Download Precalculus Notes: Unit P – Prerequisite Skills Scientific notation is a

Date
Scientific Notation
Precalculus Notes: Unit P – Prerequisite Skills
Scientific notation is a way of writing numbers that are too large or too small to be conveniently written
in standard decimal notation. Scientific notation has a number of useful properties and is commonly used
in calculators and by scientists, mathematicians and engineers.
Ex1. The number 123,000,000,000 in scientific notation is written as :
The first number 1.23 is called the coefficient. It must be greater than or equal to 1 and less than 10.
The second number is called the base . It must always be 10 in scientific notation. The base number 10 is
always written in exponent form. In the number 1.23 x 1011 the number 11 is referred to as the exponent
or power of ten.
10000 = 1 x 104
1000 = 1 x 103
100 = 1 x 102
10 = 1 x 101
1 = 100
1/10 = 0.1 = 1 x 10-1
1/100 = 0.01 = 1 x 10-2
1/1000 = 0.001 = 1 x 10-3
1/10000 = 0.0001 = 1 x 10-4
24327 = 2.4327 x 104
7354 = 7.354 x 103
482 = 4.82 x 102
89 = 8.9 x 101 (not usually done)
0.32 = 3.2 x 10-1 (not usually done)
0.053 = 5.3 x 10-2
0.0078 = 7.8 x 10-3
0.00044 = 4.4 x 10-4
Sample SAT Question(s): Taken from College Board online practice problems.
1.
When 70,000 is written as 7.0  10n , what is the value of n?
(A) 1
(A) 2
(B) 3
(C) 4
(D) 5
2.
What is the result when 436,921 is rounded to the nearest thousand and then expressed in
scientific notation?
a.
4.369  102
b.
4.369  104
c.
4.37  104
d.
4.37  104
e.
4.37  105
Page 1 of 22 Precalculus – Larson
Chapter P
Precalculus Notes: Unit P – Prerequisite Skills
You Try: Simplify using scientific notation.
 42,000,000 6,000,000,000
12,000,000
Scientific Notation: One non-zero digit left of the decimal.
 2.5  10  6  10 5  10    3  10 
4
2
1
3
Right Triangles and Circles
Pythagorean Theorem: a2 + b2 = c2
Definition: The longest side of the triangle is called the "hypotenuse", so the formal definition is:
In a right angled triangle:
the square of the hypotenuse is equal to
the sum of the squares of the other two sides.
Ex1: A "3,4,5" triangle has a right angle in it.
Let's check if the areas are the same:
32 + 4 2 = 5 2
Calculating this becomes:
9 + 16 = 25
It works ... like Magic!
Ex2: Does this triangle have a Right Angle?
Does a2 + b2 = c2 ?
Does (√3)2 + (√5)2 = (√8)2 ?
Does 3 + 5 = 8 ?
Yes, it does!
So this is a right-angled triangle
Page 2 of 22 Precalculus – Larson
Chapter P
Precalculus Notes: Unit P – Prerequisite Skills
A circle is a locus (set) of points in a plane equidistant from a fixed point
Circle whose center is at (h,k)
This will be referred to as the
"center-radius form"
It may also be referred to as
"standard form"
Circle whose center is at the origin
Equation:
Equation:
Example: Circle with center (0,0), radius 4
Example: Circle with center (2,-5), radius 3
Graph:
Graph:
Page 3 of 22 Precalculus – Larson
Chapter P
Precalculus Notes: Unit P – Prerequisite Skills
Interval Notation
If a and b are real numbers with a < b then we can use the following notation:
Inequality Notation: a < x < b
Interval notation: (a, b]
Bounded
Unbounded
Closed
[a, b]
[a, ∞)
(–∞, b]
Open
(a, b)
(a, ∞)
(–∞, b)
Half-open
[a, b)
(a, b]
Unbounded intervals are never considered
half-open
We can describe inequalities by using words, graphs and interval notation.
For examples 2-4 also note whether the interval is bounded or unbounded and open, closed or half-open.
Ex.2
Describe in words and graph the interval.
a) ( 1, ∞)
b) –4 < x < –2
Ex.3
Use inequality and interval notation to describe the
interval of real numbers.
Ex.4
Use interval notation to describe the interval of real
numbers and graph. x is between 0 and 4
[
Reflection: Describe the four types of unbounded intervals and give examples of each.
Page 4 of 22 Precalculus – Larson
)
-4 -3 -2 -1 0 1 2 3
Chapter P
4
Date:
Note: When
multiplying or
dividing by a
negative number, you
must flip the
inequality sign.
Precalculus Notes: Unit P – Prerequisite Skills
Solving Inequalities
Solving a Linear Inequality:
Ex.1: Solve and graph the solution on the number line. Express the solution in interval notation.
5x  8  2  x  3
Solving a Compound Inequality:
Ex.2
Solve and graph. Express in interval notation. 10 
Ex.3: Solve and graph. Express in interval notation. 3  
4x  2
6
2
3
x47
2
Sample SAT Question(s): Taken from College Board online practice problems.
1. If y  2 x  3 and x  2 , which of the following represents all the possible values for y?
(A) y  7
(B) y  7
(C) y  5
(D) y  5
(E) 5  y  7
2.
If 3  3t  6  18 , which of the following must be true?
(A) t  5
(B) 5  t  6
(C) 4  t  1  9
(D) 12  3t  24
(E) t  3 or t  8
You Try: Solve the inequality and graph the solution on a number line.
x
2
1  x  3
2
3
Reflection: Explain why the inequality sign must be flipped when multi. or div. by a negative number.
Page 5 of 22 Precalculus – Larson
Chapter P
Precalculus Notes: Unit P – Prerequisite Skills
Date
Exponents and Radicals
Properties of
Exponents
103
=
10
2
=
a m  a n  a mn
101
=
100
=
am
 a m n
n
a
1
1
a  n  n or  n  a n
a
a
a0  1
( a  b)  a b
m
(a )  a
m n
m m
mn
10
-1
=
10-2
=
=
=
10-3
=
=
=
m
am
a

