Download Honors Precalculus Notes Packet

Honors Precalculus
Notes Packet
2013-2014
Academic Magnet High School
Contents
Unit 1: Inequalities, Equations, and Graphs ................................................................................................. 1
Unit 2: Functions and Graphs ..................................................................................................................... 16
Unit 3: New Functions from Old ................................................................................................................. 37
Unit 4: Polynomial and Rational Functions ................................................................................................. 50
Unit 5: Graphs of Functions – Revisited ...................................................................................................... 71
Unit 6: Trigonometry – Part1 ...................................................................................................................... 80
Unit 7: Trigonometry – Part2 ...................................................................................................................... 94
Unit 8: Inverse Trigonometric Functions .................................................................................................. 104
Unit 9: Sequences and Series .................................................................................................................... 113
Unit 10: Exponential and Logarithmic Functions ...................................................................................... 122
1
AMHS Precalculus - Unit 1
Unit 1: Inequalities, Equations, and Graphs
Interval Notation
Interval notation is a convenient and compact way to express a set of numbers on the real number line.
Graphic Representation
Inequality Notation
___________________________
2
___________________________
Interval notation
x 3
1 x 4
___________________________
1 x 2
___________________________
x 2
___________________________
x
1
Inequality Properties
1. If a
2. If a
3. If a
b , then a c b c
b and c 0 , then ac bc
b and c 0 , then ac bc
Ex. 1 Solve each inequality (note that the degree is 1) and write the solution using interval notation:
a)
3x 5 12
b) 9
2x 10 5
c) 3
7 2x
3
4
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AMHS Precalculus - Unit 1
Ex. 2 Solve each inequality and write the solution using inequality notation.
a) 0
2x
3
2
b) 0
x
2
2
c) 0
2x
2
Polynomial Inequalities with degree two or more and Rational Inequalities
Solve x2
4x 7 4 by making a sign chart. Write your answer using interval notation.
1. Set one side of the inequality equal to zero.
2. Temporarily convert the inequality to an equation.
3. Solve the equation for x . If the equation is a rational inequality, also determine the values of x
where the expression is undefined (where the denominator equals zero). These are the partition
values.
4. Plot these points on a number line, dividing the number line into intervals.
5. Choose a convenient test point in each interval. Only one test point per interval is needed.
6. Evaluate the polynomial at these test points and note whether they are positive or negative.
7. If the inequality in step 1 reads 0 , select the intervals where the test points are positive. If the
inequality in step 1 reads 0 , select the intervals where the test points are negative.
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AMHS Precalculus - Unit 1
Ex. 3 Solve each inequality. Show the sign chart. Draw the solution on the number line and express the
answer using interval notation.
a)
c)
x( x 4)( x 3) 2
x 3
x2 4
0
0
b)
d)
x2 3x 4 0
3
2
x 4
x 1
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AMHS Precalculus - Unit 1
Absolute Value
x if x 0
x if x 0
x
The absolute value of a real number x is the distance on the number line that x is from 0.
Absolute value equations
Ex. 4 Solve the equation (check your answers for extraneous solutions):
a)
2x 1
x 3
4
b)
2x 3 1
Absolute value inequalities
1. if x
a , then
2. if x
a
a
0 , then x
x
a
a or x
__________________________________
a
__________________________________
Ex. 5 Solve the inequality. Express your answers in interval notation and graph the solution:
a)
4 x 1 .01
b)
2x 1
5
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AMHS Precalculus - Unit 1
c)
Equations and Graphs
Lines
mx b is a linear equation where m and b are constants. This is called SlopeIntercept form where m is the slope and b is the y-intercept.
The equation y
In general,
m 0
m 0
m 0
m is undefined
6
AMHS Precalculus - Unit 1
The slope of a Line
Point-Slope equation of a line:
Ex. 1 Find the point-slope equation of a line passing through the points (-1, -2) and (2,5).
Ex. 2 Write the equation of a line passing through the points (4,7) and (0,3).
7
AMHS Precalculus - Unit 1
Parallel and Perpendicular Lines
Two non-vertical lines are parallel iff they have the same slope.
Two lines with non-zero slopes m1 and m 2 are perpendicular iff m1 m2
1.
Ex. 3 Find the equation of the line passing through the point (-3,2) that is parallel to 5x 2 y
3.
Ex. 4 Find the equation of the line passing through (-4,3) which is perpendicular to the line passing
through (-3,2) and (1,4).
Ex. 5 A new car costs $29,000. Its useful lifetime is approximately 12 years, at which time it will be worth
an estimated $2000.00.
a) Find the linear equation that expresses the value of the car in terms of time.
b) How much will the car be worth after 6.5 years?
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AMHS Precalculus - Unit 1
Ex. 6 The manager of a furniture factory finds that it costs $2220 to manufacture 100 chairs and $4800
to manufacture 300 chairs.
a) Assuming that the relationship between cost and the number of chairs produced is linear, find
an equation that expresses the cost of the chairs in terms of the number of chairs produced.
b) Using this equation, find the factory’s fixed cost (i.e. the cost incurred when the number of
chairs produced is 0).
Ex. 7 Find the slope-intercept equation of the line that has an x-intercept of 3 and a y-intercept of 4.
9
AMHS Precalculus - Unit 1
Circles
Recall the distance formula d
( x2
x1 ) 2 ( y2
y1 ) 2
The Standard form for the equation of a circle is:
Ex.1 Write the equation of a circle with center (-1,2) and radius 3. Sketch this circle.
Ex.2 Write the equation of a circle with center at the origin and radius 1.
Ex.3 Find the equation of the circle with center (-4,1) that is tangent to the line x = -1.
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AMHS Precalculus - Unit 1
Ex. 4 Find the equation of the circle with center (4,3) and passing through the point (1,4).
Ex. 5 Express the following equations of a circle in standard form. Identify the center and radius:
a)
x2
y2
4x 6 y
3
b)
x2
2x
y2
4
4y
11
AMHS Precalculus - Unit 1
The intercepts of a graph
The x -coordinates of the x - intercepts of the graph of an equation can be found by setting y
0 and
solving for x .
The y -coordinates of the y - intercepts of the graph of an equation can be found by setting x
solving for y .
Ex. 1 Find the x and y intercepts of the line and sketch its graph:
x 2y 1
Ex. 2 Find the x and y intercepts of the circle and sketch its graph: x 2
Ex. 3 Find the intercepts of the graphs of the equations.
a)
x2
b)
y
y2
9
2 x 2 5 x 12
y2
9
0 and
12
AMHS Precalculus - Unit 1
Symmetry
In general :
A graph is symmetric with respect to the y axis if whenever ( x, y) is on a graph ( x, y) is also
a point on the graph.
A graph is symmetric with respect to the x
axis if whenever ( x, y) is on a graph ( x, y) is also
a point on the graph.
A graph is symmetric with respect to the origin if whenever ( x, y) is on a graph ( x, y) is also
a point on the graph.
Tests for Symmetry:
The graph of an equation is symmetric with respect to:
a) the y axis if replacing x by x results in an equivalent equation.
b) the x axis if replacing y by
y results in an equivalent equation.
c) the origin if replacing x and y by x and y results in an equivalent equation.
Ex. 1 Show that the equation y
x 2 3 has y axis symmetry.
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AMHS Precalculus - Unit 1
Ex. 2 Show that the equation x
Ex. 3 Show that the equation x 2
y2
y2
10 has x axis symmetry.
9 has symmetry with respect to the origin.
Ex. 4 Find any intercepts of the graph of the given equation. Determine whether the graph of the
equation possesses symmetry with respect to the x axis, y axis, or origin.
a)
x
y2
b)
y
x2
c)
y
x2 2x 2
d)
y
x 9
4
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AMHS Precalculus - Unit 1
Algebra and Limits
Difference of two squares: a 2
Difference of two cubes: a 3
b2
b3
(a b)(a b)
(a b)(a 2
(a b)(a 2
ab b 2 )
Sum of two cubes: a 3
b3
ab b 2 )
Binomial Expansion n
2 : ( a b) 2
a2
Binomial Expansion n
3 : ( a b) 3
a 3 3a 2b 3ab 2 b3
2ab b 2
Limits
Ex. 1 Estimate lim
x
2
x 2
numerically by completing the following chart:
x2 4
y
x
y
x
1.9
2.1
1.99
2.01
1.999
2.001
Conclusion: lim
x
2
x 2
=
x2 4
Properties of Limits
If a and c are real numbers, then lim c
x
a
c, lim x
x
a
a, lim x n
x
an
a
Ex. 2 Find the limit:
a)
lim( x 3
x
2
x 4)
b)
lim(2 x 7)
x
1
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AMHS Precalculus - Unit 1
Ex. 3 Find the given limit by simplifying the expression
a) lim
x
2
x2 x 6
x2 5x 6
x2 4
b) lim 3
x
2 x
8
c)
lim
x 1
d) lim
x
e)
2
lim
x
2
2
x 3
x 1
x 2 7 x 10
x 2
x 2
x2 5 3
1
f)
lim x 8
x 0
x
1
8
16
AMHS Precalculus - Unit 2
Unit 2: Functions and Graphs
Functions
A function is a rule that assigns each element in the domain to exactly one element in the range.
The domain is the set of all possible inputs for the function. On a graph these are the values of the
independent variable (most commonly known as the x values).
The range is the set of all possible outputs for the function. On a graph these are the values of the
dependent variable (most commonly known as the y values).
We use the notation f ( x) to represent the value (again, in most cases, a y - value) of a function at the
given independent value of x . For any value of x , ( x, f ( x)) is a point on the graph of the function
f ( x).
Ex. 1 Given f ( x)
x 2 , graph the function and determine the domain and range. Use interval notation
to express the domain and range.
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AMHS Precalculus - Unit 2
Ex. 2 Given f ( x)
x , graph the function and determine the domain and range. Use interval notation
to express the domain and range.
