) + 4(1.0) + 5(1.5) + … + 1(3.5)
expression to find the mean: x− = ___.
−
Reading Math
The symbol for standard
deviation is σ, the Greek
lower case letter sigma.
20
−2
2
−
2
(x1 - x) + (x2 - x) + … + 1(xn - x)
19. The formula for variance is σ2 = ___
, where
n
−
x , x , …, x are the data values and x is the mean. Supply the missing
1
2
n
terms in the numerator, and then evaluate the expression to find the
−)2 + 5(1.5 - −
1(0.5 - −
x)2 + 4(1.0 - x
x)2 + … + 1(3.5 - −
x) 2
variance: σ2 = ____
20
σ2 .
20. The standard deviation is the square root of the variance: σ = √
Find the standard deviation to the nearest thousandth.
Investigation Practice
a. Model Students were asked the survey question, “On average, how many hours
per week do you email or use the Internet? Round to the nearest hour.” The
frequency table below shows the results. Construct a histogram and a circle
graph to show the data. (A histogram is a bar graph in which adjacent bars
touch.)
Response
0–2
3–5
6–8
9–11
12 or more
Frequency
4
12
22
7
5
b. Analyze Suppose the instruction “Round to the nearest hour” were not included
in the survey question in part a. Why might it be difficult for a student to
choose a response?
c. Analyze Suppose the response choices were 0–3, 3–6, 6–9, 9–12, and 12 or more.
Why might it be difficult to interpret a histogram showing the results?
504
Saxon Algebra 2
Determine whether each question below is biased. Explain your answer. If the
question is biased, rewrite it so that it is not biased.
d. How many people live in your household?
e. Knowing that teenagers tend to be irresponsible, should people younger
than 18 be allowed to drive without an adult in the vehicle?
f. Knowing that the current sales tax rate is 5%, do you support a new rate
of 6%, beginning next year?
g. There are some scientists who believe that recent climate changes are part
of a natural long-term climate cycle rather than global warming caused
by humans. Do you believe that global warming is a serious problem?
Identify the type of sample and determine whether it is biased. Explain your
answer.
h. A utility company mails a survey to all its customers asking them to rate
customer service, enclosing a return envelope for responses.
i. A developer wants to build a store. Town council members want to know
if town residents approve. They survey all residents that live within a half
mile of the proposed site.
j. A school principal selects 2 students at random from each class in the
school to survey about students’ study habits.
k. The junior class president wants opinions on student parking rules. She
uses a list of all 255 juniors, and assigns each junior a different integer
from 1 through 255. She uses a random integer generator on a calculator
until she gets 20 different integers from 1 through 255. She chooses the 20
corresponding students to survey.
Six students were chosen at random from each grade in a high school and asked
“At what age did you have your first job?” The table below shows the data
distribution.
Age
Frequency
13
1
14
4
15
8
16
7
17
3
18
0
19
1
l. Identify the population and the type of sample. Determine whether the
sample is biased.
m. Find the mode, median, and mean.
n. Find the variance.
o. Find the standard deviation.
p. Construct your own survey question. Identify the population and the
type of sample. Determine whether the sample is biased. Find the
mode(s), median, and mean. Find the variance and standard deviation.
Investigation 7
505