Deck 6: Transformations of Functions and Their Graphs

Question 1

If the graph of the line $y=2 x+1$ is reflected about the $x$ -axis, the resulting line has the equation Y=------------------x + ------------

Answer 1

Part A: -2
Part B: -1

Question 2

An odd function is increasing and concave down in the first quadrant. How does the function behave in the third quadrant?

A) It is decreasing and concave up
B) It is increasing and concave down
C) It is increasing and concave up
D) It is decreasing and concave down

Answer 2

It is increasing and concave up

Question 3

The function $f(x)$ has odd symmetry and the function $g(x)$ has even symmetry. Is the function $h(x)=f(x) * g(x)$ odd, even, or neither?

Answer 3

odd

Question 4

The function $f(x)$ has odd symmetry and the function $g(x)$ has even symmetry. Is the function $h(x)=2 g(x+1)$ odd, even, or neither?

Question 5

The function $f(x)$ has odd symmetry and the function $g(x)$ has even symmetry. Is the function $h(x)=2 f(x)$ odd, even, or neither?

Question 6

Some of the values of $f(x)$ are given in the following table. If $f(x)$ is an odd function, what is $f(-3)$ ?
$\begin{array}{cccccc}x & 1 & 2 & 3 & 4 & 5 \\f(x) & 1 & 0 & -1 & 0 & 3\end{array}$

Question 7

The following table gives values for the functions $f$ and $g$ .

Which of the following statements are true?

A) $f$ is symmetric about the $x$ -axis.
B) $g$ is symmetric about the $x$ -axis.
C) $f$ is symmetric about the $y$ -axis.
D) $g$ is symmetric about the $y$ -axis.
E) $f$ is symmetric about the origin.
F) $g$ is symmetric about the origin.

Question 8

The following table gives some values for the function $h(x)$ . If $h(x)$ has even symmetry, what is $h(2)$ ?
$\begin{array}{ccccc}x & -3 & -2 & -1 & 0 \\h(x) & 3 & -5 & 6 & 0\end{array}$

Question 9

The graph of $f(x)$ contains the point $(1,-3)$ . What point must lie on the reflected graph if the graph is reflected about the $x$ -axis?

Question 10

The graph of $P(t)$ contains the point $(-3,-2)$ . What is the corresponding point on the graph if $P(t)$ is an even function?

Question 11

Given the following table of values for $f(x)$ , find $f(x+3)$ when $x=-3$ . If you do not have sufficient information, enter "cannot determine".

Question 12

How are the graphs of $y=g(x)=e^{7 x}$ and $y=g(-x)$ related?

A) The graph of $g(-x)$ is a reflection of the graph of $g(x)$ across the $x$ -axis.
B) The graph of $g(-x)$ is a reflection of the graph of $g(x)$ across the $y$ -axis.
C) The graph of $g(-x)$ is a reflection of the graph of $g(x)$ across the line $y=x$ .
D) The graph of $g(-x)$ is a reflection of the graph of $g(x)$ across the line $y=-x$ .
E) None of the above.

Question 13

What is the formula for $y=m(-n)$ if $m(n)=n^{2}-n+4$ ?

A) $y=-n^{2}-n-4$
B) $y=n^{2}-n+4$
C) $y=n^{2}+n+4$
D) $y=-n^{2}+n-4$

Question 14

Is the function $f(x)=x^{6}-7 x^{2}+6$ odd, even, or neither?

Question 15

If the following is the graph of $f(x)$ .

Sketch the graph of $-f(x)$ .

Question 16

The following is the graph of $f(x)$ .

Sketch the graph of $f(-x)$ .

Question 17

The following is the graph of $f(x)$ .

Sketch the graph of $-f(x)-2$ .

Question 18

The following is the graph of $f(x)$ .

Sketch the graph of $f(-x)+1$ .

Question 19

Which of the following functions are even?

A) $f(x)=3(x-5)^{2}$
B) $g(x)=3 x^{2}+75$
C) $h(x)=\frac{(x-3)^{2}}{5 x^{3}}$
D) $m(x)=\frac{3 x^{3}-5 x}{4 x^{3}}$

Question 20

Which of the following functions are odd?

