Objective 9.1A
Vocabulary to Review
completing the square [8.2A]
domain [3.2A]
range [3.2A]
quadratic function [5.2A]
New Vocabulary
parabola
input
output
input/output table
vertex of a parabola
axis of symmetry of a parabola
New Formulas

b
 b 
, f   
2
a
 2a  

b
equation of the axis of symmetry of a parabola: x  
2a
coordinates of the vertex of a parabola:  
Discuss the Concepts
1. Describe how to find the vertex of a parabola.
2. Explain why the vertex of a parabola lies on the axis of symmetry.
3. Explain how, if the coordinates of the vertex of a parabola are known, the range of the quadratic function
can be determined.
Concept Check
1. Draw the graphs of two different quadratic functions that have the line x = 3 as their line of symmetry.
Name the coordinates of the vertex. State the domain and range of the function.
2. Draw the graphs of two different quadratic functions that have the line x = -2 as their line of symmetry.
Name the coordinates of the vertex. State the domain and range of the function.
Optional Student Activity
1. The axis of symmetry of a parabola is the line x = 0. The point (-2, -3) lies on the parabola. Use the
symmetry of a parabola to find a second point on the graph. (2, -3)
2. The axis of symmetry of a parabola is the line x = 1. The point (3, 0) lies on the parabola. Use the
symmetry of a parabola to find a second point on the graph. (-1, 0)
3. The axis of symmetry of a parabola is the line x = 2. The point (4, -4) lies on the parabola. Use the
symmetry of a parabola to find a second point on the graph. (0, -4)
4. The axis of symmetry of a parabola is the line x = -1. The point (1, -1) lies on the parabola. Use the
symmetry of a parabola to find a second point on the graph.
(-3, -1)
Objective 9.1B
Vocabulary to Review
intercepts of a graph [3.3C]
discriminant [8.3A]
New Vocabulary
zero of a function
Discuss the Concepts
1. How are the x-intercepts of a graph and the real zeros of a function related?
2. When f(x) = 0, values of x for f(x) = x2 + 2x + 3 are complex numbers. What does this say about the xintercepts of the graph of f?
Concept Check
The concepts of the x-intercepts of the graph of a function, the zeros of a function, and the solutions of an
equation are different ways of discussing the same number. Use the term intercepts, zeros, or solutions to
complete the sentence.
1. If we are discussing graphing, then the _____ are our focus.
intercepts
2. If we are discussing functions, then the _____ are our focus.
zeros
3. If we are discussing equations, then the _____ are our focus.
solutions
Optional Student Activity
2
1. What effect does increasing the coefficient of x 2 have on the graph of y  ax  bx  c, a > 0?
The graph becomes thinner.
2
2. What effect does decreasing the coefficient of x 2 have on the graph of y  ax  bx  c, a > 0?
The graph becomes wider.
2
3. What effect does increasing the constant term have on the graph of y  ax  bx  c, a ≠ 0?
The graph is higher on the rectangular coordinate system.
2
4. What effect does decreasing the constant term have on the graph of y  ax  bx  c, a ≠ 0?
The graph is lower on the rectangular coordinate system.
Objective 9.1C
Vocabulary to Review
vertex [9.1A]
New Vocabulary
minimum value of a function
maximum value of a function
Discuss the Concepts
Determine whether the following statements are always true, sometimes true, or never true.
1. The axis of symmetry of a parabola passes through the vertex.
2. A parabola has two x-intercepts.
Sometimes true
3. A quadratic function has a minimum value.
Sometimes true
Concept Check
1. Which of the following functions has the highest minimum value?
a.
f  x   3x 2  4
Always true
b.
f  x   x 2  10 x  20
c.
f  x   x2  6x
a
2. What is the value of k if the vertex of the parabola given by the equation y = x2 + 4x + k is a point on the
x-axis? 4
Optional Student Activity
An equation of the form y  ax2  bx  c can be written in the form y  a  x  h   k, where h and k are
2
constants.
1. Find the vertex of each equation given below. Then use the process of completing the square to rewrite
the equation in the form y  a  x  h   k. .
2
2
a. y  x  4 x  7
(2, 3); y = (x – 2)2 + 3
2
b. y  x  2 x  2
(1, -3); y = (x – 1)2 – 3
2
c. y  x  6 x  5
(3, -4); y = (x – 3)2 – 4
2
2
d. y  x  x  2
1 7

 1 7
 ,  ; y   x  2  4


 2 4
2. Based on your answers to Exercise 1, for a parabola of the form y  a  x  h   k, what does the
2
ordered pair  h,k  represent?
The ordered pair (h, k) represents the vertex of the parabola.