 
bm
m
We define a0 = 1 & a-n =
1
, a ≠ 0.
an
00 is indeterminate.
Error allert:
( a  b) m  a m  b m
Ex.1
Examine the properties of exponents:
a. 32 · 34 =
b. (34)2 =
c. (2·4)3 =
d. 23 · 43 =
e.
35
=
32
 3
 5
2
f.   =
Ex. 2
=
 5 x 3 y 2   2 x 2 y 
 x 2 y 1    x 3 y 

 

=
3
Page 6 of 22 Precalculus – Larson
Chapter P
Precalculus Notes: Unit P – Prerequisite Skills
Write in exponential form:
Ex. 3
a. What is half of 240?
b. What is one third of 318?
Ex. 4
A pile of gravel contains 1010 stones. Take ten stones from the original pile and throw nine
onto a pile on the left and one onto a pile on the right. When the original pile is gone, how many stones
are in each of the new piles?
*
If b2 = a, then b is a square root of a.
*
The convention is to use
The symbol
-
m
n
a  a
n
.
names the negative square root.
3
64
radicand
Ex. 5 Find the principal root:
64
b)
c)
 81
d)
e)
4 16
f)
a)
n
2
names the principal square root and
index
Properties of
radicals:
instead of
3
9
m
a  n b  n a b
n
a na

n
b
b
3
125
3
27
8
Page 7 of 22 Precalculus – Larson
Chapter P
Precalculus Notes: Unit P – Prerequisite Skills
What is the
difference
between the
square root
symbol and asking
for the square
roots?
Ex.6
What are the square
16 ?
What is
roots of 16?
Ex.8 Rewrite in exponential form:
a)
b
Ex.7
5
32x5 y8
b)
a5b3
c2
3
a
Rewrite x in
exponential form.
c)
4 3
x2
7
2
3
4
Convert to radical form:
1
3
d)
2
3
4 a y
4
3
e)
1
2
3
4
b)
6
4 2
a y
f)
a y
Approximate
2 and 3 .
Rationalize the denominator.
Ex.9
a)
1
2
Simplify the expression.
Ex.10
a)
75x3
b)
3
81x7 y5
Reflection: What misconception might a student have about –n?
What is the difference between –42 and (–4)2?
Page 8 of 22 Precalculus – Larson
Chapter P
32
?
Precalculus Notes: Unit P – Prerequisite Skills
Date:
1.1
Lines in the Plane
OBJ: Students will find the slope of lines, write linear equations given points on lines and their
slopes, use slope intercept forms of linear equations to sketch lines, use slope to identify parallel and
perpendicular lines.
Slope: rate of change; steepness of a line
m
rise change in y y y1  y2 y2  y1 dy





run change in x x x1  x2 x2  x1 dx
Ex.1: Find y so that m  2 on a line through the points  5, 8 and  3, y  .
Equations of a Line