Ex. 3 For the function f ( x)
x2
2 x 4 , find and simplify:
a) a) f ( 3)
Ex. 4 For f ( x)
a) f (1)
c) f ( 2)
b) b) f ( x h)
x2 ,
x 0
2 x 1, x 0
find:
b) f ( 1)
d) f (3)
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AMHS Precalculus - Unit 2
Ex. 5 The graph of the function f is given:
a) Determine the values:
f ( 2)
f (0)
f (2)
f (4)
b) Determine the domain:
c) Determine the range:
Ex. 6 The graph of the function f is given:
a)
f ( 3)
f (0)
b) For what numbers x is f ( x)
f (4)
0?
c) What is the domain of f ?
d) What is the range of f ?
e) What is (are) the x -intercept(s)?
f)
What is the y - intercept?
g) For what numbers x is f ( x)
0?
h) For what numbers x is f ( x)
0?
19
AMHS Precalculus - Unit 2
Vertical Line Test for a Function: An equation is a function iff every vertical line intersects the graph of
the equation at most once.
Ex. 7 Determine which of the curves are graphs of functions:
a)
b)
c)
Domain (revisited)
Rule for functions containing even roots (square roots, 4th roots, etc):
Ex. 1 Determine the domain and range of f ( x)
Ex. 2 Determine the domain of f (t )
4
t 2 2t 15
x 3
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AMHS Precalculus - Unit 2
Rule for functions containing fractional expressions:
Ex. 3 Determine the domain of h( x)
x
1
Ex. 4 Determine the domain of g ( x)
x
Ex. 5 Determine the domain of h( x)
5x
3x 4
2
2
3 x
x 2
2 x 15
21
AMHS Precalculus - Unit 2
Intercepts (revisited)
The y -intercept of the graph of a function is (0, f (0)) .
The x - intercept(s) of the graph of a function f ( x) is/are the solution(s) to the equation f ( x)
These x - values are called the zeros of the function f ( x) .
Ex. 1 Find the zeros of f ( x)
x(3x 1)( x 9)
Ex. 2 Find the zeros of f ( x )
x2 5x 6
Ex. 3 Find the zeros of f ( x)
x4 1
Ex. 4 Find the x - and y - intercepts (if any) of the graph of the function f ( x)
1
x 4
2
0.
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AMHS Precalculus - Unit 2
Ex. 5 Find the x - and y - intercepts (if any) of the graph of the function f ( x)
4( x 2) 2 1
Ex. 6 Find the x - and y - intercepts (if any) of the graph of the function f ( x)
x2 4
x 2 16
Ex. 7 Find the x - and y - intercepts (if any) of the graph of the function f ( x)
3
4 x2
2
23
AMHS Precalculus - Unit 2
Transformations – Horizontal and Vertical shifts
Suppose y
1.
2.
3.
4.
y
y
y
y
f ( x) is a function and c is a positive constant. Then the graph of
f ( x) c is the graph of f shifted vertically up c units.
f ( x) c is the graph of f shifted vertically down c units.
f ( x c) is the graph of f shifted horizontally to the left c units.
f ( x c) is the graph of f shifted horizontally to the right c units.
Ex. 1 Consider the graph of a function y
f ( x) shown on the coordinates. Perform the following
transformations.
y
f ( x) 3
y
f ( x 1)
y
y
f ( x) 2
f ( x 3)
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AMHS Precalculus - Unit 2
Suppose y
1.
2.
y
y
f ( x) is a function. Then the graph of
f ( x) is the graph of f reflected over the x -axis.
f ( x) is the graph of f reflected over the y -axis.
Ex. 2 Consider the graph of a function y
f ( x) . Sketch y
f ( x 2) 3
Common (Parent) Functions
f ( x)
x
f ( x)
x2
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AMHS Precalculus - Unit 2
f ( x)
x
f ( x)
x3
f ( x)
x
f ( x)
f ( x)
or x
3
1
x
x
26
AMHS Precalculus - Unit 2
Combining common functions with transformations
Sketch the graphs of the following functions. Determine the domain and range and any intercepts.
Ex. 1 f ( x)
x 2 1
Ex.2 f ( x) 1
Ex. 3 f ( x )
( x 2)3 1
Ex. 4 f ( x)
x 2
x 1 3
27
AMHS Precalculus - Unit 2
Symmetry (revisited)
Tests for Symmetry
The graph of a function f is symmetric with respect to:
1. the y -axis if f ( x)
2. The origin if f ( x)
f ( x) for every x in the domain of the f ( x) .
f ( x) for every x in the domain of the f ( x) .
If the graph of a function is symmetric with respect to the y -axis, we say that f is an even function.
If the graph of a function is symmetric with respect to the origin, we say that f is an odd function.
In examples 1-3, determine whether the given function y
Ex. 1 f ( x)
x5
x3
Ex.2 f ( x)
x
23
Ex. 3 f ( x)
x2
2x
x
f ( x) is even, odd or neither. Do not graph.
28
AMHS Precalculus - Unit 2
Transformations – Vertical Stretches and Compressions
Suppose y
f ( x) is a function and c a positive constant. The graph of y cf ( x) is the graph of f
1. Vertically stretched by a factor of c if c 1
2. Vertically compressed by a factor of c if 0 c 1
Ex.1 Given the graph of y
a) Sketch y
2 f ( x)
f ( x)
b)
y
1
f ( x)
2
Ex. 2 Sketch the graph of the following functions. Include any intercepts.
f ( x)
x 1
f ( x) 3( x 1)
29
AMHS Precalculus - Unit 2
Quadratic Functions
A quadratic function y
f ( x) is a function of the form f ( x)
ax 2 bx c where a 0 , b and c are
constants.
The graph of any quadratic function is called a parabola.
The graph opens upward if a
0 and downward if a 0 .
The domain of a quadratic function is the set of real numbers (
, ).
A quadratic function has a vertex (which serves as the minimum or maximum of the function depending
on the value of a ), a line of symmetry, and may have zero, one or two x - intercepts.
Ex. 1 Sketch the graph of f ( x )
( x 1) 2 3 . Determine any intercepts.
30
AMHS Precalculus - Unit 2
The standard form of a quadratic function is f ( x)
parabola and x
a ( x h) 2
k where (h, k ) is the vertex of the
h is the line of symmetry.
Ex. 2 Rewrite the quadratic function f ( x )
x2
2 x 3 in standard form by completing the square.
Determine any intercepts, the vertex, the line of symmetry and sketch the graph.
Ex. 3 Rewrite the quadratic function f ( x)
4 x 2 12 x 9 in standard form by completing the square.
Determine any intercepts, the vertex, the line of symmetry and sketch the graph.
31
AMHS Precalculus - Unit 2
Ex. 4 Complete the square to find all the solutions to the equation ax2
The vertex of any parabola of the form f ( x )
ax 2 bx c is (
bx c 0
b
b
, f(
)) .
2a
2a
Ex. 5 Find the vertex of the quadratics from examples 2 and 3 directly by using (
b
b
, f(
)) .
2a
2a
Ex. 6 Find the vertex from example 2 by using the x - intercepts and the line of symmetry.
32
AMHS Precalculus - Unit 2
Ex.7 Find the intercepts and vertex of the function f ( x)
Ex. 8 Find the maximum or the minimum of the function.
1.
f ( x) 3 x 2 8 x 1
2.
f ( x)
2x2 6x 3
Ex.9 Determine the quadratic function whose graph is given.
1 2
x
2
x 1
33
AMHS Precalculus - Unit 2
Freely Falling Object - Suppose an object, such as a ball, is either thrown straight upward or downward
with an initial velocity v0 or simply dropped ( v0
0 ) from an initial height s0 . Its height, s(t ) as a
function of time t can be described by the quadratic function s(t )
1 2
gt v0t s0
2
Gravity on earth is 32 ft / sec 2 or 9.8m / sec2 .
Also, the velocity of the object while it is in the air is v (t )
gt v0
Ex. 10 An arrow is shot vertically upward with an initial velocity of 64 ft / sec from a point 6 feet above
the ground.
1. Find the height s(t ) and the velocity v(t ) of the arrow at time t
0.
2. What is the maximum height attained by the arrow? What is the velocity of the arrow at the
time it attains its maximum height?
3. At what time does the arrow fall back to the 6 foot level? What is its velocity at this time?
Ex. 11 The height above the ground of a toy rocket launched upward from the top of a building is given
by s (t )
16t 2 96t 256 .
1. What is the height of the building?
2. What is the maximum height attained by the rocket?
3. Find the time when the rocket strikes the ground. What is the velocity at this time?
34
AMHS Precalculus - Unit 2
Horizontal Stretches and Compressions
Suppose y
f ( x) is a function and c a positive constant. The graph of y
1. Horizontally compressed by a factor of
2. Horizontally stretched by a factor of
Ex.1 Given the graph of y
c) Sketch y
f (cx) is the graph of f
1
if c 1
c
1
if 0 c 1
c
f ( x)
f (2x)
Ex.2 Consider the function f ( x )
d)
x2
y
1
f ( x)
2
4
1
2
a) On the same axis, sketch f ( x), f 2 x and f ( x) . Identify any intercepts of each function.
35
AMHS Precalculus - Unit 2
b) On the same axis, sketch f ( x), 2 f x and
List the transformations on f ( x)
1
f ( x) . Identify any intercepts of each function.
2
x required to sketch f ( x)
2x 1 2
36
AMHS Precalculus - Unit 2
Silly String Activity
Objective: The use a quadratic function to model the path of silly string.
Materials: Can of silly string, tape measure, stopwatch, clear overhead transparency, TI84
Personnel: Timekeeper, Silly-String operator, assistant
Calculate the initial velocity v0 of the silly string as it exits the can.
1. Hold the can of silly string 1 foot above the ground. Have the timekeeper start the stopwatch
and say “go”. At this time, shoot a short burst of silly string towards the ceiling. Have the class
keep a casual eye on the maximum height the silly string achieves. When the silly string hits the
floor, have the timekeeper stop the stopwatch and record the elapsed time.