A) $f(x)=2(x-5)^{3}$
B) $g(x)=2 x^{3}+50$
C) $h(x)=\frac{2 x^{2}}{5 x^{3}}$
D) $m(x)=|2 x|$

Question 21

If the graph of $f(x)$ is:

Then the graph:

Represents:

A) $f(-x)$
B) $-f(x)$
C) Neither $f(-x)$ or $-f(x)$
D) Both $f(-x)$ and $-f(x)$

Question 22

If the graph of $f(x)$ is:

Then the graph:

Represents:

A) $f(-x)$
B) $-f(x)$
C) Neither $f(-x)$ or $-f(x)$
D) Both $f(-x)$ and $-f(x)$

Question 23

If the graph of $f(x)$ is:

Then the graph:

Represents:

A) $f(-x)$
B) $-f(x)$
C) Neither $f(-x)$ or $-f(x)$
D) Both $f(-x)$ and $-f(x)$

Question 24

The following figure shows the graph of $f(x)$ and the graphs of several multiples of $f(x)$ .

Suppose that
$A$ is the graph of $y=a \cdot f(x)$ ,
$B$ is the graph of $y=b \cdot f(x)$ ,
$C$ is the graph of $y=c \cdot f(x)$ ,
$D$ is the graph of $y=d \cdot f(x)$ .
Which of the following statements are true?

A) $c<0$
B)
C) $d<1$

Question 25

Below is a graph of the function $f(x)$ . If you were to sketch a graph of $y=-6 f(x)$ , what point would the point $(0.6,-0.4)$ correspond to?

Question 26

The following table gives values for $c(x)$ , the cost per foot of drilling a well at a depth of $x$ feet. The prices given are for wells drilled on a certain ranch in Texas.

Estimate the cost of drilling a well that is 40 feet deep.

Question 27

The following table gives values for $c(x)$ , the cost per foot of drilling a well at a depth of $x$ feet. The prices given are for wells drilled on a certain ranch in Texas.

The ranch sits above a dense layer of bedrock. At what depth do you estimate that the bedrock layer begins?

A) Between 40 and 60 feet.
B) Between 100 and 120 feet.
C) Between 60 and 80 feet.
D) Between 80 and 100 feet.

Question 28

Let $f(x)=-3 x^{2}+3$ . Write a formula for the following transformations of $f$ .

A) $g(x)=2 f(x)$
B) $h(x)=3 f(x-1)$
C) $m(x)=-f(x)+4$

Question 29

The amount of money in your bank account is given by $V(t)$ . Which of the following formulas matches the description "$150 more than I have in my bank account"?

A) $V(t)+150$
B) $V(t+150)$
C) $150 V(t) V(t+150)$
D) $V(t-150)$

Question 30

The amount of money in your bank account is given by $V(t)$ . Which of the following formulas matches the description "the amount of money I will have in my bank account 30 years from now"?

A) $V(t)+30$
B) $V(t+30)$
C) $30 V(t)$
D) $V(t-30)$

Question 31

The domain of $f(x)$ is $-2 \leq x \leq 6$ and the range is $-4 \leq f(x) \leq 5$ . What is the domain and range of $-2 f(x)$ ?

Question 32

The graph of $g(x)$ is the graph of $f(x)$ after it has been vertically stretched or shrunk. The point $(3,12)$ lies on the graph of $f(x)$ . The corresponding point on the graph of $g(x)$ is $(3,24)$ . What is a possible formula for $g(x)$ in terms of $f(x)$ ?

Question 33

The graphs of $f$ and $g$ are shown below.

Which of the following gives a formula for $g(x)$ in terms of $f(x)$ ?

A) $g(x)=\frac{1}{2} f(x)$
B) $g(x)=2 f(x)$
C) $g(x)=f(x-2)$
D) $g(x)=f(x)+2$

Question 34

The graph of $f(x)$ contains the point $(4,-3)$ . What corresponding point must be on the graph of $g(x)=-3 f(x-10)$ ?

Question 35

The graph of $f(x)$ contains the point $(1,-2)$ . Which of the following points must be on the graph of $g(x)=2 f(x-11)$ ?