3. Use the equation y  a  x  h   k to find the equation of the parabola that has its vertex at 1, 2  and
2
passes through the point  2,5 .
y = 3(x – 1)2 + 2
4. Use the equation y  a  x  h   k to find the equation of the parabola that has its vertex at (0, -3) and
2
passes through the point (3, -2).
y
1 2
x 3
9
Objective 9.1D
Concept Check
You might have students revisit the problem on page 500, but this time solve the equation 22 = L + W for L
instead of W. It will require them to go through the steps of solving the problem, and it will show that the
answer is the same whether the equation is solved for W or for L.
Optional Student Activity
1. The height s, in feet, of a ball thrown upward at an initial speed of 80 ft/s from a platform 50 ft high is
given by the function (s)t = -16t2 + 80t + 50, 50, where t is the time in seconds. Find the maximum height
above the ground that the ball will attain. 150 ft
2. Karen is throwing an orange to her brother Saul, who is standing on the balcony of their home. The
height h, in feet, of the orange above the ground t seconds after it is thrown is given by h(t) = -16t2 + 32t
+ 4. If Saul’s outstretched arms are 18 ft above the ground, will the orange ever be high enough so that
he can catch it? Yes
3. The fuel efficiency of an average car is given by the equation E(v) = -0.018v2 + 1.476v + 3.4, where E is
the fuel efficiency in miles per gallon and v is the speed of the car in miles per hour. What speed will
yield the maximum fuel efficiency? What is the maximum fuel efficiency?
41 mph; 33.658 mi/gal
Answers to Writing Exercises
1. A quadratic function is a function of the form f(x) = ax2 + bx + c, a ≠ 0.
2. The graph of a parabola resembles a “cup” shape (or bowl shape or U shape) and opens either up or
down.
3. The vertex of a parabola is the point with the smallest y-coordinate or the largest y-coordinate. When a >
0, the parabola opens up and the vertex of the parabola is the point with the smallest y-coordinate.
When a < 0, the parabola opens down and the vertex is the point with the largest y-coordinate.
4. The axis of symmetry of the graph of a parabola is the vertical line that passes through the vertex of the
parabola and is parallel to the y-axis.
Answers to Writing Exercises
30. a. An x-intercept of the graph of a parabola is a point at which the graph crosses the x-axis. It is a point
at which y = 0.
b. The graph of a parabola can have two, one, or no x-intercepts.
31. a. A y-intercept of the graph of a parabola is a point at which the graph crosses the y-axis. It is a point
at which x = 0.
b. The graph of a parabola has one y-intercept.
Answers to Writing Exercises
70. The minimum or maximum value of a quadratic function is the value of the function at the vertex of the
graph of the function; it is the y-coordinate of the vertex. A quadratic function has a minimum value
when a > 0 and a maximum value when a < 0 in the equation f(x) = ax2 + bx + c.
71. To find the minimum or maximum value of a quadratic function, find the x-coordinate of the vertex.
Then evaluate the function at the value of this x-coordinate.
Objective 9.2A
Vocabulary to Review
polynomial function [5.2A]
cubic function [5.2A]
domain [3.2A]
range [3.2A]
set-builder notation [1.1B]
interval notation [1.1C]
New Vocabulary
absolute-value function
radical function
vertical-line test
Optional Student Activity
Consider the graph of y = x2 + 4x + 3 shown on page 494. As a point on the graph moves from left to right,
the values of y are decreasing for x < -2. For x > -2, the values of y are increasing. The function is said to
be decreasing for x < -2 and increasing for x > -2.
For each of the following, determine the intervals on which the graph of the function is increasing and the
interval on which the graph of the function is decreasing.
1. f(x) = x2 + 4x – 1
Decreasing: (--2); Increasing: (-2, -)
Increasing: (-2); Decreasing: (2, )
2. f(x) = -x2 + 4x – 3
3. f(x) = x3 – 2
Increasing: (-)
4. f(x) = -x3 + 3x2 – 3x – 1
5. f(x) = |x – 4|
Increasing: (-0); Decreasing: (0, )
6. f(x) = -3|x| – 1
7. f  x  
Decreasing: (-)
Decreasing: (-4); Increasing: (4, )
x3
Increasing: (-3, )
8. f  x   2 x  1
9. f(x) = 2x – 3
Decreasing: (1, )
Increasing: (-)
10. f(x) = 1 – 3x
Decreasing: (-)
Optional Student Activity
Graph y 
x  2 and y 2  x  2 . Discuss the similarities and differences between the two graphs.