2. Measure the maximum height of the silly string observed by the class. Use the position equation
s(t )
1 2
gt v0t s0 with g = 32 ft / sec2
2
to calculate v0 . ( s0 = 1, get t from the timekeeper. This represents the time it took for the silly
string to reach the ground, i.e. s(t ) =0)
Now that we know g , v0 and s0 we can set up a position equation to model the height of the silly string
as a function of time. Use this equation to determine the maximum height (the vertex holds this info) of
the silly string. How does this compare to the actual height observed by the class. What factors might
have caused it to be different?
Now we are going to get the assistant to lean over the can of silly string (with the clear overhead
transparency protecting the face) in its original position 1 foot above the ground and see if the assistant
can move fast enough to avoid getting silly string in the face. Calculate the time it would take for the
silly string to reach the assistant’s face (set s(t ) = the height of the assistant’s face and solve for t )
Once the reaction time for the assistant has been calculated and discussed, see if the assistant can
actually react that quickly, i.e. avoid silly string in the face.
To date, it has never been done. Enjoy!
37
AMHS Precalculus - Unit 3
Unit 3: New Functions from Old
Piecewise – Defined Functions
A function f may involve two or more functions, with each function defined on different parts of the
domain of f . A function defined in this manner is called a piecewise-defined function.
Ex.1 Sketch the graph of the given function and find the following:
f ( x)
x if x 0
x if x 0
x
a)
f ( 1)
b)
f (2)
c) Domain:
d) Range:
Ex. 1b Express f ( x)
x 3 as a piecewise function:
Ex.2 Sketch the graph of the given function and find the following:
f ( x)
3x 1 if x
x 2 1 if x 0
a)
f ( 1)
b)
f (2)
c) Domain:
d) Range:
1
38
AMHS Precalculus - Unit 3
Ex.3 Graph the following
1
a)
f ( x)
0
if x 0
if x 0
x 1 if x
0
b)
f ( x)
x 1
x 1
Hint: write this as a piecewise function
Domain:
Domain:
Range:
Range:
39
AMHS Precalculus - Unit 3
Graphing the Absolute Value of a Function
Sketch the graph of the given functions. Include any intercepts.
Ex.1
f ( x)
Ex.2
x if x 0
x if x 0
x
Ex.3
f ( x)
f ( x)
( x 2) 2 4
Ex.4
x
2
4x 3
f ( x)
x 3 1
40
AMHS Precalculus - Unit 3
Compositions of Functions
The composition of the function f with the function g , denoted f g is defined by
( f g )( x) f ( g ( x)) . The domain of f g consists of those x values in the domain of g for which
g ( x) is in the domain of f .
Ex. 1 f ( x)
a)
x 2 and g ( x)
( f g )( x)
b) Find the domain of ( f
c)
x 2 1 . Find the following:
g )( x)
( g f )( x)
d) Find the domain of ( g
e) ( f
f )( x)
g )(2)
f)
( g f )(4)
g)
( g f )(1)
Ex.2 Write the function f ( x)
Ex.3 Write the function f ( x)
x2 3 as the composition of two functions
x
2
3
as the composition of three functions.
4x 1
41
AMHS Precalculus - Unit 3
Ex. 4 Given F ( x)
( x 4)2
x 4 ¸find functions f and g such that F ( x) ( f g )( x) .
Ex. 5 A metal sphere is heated so that t seconds after the heat had been applied, the radius r (t ) is given
by r (t )
3 .001t cm. Express the Volume of the sphere as a function of t .
Ex. 6 f ( x)
a)
x and g ( x)
( f g )( x)
b) ( g
f )( x)
c)
g (3)
d)
g (4)
e)
f (9)
f)
f (16)
x2 , ( x
0) . Find the following:
42
AMHS Precalculus - Unit 3
Inverse Functions
Suppose that f is a one-to-one function with domain X and range Y . The inverse function for the
function f is the function denoted f
f 1 ( f ( x))
x and f ( f 1 ( x))
Ex. 1 Prove that f ( x)
1
with domain Y and range X and defined for all values x
X by
x.
x 2 and g ( x)
x2
2 ( x 0) are inverse functions using composition.
Steps for Finding the Inverse of a Function:
1. Set y
f ( x)
2. Change x
y and y
3. Solve for y
4. Set y
x
f 1 ( x)
Ex. 2 Find the inverse of f ( x)
f ( x) and f 1 ( x) .
3x
and check using composition. Find the domain and range of
( x 4)
43
AMHS Precalculus - Unit 3
The graph of f
1
( x) is a reflection of the graph of f ( x) about the line y
x.
One-to-One Functions
A function is one-to-one iff each number in the range of f is associated with exactly one number in its
domain. In other words, f ( x1 )
f ( x2 ) implies x1
x2 .
Horizontal Line Test for One-to-One Functions
A function is one-to-one precisely when every horizontal line intersects its graph at most once.
Ex. 3 Determine whether the given function is one-to-one
a)
f ( x)
x3 2
b)
f ( x)
x2 2x
44
AMHS Precalculus - Unit 3
Ex.4 Given f ( x)
2x 3
Domain of f ( x) :
Domain of f
Range of f ( x) :
Range of f
Find f
1
( x) and check using composition.
Sketch the graph of f
1
( x) and f ( x) on the same axis.
1
1
( x) :
( x) :
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AMHS Precalculus - Unit 3
Translating Words into Functions
In calculus there will be several instances where you will be expected to translate the words that
describe a problem into mathematical symbols and then set up or construct an equation or a function.
In this section, we will focus on problems that involve functions. We begin with a verbal description
about the product of two numbers.
Ex.1 The sum of two nonnegative numbers is 15. Express the product of one and the square of the other
as a function of one of the numbers.
Ex.2 A rectangle has an area of 400 in2 . Express the perimeter of the rectangle as a function of the
length of one of its sides.
Ex.3 Express the area of a circle as a function of its diameter d .
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AMHS Precalculus - Unit 3
Ex. 4 An open box is made from a rectangular piece of cardboard that measures 30cm by 40cm by
cutting a square of length x from each corner and bending up the sides. Express the volume of the box
as a function of x .
Ex. 5 Express the area of the rectangle as a function of x . The equation of the line is x 2 y
4 .The
lower left-hand corner is on the origin and upper right-hand corner of the rectangle with coordinate
( x, y) is on the line.
Ex. 6 Express the area of an equilateral triangle as a function of the length s of one of its sides.
47
AMHS Precalculus - Unit 3
The Tangent Line Problem
Find a tangent line to the graph of a function f .
mtan
lim
x
f (a
0
Ex.1 Find the slope of the tangent line to the graph of f ( x )
x2
2 at x 1 .
Ex.2 Find the slope of the tangent line to the graph of f ( x )
x2
2 at x 3 .
x)
x
f (a )
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AMHS Precalculus - Unit 3
Ex.3 Find the slope of the tangent line to the graph of f ( x )
The DERIVATIVE of a function y
f '( x)
2 at x
2.
f ( x) is the function f ' defined by:
lim
x
x2
0
f (x
x)
x
f ( x)
Ex.4 Find the derivative of f ( x )
x2
Ex.5 Find the derivative of f ( x)
2 x 2 6 x 3 and use it to find the slope and then the equation of
the tangent line at x
2.
2.
49
AMHS Precalculus - Unit 3
Ex. 6 Find the slope of the tangent line to the graph of f ( x)
Ex.7 Find the derivative of f ( x)
line at x
2
and use it to find the slope and then the equation of the tangent
x
2.
Ex. 8 Find the derivative of f ( x)
2
at x 1 .
x
x 2.
50
AMHS Precalculus - Unit 4
Unit 4: Polynomial and Rational Functions
Polynomial Functions
A polynomial function y
p( x) is a function of the form
p( x) an x n an 1 xn
1
an 2 x n
2
... a2 x 2 a1 x a0
where an , an 1 ,..., a2 , a1 , a0 are real constants and are called the coefficients of p( x) .
n is the degree of p( x) and is a positive integer.
an is called the leading coefficient and a0 is the constant term of the polynomial.
The domain of any polynomial is all real numbers.
Ex. 1 Determine the degree, the leading coefficient and the constant term of the polynomial.
a)
f ( x) 5 x 4 7 x 3 3 x 7
b)
g ( x)
13x3 5 x 2 4 x
End Behavior of a Polynomial
There are four scenarios:
1) Sketch p ( x)
an
As x
As x
,
x 2 , p( x)
x 4 ( n is even,
2) Sketch p ( x)
0)
p( x)
, p( x)
even, an
As x
As x
,
0)
p( x)
, p( x)
x 2 , p( x)
x 4 ( n is
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AMHS Precalculus - Unit 4
3) Sketch p ( x)
x 5 ( n is odd,
4) Sketch p ( x)
0)
an
an
As x
As x
,
As x
and x
p ( x)
x 3 , p( x)
an x
p( x)
, p( x)
n
an 1 x
x
x
,
x3 , p( x)
x 5 ( n is odd,
0)
p( x)
, p( x)
, the graph of the polynomial
n 1
an 2 x n
2
... a2 x 2 a1 x a0 resembles the graph of y
an x n .
Ex. 2 Use the zeros and the end behavior of the polynomial to sketch an approximation of the graph of
the function.
a)
f ( x)
x3 9 x
b)
g ( x)
x4 5x2
4
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AMHS Precalculus - Unit 4
c)
f ( x)
x5
x
Repeated Zeros
If a polynomial f ( x) has a factor of the form ( x c ) k , where k
1 , then x
c is a repeated zero of
multiplicity k .
If k is even, the graph of f ( x) flattens and just touches the x -axis at x
If k is odd, the graph of f ( x) flattens and crosses the x -axis at x
c.
c.
Ex. 4: Sketch the given graphs
a)
f ( x)
x 4 3x3
2x2
b)
g ( x)
( x 1)3 ( x 2)( x 3)
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AMHS Precalculus - Unit 4
Ex. 5: The cubic polynomial p( x) has a zero of multiplicity two at x
x
2 , and p( 1)
1 , a zero of multiplicity one at
2 . Determine p( x) and sketch the graph.