A line with slope m that passes through the point  x1, y1 
Point-Slope Form
y  y1  m  x  x1 

Slope-Intercept Form
A line with slope m and y-intercept b
y  mx  b

Standard Form
A & B are not both 0; x-intercept =
C
C
, y-intercept =
A
B
Ax  By  C

Find the slope:
Ax  By  C  0
General Form
Ex.2: Write an equation of the line that passes through the points  1,3 and  5,4  in general form.
Use the slope and
one of the given
points to write the
equation in pointslope form:
Multiply by the LCD
and write in general
form:
Ex.3: Graph the line 4 x  3 y  12 .
Find the x- and y-intercepts:
Slope of Horizontal Lines: m  0
Slope of Vertical Lines: m is undefined
Page 9 of 22 Precalculus – Larson
Chapter P
Precalculus Notes: Unit P – Prerequisite Skills
Ex.4: Write an equation of the line that passes through the points  2,4 &  2,1 .
m=
Equation of the vertical line:
Parallel Lines || : lines in the same plane that do not intersect; Parallel lines have the same slope.
Perpendicular Lines    : lines in the same plane that intersect at a right angle; Perpendicular lines have
opposite reciprocal slopes.
Ex.5: Write an equation of the line parallel to 4 x  3 y  24 that passes through the point
 0, 2 .
Find the slope of the given line:
Parallel lines have the same slope, so use this slope and the given point to write the equation in pointslope form.
Rate of Change
Ex.6: A 5-minute phone call costs $0.38, and a 9-minute call costs $0.62. What is the rate of
change? Find a linear function to represent the total cost (C) of a call to the duration in minutes
(m). Then use this function to find the cost of a call that lasts 1 hour and 15 minutes.
Rate of Change =
Linear function:
1 hour and 15 minutes:
You Try: Write an equation of the line perpendicular to the line that passes through the points
 5,2 & 3, 1
in point-slope form.
Reflection: Explain why the slope of a horizontal line is zero and the slope of a vertical line is undefined
graphically and using the definition of slope.
Page 10 of 22 Precalculus – Larson
Chapter P
Precalculus Notes: Unit P – Prerequisite Skills
Date
Polynomials and Factoring
degree
constant term
leading coefficient
-6x3 + 5x – 3
coefficients
Degree: the largest sum of exponents in a term
Polynomial: anxn + an-1xn-1 + … + a1x + a0 where an ≠ 0
Ex.1
Identify the coefficients & degree of 4 – x + 2x3.
Coefficients:
Degree:
Perform the operation on the polynomials:
Ex.2 (3x4 + 2x2 – 5x – 16) – (4x4 + x3 – 7x + 5)
Ex.3
a) (2x + 5)(x – 6)
b) (x2 – 3x – 6)( 2x2 – 4x + 1)
Page 11 of 22 Precalculus – Larson
Chapter P
Precalculus Notes: Unit P – Prerequisite Skills
Product of a Sum & Diff. = (ax – b)(ax + b) =
Ex.4
a) (3x – 2)(3x + 2)
b)
 x 3  x 3 
c)
 13 1  13 1 
 x  y 2  x  y 2 