The range of y 
x  2 is { y | y  0} . The range of y 2  x  2 is { y | y  real numbers}. The graphs are
2
the same for y ≥ 0 but the graph of y  x  2 includes the reflection of the graph of y 
axis.
x  2 over the x-
Objective 9.3A
New Formulas
operations on functions:
(f + g)(x) = f(x) + g(x)
(f – g)(x) = f(x) – g(x)
(f ∙ g)(x) = f(x) ∙ g(x)
 f 
f ( x)
, g ( x)  0
  ( x) 
g
g
( x)
 
Concept Check
1. Given f  x   4 x1/ 2 and g  x   2 x1/ 2 , find a. the sum of the functions, b. the difference of the
functions, and c. the domains of the sum and difference.
a. 2x1/2
b. 6x1/2
c. The domain of the sum is {x| x ≥ 0}; the domain of the difference is {x| x ≥ 0}.
2. Given f  x   5x and g  x   x1/ 4 , find a. the product of the functions, b. the quotient of the functions,
and c. the domains of the product and quotient.
a. 5x5/4
b. 5x3/4
c. The domain of the product is {x| x ≥ 0}; the domain of the quotient is {x| x > 0}.
Optional Student Activity
The values of some functions may depend on several variables. For instance, the perimeter of a rectangle
depends on the length L and the width W. We can write this as P  L,W   2L  2W . To evaluate this
function, we need to be given the values of L and W. For example, to find the perimeter of a rectangle
whose length is 5 ft and whose width is 3 ft, evaluate P  L,W  when L  5 and W  3.
P  L,W   2 L  2W
P  5,3  2  5   2  3
 10  6  16
The perimeter is 16 ft.
1. Evaluate f  a, b   2a  3b when a  3 and b  4.
2. Evaluate R  s, t   2st  t 2 when s = -1 and t  2.
18
28
3. Although we normally do not think of addition as a function, it is a function of two variables. If we define
the function Add as Add  a, b   a  b, find Add  3,7  .
10
4. Write the area of a triangle as a function of two variables.
A(b, h) 
1
bh
2
5. Write the length of the hypotenuse of a right triangle as a function of two variables.
H  a, b   a 2  b2
6. Give an example of a function whose value depends on more than one variable.
Answers will vary. One example is the calculation of simple interest (I) as a function of principal, interest
rate, and time: I(p, r, t) = prt.
Objective 9.3B
New Vocabulary
composition of two functions
composite function
New Symbols
 f + g  x  
f  g  x  
Concept Check
Sets R, S, and T are shown in the figure below, along with functions f and g.
1. Evaluate f  4  .
16
2. Evaluate g  9  .
5
3. Evaluate  g
f  2 . 0
4. Evaluate  g
f 1 .
-3
5. Write a rule for f.
The function f squares an element in set R.
6. Write a rule for g.
The function g subtracts 4 from an element in set S.
7. Write a rule for g f . The composite of functions f and g first squares an element in set R and then
subtracts 4 from the result.
8. Create your own two functions f and g, diagram them as shown below, and find the composition of at
least two elements of g f .
1
2
3
4
5
1
4
9
16
25
-3
0
5
12
21
R
S
T
Optional Student Activity
It is sometimes possible to “decompose” a composite function f + g into two functions f and g. In the
example below, we use the fact that
f
g  x   f  g  x   .
Example:
Determine two functions f and g such that
f
g  x   8 x  3.
Solution:
Let g  x  
x and f  x   8x - 3.
Check:
f
f
g  x   f  g  x   
 x 8
x 3
1. Determine two functions f and g such that
Let g(x) =
x1/4 and
f
g  x   5x1/ 4  6.
f
g  x   2x3  7.
f
g  x   x4  x2  9.
f(x) = 5x + 6.
2. Determine two functions f and g such that
Let g(x) = x3 and f(x) = -2x – 7.
3. Determine two functions f and g such that
Let g(x) = x2 and f(x) = x2 + x + 9.
Answers to Writing Exercises
22. f  g  3   means to evaluate the function f at the value of the function g when g is evaluated at 3.
23. The expression
f
g  2 means to evaluate the function f at g(-2).
Objective 9.4A
Vocabulary to Review
function
[3.2A]
New Vocabulary
one-to-one function
horizontal-line test
Discuss the Concepts
1. How can you determine whether a graph is the graph of a 1–1 function?
2. Why is a linear function always a 1–1 function?
3. Why is a quadratic function never a 1–1 function?
Concept Check
Which of the following functions are 1–1 functions?
a. f  x   0.5 x
Yes
b. f(x) = -3x + 1 Yes
c. f(x) = x2 + 3x – 4
No
d. f(x) = |x| + 2 No
e. f(x) = 2x3 Yes
f. f(x) = -x3 + 4x2 – 4x + 1
No
Optional Student Activity
The greatest-integer function is defined by the equation f ( x)   x  , where the symbol  x  means the
greatest integer less than or equal to x. Here are some examples.
4.2  4
5.9999  5
   3
2.3  3
2  2
In each of these examples, the value of the function is an integer that is less than or equal to the number in
the symbol   .