Ex. 6: An open box is to be made from a rectangular piece of cardboard that is 12 by 6 feet by cutting
out squares of side length x feet from each corner and folding up the sides.
a) Express the volume of the box v( x) as a function of the size x cut out at each corner.
b) Use your calculator to approximate the value of x which will maximize the volume of the box.
Ex. 7: The product of two non-negative numbers is 60. What is the minimum sum of the two numbers?
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AMHS Precalculus - Unit 4
The Intermediate Value Theorem
Suppose that f is continuous on the closed interval [a, b] and let N be any number between f (a) and
f (b) , where f (a)
f (b) . Then there exists a number c in (a, b) such that f (c) N .
Ex. 1: Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of
c guaranteed by the theorem.
f ( x)
x2
x 1 , [0,5], f (c) 11
Ex. 2: Show that there is a root of the equation x3
2x 1 0 in the interval (0,1).
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AMHS Precalculus - Unit 4
The Division Algorithm
Let f ( x) and d ( x)
0 be polynomials where the degree of f ( x) is greater than or equal to the degree
of d ( x) . Then there exists unique polynomials q( x) and r ( x) such that
f ( x)
d ( x)
q ( x)
r ( x)
or f ( x) d ( x)q( x) r ( x) .
d ( x)
where r ( x) has a degree less than the degree of d ( x) .
Ex. 1: Divide the given polynomials.
a)
6 x3 19 x 2 16 x 4
x 2
c)
3x3 x 2 2 x 6
x2 1
b)
x3 1
x 1
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AMHS Precalculus - Unit 4
Remainder Theorem
If a polynomial f ( x) is divided by a linear polynomial x c , then the remainder r is the value of f ( x)
at x
c . In other words , f (c)
r
Ex. 2: Use the Remainder Theorem to find r when f ( x)
Ex. 3: Use the Remainder Theorem to find f (c) for f ( x )
4 x3
x2
3x 4 5 x 2
4 is divided by x 2 .
27 when c
1
2
Synthetic Division
Synthetic division is a shorthand method of dividing a polynomial p( x) by a linear polynomial x c . It
uses only the coefficients of p( x) and must include all 0 coefficients of p( x) as well.
Ex. 4: Use synthetic division to find the quotient and remainder when
a)
f ( x)
x 3 1 is divided by x 1
b)
f ( x)
x 4 14 x 2 5 x 9 is divided by x 4
c)
8x4 30x3 23x2 8x 3 is divided by x
1
4
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AMHS Precalculus - Unit 4
Ex. 5: Use synthetic division and the Remainder Theorem to find f (c) for
f ( x)
3x6
4 x5
x 4 8 x3 6 x 2 9 when c
2.
Ex. 6: Use synthetic division and the Remainder Theorem to find f (c) for
f ( x)
x 3 7 x 2 13x
15 when c 5 .
The Factor Theorem
A number c is a zero of a polynomial p( x) ( p(c)
Ex. 1: Determine whether
x4 5x2
a)
x 1is a factor of f ( x)
b)
x 2 is a factor of x3 3x2 4
6x 1
0 ) if and only if ( x c) is a factor of p( x) .
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AMHS Precalculus - Unit 4
Fundamental Theorem of Algebra
A polynomial function p( x) of degree n
0 has at least one zero.
In fact, every polynomial function p( x) of degree n
0 has at exactly n zeros.
Complete Factorization Theorem
Let c1 , c2 ,...cn be the n (not necessary distinct) zeros of the polynomial function of degree n
p( x) an x n an 1 xn
1
an 2 x n
2
... a2 x 2 a1 x a0 .
Then p( x) can be written as the product of n linear factors
p( x)
an ( x c1 )( x c2 )
( x cn ).
Ex.1: Give the complete factorization of the given polynomial p( x) with given information:
a)
p( x)
2 x3 9 x 2 6 x 1 ; x
b)
p( x)
4 x 4 8 x 3 61x 2
1
is a zero.
2
2 x 15 ; x
3, x 5 are both zeros.
0:
59
AMHS Precalculus - Unit 4
x3 6 x 2 16 x 48 ; ( x 2) is a factor.
c)
p( x)
d)
p( x) 3x 4 7 x 3 5 x 2
x ; x(3x 1) is a factor.
Ex. 2: Find a polynomial function f ( x) of degree three, with zeros 1,-4, 5 such that the graph possesses
the y - intercept (0,5) .
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AMHS Precalculus - Unit 4
The Rational Zero Test
Suppose
p
is a rational zero of f ( x) an xn an 1 x n
q
where a0 , a1......, an are integers and an
1
an 2 x n
2
... a2 x 2 a1 x a0 ,
0 . Then p divides a0 and q divides an .
The Rational Zero Test provides a list of possible rational zeros.
Ex. 1: Find all the rational zeros of f ( x) then factor the polynomial completely.
a)
f ( x) 3x 4 10 x 3 3x 2 8 x 2.
b)
f ( x)
x 4 3x3
x 2 3x 2
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AMHS Precalculus - Unit 4
Complex Roots of Polynomials
Consider factoring the function:
f ( x)
x3 1
The Square Root of -1
We define i
1 so that i 2
1.
Complex Numbers
A complex number is a number of the form a bi where a and b are real numbers. The number a is
called the real part and the number b is called the imaginary part.
Complex Arithmetic
Ex. 1
a)
(2 3i) (6 i)
b) (2 3i)(4 i)
c)
(3 6i)(3 6i)
d) (4 5i)(4 5i)
Complex Conjugates
The complex conjugate for a complex number z
In general, (a bi)(a bi)
a bi is z
a bi .
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AMHS Precalculus - Unit 4
Ex. 2: Simplify.
a)
(2 3i)
(1 6i)
(2 i)
(1 7i)
Ex. 3: Simplify.
a)
b)
4
Ex. 4: Determine all solutions to the equation x2
Ex. 5: Completely factor f ( x )
x 3 1.
4x 13 0
8
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AMHS Precalculus - Unit 4
Ex. 6: Find the complete factorization of f ( x)
x 4 12 x3 47 x 2 62 x 26 given that 1 is a zero of
multiplicity two.
Conjugate Pairs of Zeros of Real Polynomials
If the complex number z
conjugate z
a bi is a zero of some polynomial p( x) with real coefficients, then its
a bi is also a zero of p( x) .
Ex. 7: Find a 3rd degree polynomial g ( x) with real coefficients and a leading coefficient of 1 with zeros 1
and 1 i .
Ex. 8: 1 2i is a zero of f ( x)
complete factorization of f ( x) .
x 4 2 x 3 4 x 2 18 x 45. Find all other zeros and then give the
64
AMHS Precalculus - Unit 4
Rational Functions
A rational function y
f ( x) is a function of the form f ( x)
p ( x)
, where p and q are polynomial
q ( x)
functions.
Ex. 1: Recall the parent function f ( x)
1
. Use transformations to sketch g ( x)
x
2
x 1
Asymptotes of Rational Functions
The line x
a is a vertical asymptote of the graph of f ( x) if f ( x)
(from the right) or x
or f ( x)
as x
a
a (from the left).
Vertical Asymptotes
p ( x)
has vertical asymptotes at the zeros of q( x) after all of the common factors
q ( x)
of p( x) and q( x) have been canceled out; the values of x where q( x) 0 and p( x) 0 .
The graph of f ( x)
Holes
The graph of f ( x)
p ( x)
has a hole at the values of x where q( x) 0 and p( x) 0 .
q ( x)
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AMHS Precalculus - Unit 4
Horizontal Asymptotes
The line y
b is a horizontal asymptote of the graph of f ( x) if f ( x)
In particular, with a rational function f ( x)
p( x)
q( x)
b when x
or x
.
an x n an 1 x n 1 ... a1 x a0
bm x m bm 1 x m 1 ... b1 x b0
There are three cases:
1. If n
m , then y
Ex: f ( x)
2. If n
3. If n
3x
13x3 7 x
m , then y
Ex: f ( x)
0 is the horizontal asymptote.
an
is the horizontal asymptote.
bm
3x3 6 x
4 x3 x 2 3
m , then there is no horizontal asymptote.
Ex: f ( x)
3 x 4 2 x3 5 x 2 1
3x 2 4 x
Slant Asymptote
If the degree of numerator is exactly one more than the degree of the denominator, the graph of f ( x)
has a slant asymptote of the form y
mx b . The slant asymptote is the linear quotient found by
dividing p( x) by q( x) and essentially disregarding the remainder.
Ex: f ( x)
x3 8 x 12
x2 1
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AMHS Precalculus - Unit 4
Ex. 2: Find all asymptotes and intercepts and sketch the graphs of the given rational functions:
a)
f ( x)
2
x 1
Domain:
Range:
Equation(s) of vertical asymptotes:
Equation(s) of horizontal asymptotes:
Equation of slant asymptote:
x - intercepts:
y - intercept:
b)
f ( x)
3x 2
2x 4
Domain:
Range:
Equation(s) of vertical asymptotes:
Equation(s) of horizontal asymptotes:
Equation of slant asymptote:
x - intercepts:
y - intercept:
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AMHS Precalculus - Unit 4
c)
f ( x)
x 3
( x 2)( x 5)
Domain:
Range:
Equation(s) of vertical asymptotes:
Equation(s) of horizontal asymptotes:
Equation of slant asymptote:
x - intercepts:
y - intercept:
d)
f ( x)
x2 x
x 1
Domain:
Equation(s) of vertical asymptotes:
Equation(s) of horizontal asymptotes:
Equation of slant asymptote:
x - intercepts:
y - intercept:
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AMHS Precalculus - Unit 4
e)
f ( x)
x 1
x x 2
2
Domain:
Range:
Equation(s) of vertical asymptotes:
Equation(s) of horizontal asymptotes:
Equation of slant asymptote:
x - intercepts:
y - intercept:
f)
f ( x)
(3x 1)( x 2)
( x 2)( x 1)
Domain:
Range:
Equation(s) of vertical asymptotes:
Equation(s) of horizontal asymptotes:
Equation of slant asymptote:
x - intercepts:
y - intercept:
Ex. 3: Sketch the graph of a rational function that satisfies all of the following conditions:
as x 1 and f ( x)
as x 1
f ( x)
as x
as x
f ( x)
2 and f ( x)
2
f ( x) has a horizontal asymptote y 0
f ( x) has no x -intercepts
Has a local maximum at ( 1, 2)
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AMHS Precalculus - Unit 4
Honors Precalculus – Academic Magnet High School
Name_____________________
Mandelbrot Set Activity
using Fractint fractal generator
STEP 1 - CREATE, SAVE, and PRINT an inspirational, visually pleasing area of the Mandelbrot set.