=
=
Square of a sum or difference:
(ax – b)2
(ax + b)2 =
Cube of a sum: (ax+b)3 =
Ex.5 (x – 3)3 =
GCF (always look for the GCF 1st):
Ex.6 a) 9x3 – 12x2 + 6x
b) 8w2y – 4wy
Page 12 of 22 Precalculus – Larson
Chapter P
Precalculus Notes: Unit P – Prerequisite Skills
Difference of Two Squares (2 terms):
Ex.7
a) x2 – 196
b) 9x4 – 121
c)
x2 + 9
d) 25x2 – (x – 1)2
Sum & Difference of Two Cubes (2 terms):
a3 – b3 =
a3 + b3 =
Ex.8
b) 27x6 – 1
a) x3 + 125
Grouping (4 terms):
Ex.9
a) x3 – 3x2 + 3x – 9
b) ax2 – bx2 – a + b
Trinomials:
Split the middle: a technique for factoring trinomials.
Factor: 2x2 + 11x + 12
Step #1
Find the product ac.
2  12 = 24
Find the pairs of factors of the product ac.
Which pair of factors has a sum that equals b?
1 + 24 = 25, 2 + 12 = 14, 3 + 8 = 11, 6 + 4 = 10
Page 13 of 22 Precalculus – Larson
Chapter P
Precalculus Notes: Unit P – Prerequisite Skills
Step #2
Rewrite bx & then group the 1st & last two terms:
2x2 +
11x
+ 12
=2x2 + (3x + 8x) + 12
=(2x2 + 3x) + (8x + 12)
Step #3
Factor out the GCF of each group:
=(2x2 + 3x) + (8x + 12)
=x(2x + 3) + 4(2x + 3)
Step #4
Factor out the binomial
to get your final answer:
=(2x + 3)(x + 4)
Always check to see if you can factor again!
Ex.10
a) x2 + 5x – 14
b) 2x2 – 7x – 15
Ex.11 a) 24x2 + 22xy – 30y2
b) x4 – 16
Quadratic Form:
Ex.12 a) x + 5
x
b) 2x6 – 15x3 – 8
+6
Page 14 of 22 Precalculus – Larson
Chapter P
Precalculus Notes: Unit P – Prerequisite Skills
Date:
Solving Equations
Linear Equation: an equation that can be written in the form ax  b  0, a  0
Equations are equivalent if they have the same solutions.
Ex.1: Show that z  2 is not a solution of 2  4 z  3  3  z  1  3z  1 . Then find the
solution.
Substitute z  2 into the equation:
Solve for z:
Ex.2: Solve the equation for t.
t 1
1
 3  t  5 
4
6
Wipe out the fractions by multiplying by the least common denominator, in this case, _____.
Ex.3: Find the volume of material that makes up the Earth’s crust, which is ten miles thick.
(Earth’s radius  3960 miles) Volume of a sphere: V 
4 3
r
3
Volume of the crust = volume of the earth minus volume of the earth not including the crust
Solving Radical Equations:
Ex.4
c)
a)
x  x  7 5
x 10  5 3 x
b)
d)
4  3 x  2 8
x 1  1 2x 12
Page 15 of 22 Precalculus – Larson
Chapter P
 a, a  0
 a a  0
Precalculus Notes: Unit P – Prerequisite Skills
Absolute Value: a  
4 7
4  7  0 , so by the definition, 4  7  4  7
4  17
4  17  0 , so by the definition, 4  17  4  17
x
Ex.5: Rewrite without an absolute value symbol. x
for a.) x  0
Solving Absolute Value Equations:
Ex.5: Find the solution set for the equation. 3 2 x  1  8  16
Isolate the absolute value expression.
There are two values that have an absolute value of 8, _______________
So
Solve each equation for x.
Solutions:
Sample SAT Question(s): Taken from College Board online practice problems.
1.
If x  k  12 and p  x  k   36 , what is the value of p?
a.
b.
c.
d.
e.
3
4
6
9
12
Page 16 of 22 Precalculus – Larson
Chapter P
b.) x  0
Precalculus Notes: Unit P – Prerequisite Skills
Date:
Solving Equations Continued
Quadratic Equation: an equation of the form ax 2  bx  c  0 , with a  0
Methods for Solving a Quadratic Equation
 Square Roots
Ex.1: Solve the equation 2  x  1  5 by extracting square roots.
2
Isolate the expression that is squared:
Square root both sides of the equation:
Reminder:
Solve for x:
Set Notation:
(rationalized)

Completing the Square Note: a must equal 1 in order to complete the square.
Ex.2: Solve the equation x 2  8 x  1  0 by completing the square.
Move c to the other side of the equation:
b
Complete the square by adding  
2
2
to both sides:
Rewrite the perfect square trinomial as a binomial squared:
Solve by square roots:
Page 17 of 22 Precalculus – Larson
Chapter P
x2  x
Precalculus Notes: Unit P – Prerequisite Skills

b  b  4ac
2a
2
o Discriminant: b  4ac
b2  4ac  0  two real solutions
Quadratic Formula
x
2
b2  4ac  0  no real solutions (2 complex solutions)
b2  4ac  0  one real solution
Ex.3: Solve the equation using the quadratic formula. x 2  8 x  6
Rewrite in standard form and determine a, b, and c.
Find the discriminant to determine the number of solutions.
Use the quadratic formula to solve.
Solution set:

Factoring: use the zero product property
Ex.4: Solve the equation by factoring. 3x 2  20 x  7
Set the equation equal to zero.
Factor the quadratic. (ac method is shown)
Split the middle term:
Factor by grouping:
Set each factor equal to zero and solve:
Page 18 of 22 Precalculus – Larson
Chapter P
Precalculus Notes: Unit P – Prerequisite Skills
Solving Rational Equations
Ex.5: Find the value(s) of x that make the equation true.
3
8
1  2
x7
x  9 x  14
Factor and find the LCD.
LCD:
Multiply each term by the LCD.
Simplify and solve the resulting equation.
Solving an Equation Graphically
Method 1: Find the zeros (roots).
Ex.6: Solve the equation graphically. x 4  2 x 2  3x  2
Set the equation equal to zero.
x 4  2 x 2  3x  2  0
Graph the function and find the zero(s).
y  x 4  2 x 2  3x  2
Solutions:
Method 2: Finding the point(s) of intersection.
Ex.7: Solve the equation graphically. x 2  4 x  3  x 3  x  6
Graph both sides of the equation as two separate functions. Then find the point(s) of intersection.
Solution:
Think About It: What does the Y-value represent in the point of intersection?
Page 19 of 22 Precalculus – Larson
Chapter P
Precalculus Notes: Unit P – Prerequisite Skills
Sample SAT Question(s): Taken from College Board online practice problems.
1. If a  0 and
5 5 a

, what is the value of x?
x xa
(A) −5
(B) −1
(C) 1
(D) 2
(E) 5
2. If x 2  x  6 , which of the following must be true?
(A) x  6
(B) x  3
(C) x  0
(D) x 2  x
(E) x 2  x
You Try: Solve for x in the equation ax 2  bx  c  0 by completing the square.
Reflection: True or False: An absolute value equation always has two solutions. Explain your answer.
Page 20 of 22 Precalculus – Larson
Chapter P
Precalculus Notes: Unit P – Prerequisite Skills
Date:
Distance Formula and Midpt.
Distance between a and b: a  b
Number Line Formulas:
Midpoint:
ab
2
Ex.2: Find the distance between the points and the midpoint of the line segment that is
formed by them.
a.
3,5.4
Distance =
Midpoint =
b.
1 3
,
6 4
Distance =
Midpoint =
Ex. 3: Write the statement using absolute value notation: The distance between x and −5 is
less than 4.
Coordinate Plane Formulas:
Distance Formula (derived from the Pythagorean Theorem)
(x2 , y2)
(4, 3)
(x2 , y1)
(x1 , y1)
(0, 0)
(4, 0)
Ex.4: Determine if the points form an isosceles triangle:
 2, 3 ,  2,0 ,  5,1
Use the distance formula to find the lengths of the three sides:
Page 21 of 22 Precalculus – Larson
Chapter P
Precalculus Notes: Unit P – Prerequisite Skills
Midpoint Formula: average of the x-coordinates, average of the y-coordinates
 x1  x2 y1  y2 
,

2 
 2
Midpoint of the line segment connecting  x1, y1  &  x2 , y2  : M  
Ex. 5: The midpoint of AB is M  2,4 , and the coordinates of A are  1,9  . Find B.
A circle is the locus of points  x, y  equidistant (r) from a given point.
Use the distance formula to find the equation of a circle centered at the origin. Then find the
equation of a circle centered at  h, k  .
Ex.6: Find the equation of a circle that has  1, 2 and  5,3 as the endpoints of one of its
diameters.
The center will be at the midpoint of the diameter:
The radius is equal to half the length of the diameter:
Equation of the circle:
You Try: Find the center and diameter of the circle.
 x  9
2
 y 2  24
Reflection: Write the standard equation of a circle, label the variables, and describe how it is derived.
Page 22 of 22 Precalculus – Larson
Chapter P