Graph the greatest-integer function. Explain why the greatest-integer function is sometimes called a step
function. Is the greatest-integer function a 1–1 function? Why or why not?
The greatest-integer function is sometimes called a step function because its graph resembles steps. The
greatest-integer function is not a 1–1 function because it does not pass the horizontal-line test.
Objective 9.4B
Vocabulary to Review
domain [3.2A]
range [3.2A]
New Vocabulary
inverse of a function
New Symbols
f -1
New Properties
Composition of Inverse Functions Property:
f -1 [f(x)] = x and f[f -1(x)] = x
Discuss the Concepts
Determine whether the statement is always true, sometimes true, or never true.
1. A function has an inverse if and only if it is a 1–1 function.
Always true
2. The inverse of the function {(2, 3), (4, 5), (6, 3)} is the function {(3, 2), (5, 4), (3, 6)}.
3. The inverse of a function is a relation.
Always true
4. The inverse of a function is a function.
Sometimes true
Never true
Concept Check
One length of an Olympic swimming pool is 50 m; therefore, swimming one lap is equal to swimming 100 m.
We can refer to the distance swum as a function of the number of laps completed. Some of the ordered
pairs of this function are {(0, 0), (1, 100), (2, 200), (3, 300), (4, 400)}. Provide some ordered pairs of the
inverse function and state what the inverse function represents.
The equation that describes the distance swum as a function of the number of laps completed is f(x) =
100x. Write the inverse function. Some ordered pairs of the inverse function are {(0, 0), (100, 1), (200, 2),
(300, 3), (400, 4)}. This function gives the number of laps completed as a function of the distance swum.
1
The inverse function is f ( x ) 
x
.
100
Optional Student Activity
1. To convert square feet to acres, we can use the equation A 
s
, where A is the number of acres
43,560
and s is the number of square feet.
a. Write an equation for the inverse function. s = 43,560A
b. What does the inverse function represent?
It represents the equation that converts acres to square feet.
2. On December 2, 2003, the exchange rate for converting euros to U.S. dollars was given by the equation
d = 1.21E where d is the number of U.S. dollars and E is the number of euros.
a. Write an equation for the inverse function. Write the coefficient as a decimal. Round to the nearest
thousandth.
E = 0.826d
b. What does the inverse function represent?
It represents the exchange rate on December 2, 2003, for converting U.S. dollars to euros.
Answers to Writing Exercises
1. A function is a 1–1 function if, for any a and b in the domain of f, f(a) = f(b) implies a = b.
Here is an alternative definition: A function is a set of ordered pairs in which no two ordered pairs that
have the same first coordinate have different second coordinates. This means that given any x, there is
only one y that can be paired with that x. A 1–1 function satisfies the additional condition that given any
y, there is only one x that can be paired with that y.
2. The horizontal-line test is used to determine whether the graph of a function is the graph of a 1–1
function.The horizontal-line test states that the graph of a function represents the graph of a 1–1
function if any horizontal line intersects the graph at no more than one point.
Answers to Writing Exercises
15. The coordinates of each ordered pair of the inverse of a function are in the reverse order of the
coordinates of the ordered pairs of the original function. For example, if (-1, 5) is an ordered pair of the
original function, then (5, -1) is an ordered pair of the inverse of a function.
16. Not all functions have an inverse function because sometimes reversing the coordinates of the ordered
pairs results in a set of ordered pairs that does not represent a function. For example, consider the
constant function f(x) = 5. Some ordered pairs of this function are (0, 5), (-3, 5), and (2, 5). Reversing
the coordinates of these ordered pairs gives (5, 0), (5, -3), and (5, 2). These ordered pairs do not
satisfy the conditions of a function because they are ordered pairs with the same first coordinate and
different second coordinates.
Answers to Writing Exercises
63. No, the inverse of the grading scale is not a function because each grade is paired with more than one
score.
64. No, the inverse of the first-class postage rates is not a function because each cost is paired with more
than one weight.
71. A constant function is defined as y = b where b is a constant. The inverse of this function would be x =
a, where a is a constant. This is not a function. (See also the answer to Writing Exercise 16 on page
528.)
72. Yes, all functions given by f(x) = mx + b, m ≠ 0 are 1–1 functions. These are linear functions, and all
linear functions of this form pass the horizontal-line test and the vertical-line test.
Answers to Projects and Group Activities: Finding the Maximum or
Minimum of a Function Using a Graphing Calculator
1. Maximum value at 4.25
2. Minimum value at -6
3. Maximum value at 3.744716
4. Minimum value at -7.229744
5. 100 lenses
6. a. 48 years old
b. $148,000
Answers to Projects and Group Activities: Business Applications of
Maximum and Minimum Values of Quadratic Functions
1. a. 625 DVDs
b. $74,125
2. $3050; $15