Important Menu Items:
VIEW- Image Settings, Zoom In/ Out box, Coordinate Box
FRACTALS- Fractal Formula, Basic Options, Fractal Parameters
COLORS- Load Color- Map
FILE- Save As
1) Start Fractint by clicking on the desktop icon.
Fractint always starts with the Mandelbrot set, but in case things get weird, ALWAYS
make sure “mandel” is selected in the Fractal-Fractal Formula menu item.
Use the Image Settings box to set the size of the picture (800 x 600 should work fine).
2) Use the Zoom In/Out feature along with the Colors-Load Color Map to create a variation of
the Mandelbrot set.
If the color palettes do not load, double click on the box that is labeled Pallette Files
(*.Map)
If you zoom in a few times you lose detail, you can increase the iterations in the
Fractals-Basic Options Box- Remember that the more iterations the computer has to
perform, the longer it will take
3) Use the Fractals-Fractal Params …window to record the x and y mins and maxs of the
viewing rectangle on the imaginary plane.
4) Using the Coordinates box, point your arrow to a point you think is in the Mandelbrot set and
record the x and y values.
5) Repeat #4 for a point you think is NOT in the set.
6) SAVE the fractal. Write down the coordinates (x and y mins and maxs) and number of
iterations of your current position in the Mandelbrot set.
7) Print your fractal.
STEP 2 - Create a typed text document (1 page or so) including, but not limited to:
The NAME of your group’s fractal and the name of everyone in your group
A short story about your creation (what it makes you think of, color, choice, etc.)
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AMHS Precalculus - Unit 4
STEP 3 – Typed:
1) List the x and y mins and maxs for your viewing rectangle from Step 1
2) Recall the coordinates of the point you thought was in the Mandelbrot set from Step 1.
Let x = a and y = b for the complex number a + bi
Let this number a + bi = c iterate this value 100 or more times using the Mandelbrot
sequence:
x0 = c
x1 = x02 + c
x2 = x12 + c
Etc…
You will be using decimals and your calculator. Unlike the fractals, these calculations will
not be pretty. Let your TI-84 do the work for you (i is above the decimal point).
3) Record the last 20 iterations for analysis. Remember that you may need to scroll the TI- 84
to the right to get the entire number
4) Were your predictions right about this point? Do you need more information to determine if
it is in the set?
5) Repeat for the point you thought was not in the set.
6) Summarize your findings.
TURN IN ALL 3 STEPS PAPER-CLIPPED together in order.
Extra Credit: Create your own color map.
http://www.nahee.com/spanky/www/fractint/fractint.html - for info on Fractint
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AMHS Precalculus - Unit 5
Unit 5: Graphs of Functions – Revisited
Solving Equations Graphically
The Intersection Method
To solve an equation of the form f ( x)
1. Graph y1
f ( x) and y2
g ( x) :
g ( x) on the same screen.
2. Find the x - coordinate of each point of intersection.
Ex. 1: Solve.
a)
2x 1
x 3
c)
x2 4 x 3
b)
4
x3
x2
3x 4
x 6
The x - intercept Method
To solve an equation of the form f ( x)
g ( x) :
1. Write the equation in the equivalent form f ( x)
2. Graph y
0.
f ( x) .
3. The x - intercepts of the graph are the real solutions to the equation.
Ex.2: Solve.
a)
2x 1
x 3
c)
x2 4 x 3
4
x3
x 6
b)
x2
3x 4
d)
x5
x2
x3 5
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AMHS Precalculus - Unit 5
Technological Quirks
1. Solve
2. Solve
f ( x)
f ( x)
g ( x)
0 by solving f ( x) 0 .
0 by solving f ( x) 0 (eliminate any values that also make g ( x) 0 ).
Ex. 3: Solve.
x4
a)
x2 2x 1 0
2 x2 x 1
b)
9x2 9 x 2
0
Applications
Ex. 1: According to data from the U.S. Bureau of the Census, the approximate population y (in millions)
of Chicago and Los Angeles between 1950 and 2000 are given by:
Chicago: y
.0000304 x 3 .0023 x 2 .02024 x 3.62
Los Angeles: y
where x
.0000113x 3 .000992 x 2 .0538 x 1.97
0 corresponds to 1950. In what year did the two cities have the same population?
Ex. 2: The average of two real numbers is 41.125, and their product is 1683. Find the two numbers.
Ex. 3: A rectangle is twice as wide as it is high. If it has an area of 24.5 square inches, what are the
dimensions of the rectangle?
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AMHS Precalculus - Unit 5
Ex. 4: A rectangular box with a square base and no top is to have a volume of 30,000 cm3 . If the surface
area of the box is 6000 cm2 , what are the dimensions of the box?
Ex. 5: A box with no top that has a volume of 1000 cubic inches is to be constructed from a 22 x 30-inch
sheet of cardboard by cutting squares of equal size from each corner and folding up the sides. What size
square should be cut from each corner?
Ex. 6: A pilot wants to make 840-mile trip from Cleveland to Peoria and back in 5 hours flying time.
There will be a headwind of 30 mph going to Peoria, and it is estimated that there will be a 40-tail wind
on the return trip. At what constant engine speed should the plane be flown?
74
AMHS Precalculus - Unit 5
Solving Inequalities Graphically
1. Rewrite the inequality in the form f ( x)
0 or f ( x) 0 .
2. Determine the zeros of f .
3. Determine the interval(s) where the graph is above ( f ( x)
0 ) or below ( f ( x) 0 ) the x -axis.
Ex. 1: Solve each inequality graphically. Express your answer in interval notation.
a)
c)
x( x 4)( x 3) 2
x 3
x2 4
0
b)
d)
0
e)
f)
x 2 3x
4
3
2
x 4
x 1
x 4 6 x3 2 x 2
5x 2
Ex. 2: A company store has determined the cost of ordering and storing x laser printers is:
c
2x
300, 000
x
If the delivery truck can bring at most 450 printers per order, how many printers should be ordered at a
time to keep the cost below $1600.00?
75
AMHS Precalculus - Unit 5
Increasing, Decreasing and Constant Functions
A function f is increasing on an interval when, for any x1 and x2 in the interval, x1 < x2 implies
f ( x1 )
f ( x2 ).
A function f is decreasing on an interval when, for any x1 and x2 in the interval, x1 < x2 implies
f ( x1 )
f ( x2 ).
A function f is constant on an interval when, for any x1 and x2 in the interval, f ( x1 )
Ex.1: Determine the open intervals on which each function is increasing, decreasing or constant.
a)
f ( x)
x 1
x 3
b)
f ( x)
x3 3x
c)
f ( x)
x3
f ( x2 ).
76
AMHS Precalculus - Unit 5
Relative Minimum and Maximum Values (Relative Extrema)
A function value f (a) is called relative minimum of f when there exists an interval ( x1 , x2 )
that contains a such that x1
x
x2 implies f (a)
f ( x).
A function value f (a) is called relative maximum of f when there exists an interval ( x1 , x2 )
that contains a such that x1
x
x2 implies f (a)
f ( x).
Ex. 2: Determine the relative minimum and x -intercepts of f ( x)
3x 2
4x 2
Ex. 3: Use a graphing utility to determine the relative minimum and x -intercepts of
f ( x) 3x 2
4x 2
Ex. 4: Use a graphing utility to determine any relative minima or maxima for f ( x)
x3
x
77
AMHS Precalculus - Unit 5
Ex. 5: During a 24-hour period, the temperature t ( x) (in degrees Fahrenheit) of a certain city can be
approximated by the model t ( x)
.026 x 3 1.03x 2 10.2 x 34 , 0
x 24
where x represents the time of day, with x 0 corresponding to 6 A.M.
Approximate the maximum and minimum temperatures during this 24-hour period.
Optimization: Translating Words into Functions – revisited
Ex.1: The sum of two nonnegative numbers is 15. Express the product of one and the square of the
other as a function of one of the numbers. Use a graphing utility to find the maximum product.
Ex.2: A rectangle has an area of 400 in2 . Express the perimeter of the rectangle as a function of the
length of one of its sides. Use a graphing utility to find the minimum perimeter.
Ex. 3: An open box is made from a rectangular piece of cardboard that measures 30cm by 40cm by
cutting a square of length x from each corner and bending up the sides. Express the volume of the box
as a function of x . Use a graphing utility to find the dimensions of the box with the maximum volume.
78
AMHS Precalculus - Unit 5
Ex. 4: Express the area of the rectangle as a function of x . The equation of the line is x 2 y
4 .The
lower left-hand corner is on the origin and upper right-hand corner of the rectangle with coordinate
( x, y) is on the line. Use a graphing utility to find the rectangle with the maximum area.
Concavity and Inflection Points
Concavity is used to describe the way a curve bends. For any two points in a given interval that lie on a
curve, if the line segment that connects them is above the curve, then the curve is said to be concave up
over the given interval. If the segment is below the curve, then the curve is said to be concave down
over the interval. A point where the curve changes concavity is called an inflection point.
79
AMHS Precalculus - Unit 5
Ex. 1 For the following functions, estimate the following:
1.
2.
3.
4.
All local maxima and minima (relative extrema) of the function
Intervals where the function is increasing and/or decreasing
All inflection points of the function
Intervals where the function is concave up and when it is concave down
a)
f ( x)
c)
f ( x)
2 x3 6 x 2
3
( x 2)2
x 3
b)
g ( x)
d)
f ( x)
x3
4x 2
x2 x
x 1
80
AMHS Precalculus - Unit 6
Unit 6: Trigonometry – Part1
Right Triangle Trigonometry
a) Sine
Hypotenuse
Opposite
d) Cosecant
sin( )
csc( )
b) Cosine
e) Secant
cos( )
Adjacent
c) Tangent
tan( )
Ex. 1: Find the values of the six trigonometric functions of the angle .
7
3
Ex.2: Find the exact values of the sin,cos, and tan of 45
45˚
sec( )
f)
Cotangent
cot( )
81
AMHS Precalculus - Unit 6
Ex. 3: Find the exact values of the sin,cos, and tan of 60 and 30
30˚
Ex. 4: Find the exact value of x (without a calculator).
5
30˚
x
Ex. 5: Find all missing sides and angles (with a calculator).
33˚
21
82
AMHS Precalculus - Unit 6
Applications
An angle of elevation and an angle of depression can be measured from a point of reference and a
horizontal line. Draw two figures to illustrate.
Ex. 6: (use a calculator) A surveyor is standing 50 feet from the base of a large building. The surveyor
measures the angle of elevation to the top of the building to be 71.5˚. How tall is the building? Draw a
picture.
Ex. 7: A ladder leaning against a house forms a 67˚angle with the ground and needs to reach a window
17 feet above the ground. How long must the ladder be?
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AMHS Precalculus - Unit 6
Angles – Degrees and Radians.
An angle consists of two rays that originate at a common point called the vertex. One of the rays is
called the initial side of the angle and the other ray is called the terminal side.
Angles that share the same initial side and terminal side are said to be coterminal.
To find a co-terminal angle to some angle
(in degrees):
Radians – the other angle measure.
One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius
of the circle.
84
AMHS Precalculus - Unit 6
Conversion between degrees and radians
360 =
radians.
Therefore,
a) 1 radian =
b) 1 =
degrees
radians
Ex. 1: Convert the following radian measure to degrees:
a)
5
6
b)
c)
4
d) 3
10
Ex. 2: Convert the following degree measure to radians:
a)
400
To find a co-terminal angle to some angle
b) - 120
(in radians):
Ex. 3: Find a coterminal angle, one positive and one negative, to
5
.
3
85
AMHS Precalculus - Unit 6
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AMHS Precalculus - Unit 6
We use the unit circle to quickly evaluate the trigonometric functions of the common angle found on it.
To summarize how to evaluate the Sine and Cosine of the angles found on the unit circle:
1. sin(
)=
2. cos(
)=
Ex. 1: Find the exact value
a) sin( )
4
d) cos( )
b) cos( )
2
e) sin(150 )
3
g) sin(
3
)
2
h) cos(
5
)
3
c)
sin( )
6
f)
11
cos(
)
6
i)
sin(330 )
Ex. 2: Find the exact value by finding coterminal angles that are on the unit circle.
13
)
4
a) sin(
b) sin(
7
)
6
c)
sin( 300 )
87
AMHS Precalculus - Unit 6
Cosine is an even function.
Sine is an odd function.
cos(
sin(
) cos( )
)
sin( )
Ex. 3: Find the exact value.
a) cos(
7
)
6
b) sin(
3
)
4
c)
cos(
13
)
4
Reference Angles
For any angle in standard position, the reference angle ( ' ) associated with
formed by the terminal side of and the x - axis.
Ex. 1: Find the reference angle
a)
2
3
b)
is the acute angle
' for the given angles.
2.3
c)
5
4
d)
5
3
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AMHS Precalculus - Unit 6
The signs (+ or – value) of the Sine, Cosine and Tangent functions in the four quadrants of the Euclidean
plane can be summarized in this way:
Ex.2: Find the exact value.
a) sin(
2
)
3
d) cos(
3
b) cos(
e) cos( 300 )
)
Ex.3: Find all values of
a) sin( )
Ex. 5: If sin(t )
5
)
6
2
and
3
sin(
f)
sin(150 )
in the interval [0, 2 ] that satisfy the given equation
2
2
b) cos( )
t
3
, find the value of cos(t ) .
2
5
)
3
c)
3
2
89
AMHS Precalculus - Unit 6
Arc Length
In a circle of radius r , the length s of an arc with
angle radians is: s r
Ex. 1: Find the length of an arc of a circle with radius 5 and an angle
5
.
4
Ex. 2: Find the length of an arc of a circle with radius 13 and an angle 30 .
Ex. 3: The arc of a circle of radius 3 associated with angle
has length 5. What is the measure of
Area of a Circular Sector
In a circle of radius r , the area A of a circular sector formed by an angle of
A
1 2
r
2
Ex. 1: Find the area A of a sector with angle 45 in a circle of radius 4.
radians is
?
90
AMHS Precalculus - Unit 6
Graphs of the Sine and Cosine Functions
Ex. 1: Graph f ( x)
sin x
Domain:
Range:
x -intercepts:
Period:
Amplitude:
Even or odd?
Ex. 2: Graph f ( x)
cos x
Domain:
Range:
x -intercepts:
Period:
Amplitude:
Even or odd?
91
AMHS Precalculus - Unit 6
Ex.3: Graph one period of each function
a)
f ( x)
c)
f ( x) cos( x
2cos( x)
2
)
b)
f ( x) 1 sin( x)
d)
f ( x) sin(2x)
92
AMHS Precalculus - Unit 6
Graphs of f ( x)
A sin( Bx C) D and f ( x)
Amplitude:
Period:
Horizontal shift (Phase shift):
Vertical shift:
Ex.1: Graph one period of each function.
a)
f ( x)
2sin( x
3
)
Amplitude:
Period:
Horizontal Shift:
End Points:
b)
1
f ( x) 3sin( x
2
Amplitude:
Period:
Horizontal Shift:
End Points:
4
)
A cos( Bx C) D where A 0 and B 0 have:
93
AMHS Precalculus - Unit 6
c)
f ( x) 1 2cos(2 x
4
)
Amplitude:
Period:
Horizontal Shift:
End Points:
Ex. 2: Write a Sine or Cosine function whose graph matches the given curve.
a) x -scale is
4
b) x -scale is
Ex. 3: Write a Sine and Cosine function whose graph matches the given curve. x -scale is
3
94
AMHS Precalculus - Unit 7
Unit 7: Trigonometry – Part2
Revisiting Tangent, Cotangent, Secant and Cosecant
These are called the Quotient Identities:
a)
tan( )
sin( )
cos( )
b) cot( )
cos( )
sin( )
The following are called the Reciprocal Identities:
1
sin( )
1
c) cot( )
tan( )
1
e) sin( )
csc( )
a) csc( )
1
cos( )
1
d) tan( )
cot( )
1
f) cos( )
sec( )
b) sec( )
Ex. 1 Evaluate all six trigonometric functions at the following values of
a)
6
b)
2
:
95
AMHS Precalculus - Unit 7
The Pythagorean Identities:
a) sin 2 ( x) cos 2 ( x) 1
Using the Quotient and Reciprocal identities we can derive the other two Pythagorean Identities:
sin 2 ( x) cos 2 ( x) 1
sin 2 ( x) cos 2 ( x) 1
Conclusion – The other two identities are:
b)
c)
Ex. 2 Find the values of all six trigonometric functions from the given information:
a) sin( )
4
,
5
is in the first quadrant.
b) csc( )
5,
3
2
2
96
AMHS Precalculus - Unit 7
Graphs of the Tangent and Cotangent Functions
Ex. 1 Graph f ( x)
tan x
Domain:
Range:
x -intercepts:
Period:
Even or odd?
Ex. 2 Graph f ( x)
cot x
Domain:
Range:
x -intercepts:
Period:
Even or odd?
97
AMHS Precalculus - Unit 7
Ex.3 Sketch one period of each function
e)
f ( x) 2 tan( x)
f)
f ( x) cot( x
4
)
Ex. 4 Find the period of the following functions:
f ( x) tan(2x)
f( )
tan( )
2
f ( x)
Ex. 5 Find all the values of t in the interval [0, 2 ] satisfying the given equation:
a)
tan(t ) 1 0
b) cot(t )
3
0
cot(
x
)
3
98
AMHS Precalculus - Unit 7
Graphs of the Secant and Cosecant Functions
Ex. 1 Graph f ( x)
sec x
Domain:
Range:
x -intercepts:
Period:
Even or odd?
Ex. 2 Graph f ( x)
csc x
Domain:
Range:
x -intercepts:
Period:
Even or odd?
99
AMHS Precalculus - Unit 7
Ex.3 Graph one period of each function
g)
f ( x)
1
sec( x
2
4
h)
)
f ( x)
2csc( x
2
)
More on Trigonometric Identities
Ex. 1 Use the identities you have learned so far to verify the following:
a) sin( ) cos 2 ( ) sin( )
sin 3 ( )
b) (1 (cos x) 2 )(sec x) 2
(tan x) 2
100
AMHS Precalculus - Unit 7
c) 1 2sin
cos
(sin
cos ) 2
d) cot x
tan x
sec x csc x
Sum and Difference formulas for Sine and Cosine
sin(
sin(
) sin cos
) sin cos
cos sin
cos sin
cos(
cos(
) cos cos
) cos cos
sin sin
sin sin
Ex. 2 Use the sum and difference formulas to determine the value of the following trigonometric
functions.
a) sin(
6
b) cos(
3
)
4
7
)
12
Ex. 3 Verify the identity:
sin(t
2
) cos t
101
AMHS Precalculus - Unit 7
We use the sum and difference formula to derive the double angle formulas.
1.
sin(2 x) 2sin( x)cos( x)
2. cos(2 x) cos 2 ( x) sin 2 ( x)
Verification:
We can then use the Pythagorean identities to derive two other versions of the double angle formula for
cosine.
3.
cos(2 x) 1 2sin 2 ( x)
4.
cos(2 x)
2 cos 2 ( x) 1
Verification:
Power-Reducing formulas
If we solve for sin 2 ( x ) and cos 2 ( x) in #3 and #4 above, we get:
a) cos 2 ( x)
cos(2 x) 1
2
b) sin 2 ( x)
1 cos(2 x)
2
These formulas should be memorized and are very useful in integral calculus.
102
AMHS Precalculus - Unit 7
Ex. 4 If sin t
3
,
5
t
3
¸ find cos(2t ),sin(2t ) and tan(2t )
2
Ex. 5 Verify the identities
a)
cos x sin x sin 2x cos x cos 2x
b) cot x tan x
2cot 2x
Ex. 6 Find all the values of x in the interval [0, 2 ] that satisfy the given equation.
a) sin 2 x
sin x
b) (cos x) 2
3sin x 3 0
103
AMHS Precalculus - Unit 7
Ex. 7 Find all values of t that satisfy the given equation.
a)
c)
2cos t
2
(cos t ) 2 cos t
e) sin(3t )
0
3
0
1
2
d) 2sin 2 t sin t 1 0
f)
2
g) 3tan(2t ) 3
b) sin t
csc t
h) sin t
2
cos t
104
AMHS Precalculus - Unit 8
Unit 8: Inverse Trigonometric Functions
Inverse Trigonometric Functions
The Inverse Sine Function
In order for the sine function to have an inverse that is a function, we must first restrict its domain to
[
, ] so that it will be one-to-one and therefore have an inverse that is a function.
2 2
y sin( x)
Domain: [
y
, ]
2 2
Range:
sin 1 ( x)
Domain:
Range:
The range of the arcsine function can be visualized by:
The arcsine function ( arcsin ( x )), or inverse sine function ( sin 1 ( x) ), is defined by
y arcsin( x) iff x sin( y) where 1 x 1 and
2
y
2
.
In other words, the arcsine of the number x is the angle y where
2
y
2
whose sine is x .
105
AMHS Precalculus - Unit 8
Ex. 1 Find the exact value of the given expression.
a)
1
arcsin ( )
2
c)
sin 1 (-1)
b) sin 1 (
3
)
2
2
)
2
1
cos(arcsin( ))
2
d) arcsin (
e) arcsin(sin(
3
))
4
f)
The Inverse Cosine Function
In order for the cosine function to have an inverse that is a function, we must first restrict its domain to
[0, ] .
y cos( x)
Domain: [0, ]
y
Range:
cos 1 ( x)
Domain:
The range of the arccosine function can be visualized by:
Range:
106
AMHS Precalculus - Unit 8
The arccosine function ( arccos ( x )), or inverse cosine function ( cos 1 ( x ) ), is defined by
y arccos( x) iff x cos( y) where 1 x 1 and 0
.
y
In other words, the arccosine of the number x is the angle y where 0
y
whose cosine is x .
Ex. 2 Find the exact value of the given expression.
a)
arccos (
c)
cos 1 (-1)
1
)
2
e) arccos(cos(
b) cos 1 (
3
)
2
2
)
2
1
cos(arcsin( ))
3
d) arccos (
5
))
4
f)
The Inverse Tangent Function
In order for the Tangent function to have an inverse that is a function, we must first restrict its domain
to (
, ).
2 2
y tan( x)
Domain: (
y
, )
2 2
Range:
tan 1 ( x)
Domain:
Range:
107
AMHS Precalculus - Unit 8
The range of the arctangent function can be visualized by:
The arctangent function ( arctan ( x )), or inverse tangent function ( tan 1 ( x) ), is defined by
y arctan( x) iff x tan( y) where 1 x 1 and
2
y
2
.
In other words, the arctangent of the number x is the angle y where
2
Ex. 3 Find the exact value of the given expression.
a)
tan 1 (1)
b) arctan(
c)
arctan(tan )
7
d) sin(arctan( ))
Ex. 4 Write the given expression as an algebraic expression in x .
a) sin(tan 1 x)
3)
1
4
y
2
whose tangent is x .
108
AMHS Precalculus - Unit 8
“Algebraic” solutions to Trigonometric Equations
Solutions for basic Trigonometric equations.
1. cos( x)
c , ( 1 c 1)
Solve : cos x
x
cos 1 (c) 2 n and x
x
sin 1 (c) 2 n and x
cos 1 (c) 2 n
.6
Solve : 8cos x 1 0
2. sin( x)
c , ( 1 c 1)
Solve : sin x
Solve:
.75
3sin 2 x sin x 2 0
(
sin 1 (c)) 2 n
109
AMHS Precalculus - Unit 8
3.
tan( x) c
Solve: tan x
x
tan 1 (c)
n
3.6
Solve: sec2 x 5tan x
2
Angle of inclination
If L is a nonvertical line with angle of inclination
Ex. 1 Find the angle of inclination of a line of slope
(0
5
.
3
Ex. 2 Find the angle of inclination of a line of slope -2.
180 ), then tan = the slope of L .
110
AMHS Precalculus - Unit 8
Law of Sines and Law of Cosines – techniques for solving general triangles.
When we are given two angles and an included side (ASA), two angles and a non-included side (AAS), or
two sides and a non-included angle (SSA), we can find the remaining sides and angles using the Law of
Sines.
Law of Sines
sin
a
sin
b
sin
c
Ex1. A telephone pole makes an angle of 82 with the ground. The angle of elevation of the sun is 76 .
Find the length of the telephone pole if its shadow is 3.5m. (assume that the tilt of the pole is away
from the sun and in the same plane as the pole and the sun).
SSA – The ambiguous case. When given two sides and a non-included angle, there are three different
scenarios:
a) No triangle
b) One, unique triangle
c) Two different triangles (since you will be solving for an angle with SSA, see if another triangle is
possible by subtracting the acute angle found with arcsine from 180 )
111
AMHS Precalculus - Unit 8
Ex. 2 Solve the triangle: a
2, c 1,
50
Ex. 3 Given a triangle with a = 22 inches, b =12 inches and
Ex. 4 Solve the triangle: a
6, b 8,
35 .
= 35 , find the remaining sides and angles.
112
AMHS Precalculus - Unit 8
Law of Cosines
We use the law of cosines when we are given three sides (SSS) or two sides and an included angle (SAS).
a2
b2
c2
b2 c2 2bc cos
a 2 c 2 2ac cos
a 2 b 2 2ab cos
Ex. 5 Find all the missing angles of a triangle with sides a
8, b 19, c 14 .
Ex. 6 A ship travels 60 miles due east and then adjusts its course 15 northward. After traveling 80
miles in that direction, how far is the ship from its departure?
113
AMHS Precalculus - Unit 9
Unit 9: Sequences and Series
A sequence is an ordered list of numbers and is formally defined as a function whose domain is the set
of positive integers.
It is common to use subscript notation rather than the standard function notation.
For example, we could use a1 to represent the first term of the sequence, a2 to represent the second
term, a3 the third term and so forth where a n would represent the “nth term”
Note: Occasionally, it is convenient to begin a sequence with something other than a1 , such as a0 , in
which case the sequence would be a0 , a1 , a2 , a3 , … , an
2
,
an
1
,
an an 1 , … (note how an 1 is the term
before a n and so forth)
Ex. 1 List the first 5 terms of:
a)
an
b) bn
c)
cn
3 ( 1)n
n
1 2n
n2
2n 1
d) The recursively defined sequence an
1
an 5, a1
25
,
114
AMHS Precalculus - Unit 9
The symbol n! (read “ n factorial”) is defined as n! n (n 1) (n 2)
For example, 6! 6 5 4 3 2 1 720
Ex. 2 Simplify the ratio of factorials
a)
25!
23!
c)
(2n)!
(2n 2)!
b)
Pattern Recognition for sequences
Ex.3 Find a sequence an whose first five terms are
a)
b)
2 4 8 16 32
, , , , , ...
1 3 5 7 9
2 8
, ,
1 2
c) 1, x,
26 80 242
, ,
, ...
6 24 120
x 2 x3 x 4
, , , ...
2 6 24
8.1 1-85 e.o.o
(n 2)!
n!
4 3 2 1 where 0! 1 .
115
AMHS Precalculus - Unit 9
Series
Definition of a Series
Consider the infinite sequence a1 , a2 , a3 , … ai , …
1. The sum of the first n terms of the sequence is called a finite series or the partial sum of the
sequence and is denoted by
n
a1 a2
a3 ... an
ai
i 1
where i is called the index of summation, n is the upper limit of summation and 1 is the lower
limit of summation
2. The sum of all the terms of an infinite sequence is called an infinite series and is denoted by
a1 a2
a3 ... ai ...
ai
i 1
Ex.3 Find the sum:
5
3i
a)
i 1
6
(1 k 2 )
b)
k 3
8
(
c)
n 0
1
)
n!
116
AMHS Precalculus - Unit 9
3
d)
(
i 1
3
)
10i
Ex. 4 Use sigma notation to write the sum
1
3(1)
1
3(2)
1
3(3)
1
3(9)
8.1 87-117
Arithmetic Sequences and Partial Sums
What behavior do the first two sequences have that the third one does not?
1. 7, 11, 15, 19, ….
2. 2, -3, -8, -13, -18 …
3. 1, 4, 9, 16, …
A sequence is arithmetic when the difference between consecutive terms is constant. We call this
difference the common difference ( d ) where d
an
an 1 .
Ex.1 Find a formula for the nth term of the arithmetic sequence 7, 11, 15, 19, …. where a1 is the first
term.
The nth term of an arithmetic sequence has the form
an
a1 (n 1)d where d is the common difference and a1 is the first term.
Ex.2 Find a formula for the nth term of the arithmetic sequence whose common difference is 3 and
whose first term is 2. List the first five terms of this sequence.
117
AMHS Precalculus - Unit 9
Ex.3 The fourth term of an arithmetic sequence is 20, and the 13th term is 65. Write the first several
terms of this sequence.
Ex. 4 Find the ninth term of an arithmetic sequence whose first two terms are 2 and 9.
8.2 1-69 e.o.o.
The sum of a finite arithmetic sequence with n terms is given by:
Sn
n
(a1 an )
2
Note:
Ex.5 Find the sum: 1+3+5+7+9+11+13+15+17+19
Ex.6 Find the 150th partial sum of the sequence 5,16,27,38,49, ….
500
Ex. 7
(2n 8)
n 1
118
AMHS Precalculus - Unit 9
50
Ex. 8
(50 3n)
n 0
8.2 71-81, 84
Geometric Sequences and Series
List the first five terms of the geometric sequence an
2n
A sequence is geometric when the ratios of consecutive terms are constant. We call this constant the
common ratio(r) where r
an
.
an 1
Ex.1 Determine if the following sequences are geometric and if so, determine the common ratio.
a) 12, 36, 108, 324, …
b)
1 1
, ,
3 9
1 1
, ,
27 81
c) 1, 4, 9, 16, …..
1
, ...
243
119
AMHS Precalculus - Unit 9
The nth term of a geometric sequence has the form
an
a1r n 1 where r is the common ratio and a1 is the first term.
Ex. 2 Write the first five terms and the general term of the geometric sequence whose first term is
a1
3 and whose common ratio is r
2.
Ex. 3 Find the 15th term of the geometric sequence whose first term is 20 and whose common ratio is
1.05.
Ex. 4 Find a formula for the nth term of the following geometric sequence. What is the ninth term of the
sequence?
5, 15, 45…
Ex. 5 The fourth term of a geometric sequence is 125 and the 10th term is
(assume all terms are positive).
8.3 1-45 e.o.o.
125
. Find the 14th term
64
120
AMHS Precalculus - Unit 9
The nth partial sum of a geometric sequence with common ratio of r and first term of a1 is given by
n
a1r i
Sn
1
a1 (
i 1
1 rn
)
1 r
Note:
Ex. 6 Find the sum:
12
(4(.3) n )
a)
n 1
7
2n
b)
1
n 0
The Sum of an Infinite Geometric Series (or simply Geometric Series)
If | r | 1 , then the infinite geometric series has the sum
a1 (r i )
S
i 0
a1
1 r
If | r | 1 , then the series does not have a sum and diverges to infinity.
121
AMHS Precalculus - Unit 9
Ex. 7 Find the sum:
(4(.6) n )
a)
n 0
b) 3 + 0.3 + 0.03 + 0.003 + …
8.3 55-93 e.o.o
122
AMHS Precalculus - Unit 10
Unit 10: Exponential and Logarithmic Functions
Rational exponents
If b is a real number and n and m are positive and have no common factors, then
n
b m = m bn
( m b )n
Laws of exponents
a)
b)
c)
d)
e)
f)
g)
Ex. 1 Simplify
8 13
a) ( )
27
b) 9
5
2
Exponential Function
If b 0 and b 1, then an exponential function y f ( x) is a function of the form f ( x)
The number b is called the base and x is called the exponent.
bx .
123
AMHS Precalculus - Unit 10
Ex. 2 Graph each of the given functions
2 x and f ( x)
a)
f ( x)
b)
f ( x) 2
x
4x
1
( ) x and f ( x) 4
2
x
1
( )x
4
124
AMHS Precalculus - Unit 10
In general …
f ( x) b x
f ( x) b
Domain:
Range:
Intercept:
Horizontal Asymptote:
x
1
( )x
b
Domain:
Range:
Intercept:
Horizontal Asymptote:
Ex. 3 Sketch each of the given functions
a)
h( x )
3x
1
b) h( x)
5
x
2
125
AMHS Precalculus - Unit 10
The Natural Base e
Use you calculator to explore lim(1
n
1 n
) .
n
Conclusion:
Ex. 4 Sketch each of the given functions
a)
f ( x)
ex
c)
f ( x)
2 e
b) h( x )
x
e
x
For c) State the Domain and Range
126
AMHS Precalculus - Unit 10
Ex. 5 Solve
a)
2x
c)
4x
3
2
8x
1
b) 72( x
1)
23 x
2
d) 64 x 10(8 x ) 16
343
0
Compound Interest
The amount of money A(t ) at some time t (in years) in an investment with an initial value, or principle
of P with an annual interest rate of r (APR – given as a decimal), compounded n times a year is:
A(t )
r
P 1
n
nt
Ex. 6 Determine the value of a CD in the amount of $1000.00 that matures in 6 years and pays 5% per
year compounded
a) Annually
b) Monthly
c) Daily
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AMHS Precalculus - Unit 10
Continuously Compounded Interest.
If the interest is compounded continuously ( n
A(t )
), then the amount of money after t years is:
Pe rt
Ex. 7 Determine the amount in the CD from example 6 if the interest is compounded continuously.
Ex. 8 Which interest rate and compounding period gives the best return?
a) 8% compounded annually
b) 7.5% compounded semiannually
c) 7% compounded continuously
Ex. 9 What initial investment at 8.5 % compounded continuously for 7 years will accumulate to $50,000?
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AMHS Precalculus - Unit 10
Logarithmic Functions
Set up
Sketch f ( x)
2 x . Give the domain and range. Then find f 1 ( x) .
f 1 ( x) =
Domain:
Range:
Intercept:
V.A.:
Definition
For each positive number a
0 and each x in (0, ) , y log a x if and only if x a y .
x a y is the corresponding “exponential form” of the given “logarithmic form” y log a x .
Ex. 1 Evaluate each expression.
a) log10 1000
b) log10 0.1
c)
d) log 2 4
log 2 32
e) log 8 8
g)
log5 52
f)
log 3 1
h) 3log3 8
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AMHS Precalculus - Unit 10
Properties of the Logarithm function with base a .
a)
log a 1 0
c)
log a a x
x
b) log a a
1
d) aloga x
x
Ex. 2 On the same coordinate plane, sketch the following functions.
f ( x)
3x and g ( x)
log 3 x
1
f ( x) ( ) x and g ( x) log1/2 x
2
In general ….
g ( x)
log a x , a 1
Domain:
Range:
Intercept:
V.A.:
g ( x)
log a x , 0 a 1
Domain:
Range:
Intercept:
V.A.:
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AMHS Precalculus - Unit 10
Ex. 3 Sketch the following functions.
g ( x)
log 3 ( x 2)
g ( x) log1/2 x 1
The Natural Logarithm Function
The function defined by f ( x)
log e x
ln x and y ln x iff x
Ex. 4 On the same coordinate plane, sketch the following functions.
f ( x)
e x and g ( x) ln x
ey .
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AMHS Precalculus - Unit 10
Properties of the Logarithm function with base e .
a)
b)
c)
d)
Arithmetic Properties of Logarithms
For each positive number a
we have:
1 , each pair of positive real numbers U and V , and each real number n
Base a Logarithm
Natural Logarithm
a)
a)
b)
b)
c)
c)
Ex. 5 Evaluate each expression
a)
ln e4
c)
ln
b) eln 45
1
e
d) e
e) e3ln8
f)
Change-of-Base Formula
For a
0, a 0, x 0 ... log a x
log x
log a
ln x
ln a
Ex. 6 Use your calculator to evaluate log 6 13 .
(1/2)ln16
log 2 6 log 2 15 log 2 20
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AMHS Precalculus - Unit 10
Ex.7 Use the properties of logarithms to simplify each expression so that the ln y does not contain
products, quotients or powers.
a)
y
(2 x 1)(3 x 2)
4x 3
b)
y 64 x6 x 1 3 x2 2
Solving Exponential and Logarithmic Equations
Ex. 8 Solve each of the given equations
a)
ex
c)
2x 13
83
e) ln( x 1) ln( x 3) 1
b) 4e2 x
7
d) 2ln(3x)
f)
6
ln( x 2) ln(2x 3) 2ln x
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AMHS Precalculus - Unit 10
g)
log 2 ( x 3)
i)
ex x2 ex x 6ex
Ex. 9 Given the function f ( x)
function.
h) 2x
4
j)
e3 x
1
1
e5 x
3
x ln x x 0
5 , Find f 1 ( x) and state the domain and range of the inverse
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AMHS Precalculus - Unit 10
Exponential Growth and Decay
In one model of a growing (or decaying) population, it is assumed that the rate of growth (or decay) of
the population is proportional to the number present at time t (rate of growth = kP(t ) ). Using calculus,
it can be shown that this assumption gives rise to:
P(t )
P0ekt where k is the rate of growth ( k 0 ) or decay ( k 0 ).
Ex.1 The number of a certain species of fish is given by n(t ) 12e0.012t where t is measured in years and
n(t ) is measured in millions.
a) What is the relative growth rate of the population?
b) What will the fish population be after 15 years?
Ex.2 A bacteria culture starts with 500 bacteria and 5 hours later has 4000 bacteria. The population
grows exponentially.
a) Find a function for the number of bacteria after t hours.
b) Find the number of bacteria that will be present after 6 hours.
c) When will the population reach 15000?
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AMHS Precalculus - Unit 10
Ex. 3 A culture of cells is observed to triple in size in 2 days. How large will the culture be in 5 days if the
population grows exponentially?
Ex. 4 Carbon-14, one of the three isotopes of carbon, has a half-life of 5730 years. If 10 grams were
present originally, how much will be left after 2000 years? When will there be 2 grams left?
Ex. 5 On September 19th, 1991, the remains of a prehistoric man were found encased in ice near the
border of Italy and Switzerland. 52.4% of the original carbon 14 remained at the time of the discovery.
Estimate the age of the Ice Man.
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AMHS Precalculus - Unit 10
Ex. 6 The radioactive isotope strontium 90 has a half-life of 29.1 years.
a) How much strontium 90 will remain after 20 years from an initial amount of 300 kilograms?
b) How long will it take for 80% of the original amount to decay?