A) $(12,-4)$
B) $(1,-4)$
C) $(-10,-4)$
D) $(1,-15)$

Question 36

The function $g(x)$ is obtained from $f(x)$ by a single transformation. Use the tables below to find a formula for $g(x)$ in terms of $f(x)$ .

Question 37

If $f(x)$ is an odd function, determine whether $g(x)=-6 f(x)$ is odd, even, or neither. State "cannot tell" if there is not enough information to determine.

Question 38

If $f(x)$ is an even function, determine whether $g(x)=13 f(x-2)$ is odd, even, or neither. State "cannot tell" if there is not enough information to determine.

Question 39

The domain of $f(x)$ is $-8 \leq x \leq 6$ and the range is $3 \leq y \leq 14$ . If $g(x)=5 f(x-4)$ , what is the domain and range of $g(x)$ ?

Question 40

Suppose the range of $f(x)$ is $3 \leq y \leq 14$ . Let $g(x)=5 f(x-4)$ .
There exists some $p$ such that $g(p)=4$ .

Question 41

Graph $g(x)=f(x+3)$ if $f(x)=x^{3}$ .

Question 42

Graph $g(x)=-2 f(x)$ if $f(x)=e^{x}$ .

Question 43

The average rate of change of $f(x)$ over $1 \leq x \leq 7$ is -3 . Find the average rate of change of $y=4 f(x)+14$ over $1 \leq x \leq 7$ .

Question 44

If the graph of $f(x)$ is:

Then the graph:

Represents:

A) $f(2 x)$
B) $f(x+2)$
C) $2 f(x)$
D) $f(x-2)$

Question 45

If the graph of $f(x)$ is:

Then the graph:

Represents:

A) $f(4 x)$
B) $f(x+4)$
C) $4 f(x)$
D) $f(x-4)$

Question 46

Let 5 and -4 be the zeros of the quadratic function $f(x)$ . What are the zeros of the function $g(x)=f(5 x)$ ?

A) 5
B) -4
C) 1
D) -0.8
E) 25
F) -20

Question 47

Let $y=p(x)$ be defined by the following graph, and let $y_{0}=p\left(x_{0}\right)$ and $y_{1}=p\left(x_{1}\right)$ .

Arrange the following values in order from least to greatest by placing a " 1 " next to the least, a "2" next to the next least, etc.
A. $p\left(x_{0}\right)$
B. $-p\left(x_{0}\right)$
C. $p\left(2 x_{0}\right)$

Question 48

The graph of $h(x)$ contains the point $(4,-1)$ . What is the corresponding point on the graph of $y=h(-2 x)$ ?

Question 49

Given $f(x)=2^{x}+x^{2}$ , the following figure shows the graphs of $f(x), f(-0.5 x)$ , and $f(2 x)$ . Which curve is the graph of $f(-0.5 x)$ ?

Question 50

Using the following table, find $f(2 x)$ for $x=-0.5$ .

Question 51

Let $f(x)$ be the height of a box in inches. Find a formula for the height $g(t)$ in feet?

Question 52

The graphs of $f$ and $g$ are shown at the right.

Which of the following gives a formula for $g(x)$ in terms of $f(x)$ ?

A) $g(x)=f\left(\frac{x}{2}\right)$
B) $g(x)=f(2 x)$
C) $g(x)=\frac{1}{2} f(x)$
D) $g(x)=2 f(x)$

Question 53

Suppose $f(x)$ is an even function. Determine whether $g(x)=f\left(\frac{x}{4}\right)$ is odd, even, or neither. Write "cannot tell" if there is not enough information to determine.

Question 54

Suppose $f(x)$ is an odd function. Determine whether $g(x)=f(2 x)+16$ is odd, even, or neither. Write "cannot tell" if there is not enough information to determine.

Question 55

Let $V$ represent the volume of a box with square base of side length $x$ and height 8 . By what percent does the volume increase if the side length is increased by $15 \%$ ?

Question 56

Let $V(x)$ represent the volume in cubic feet of a box with square base of side length $x$ feet and height 7 feet. Let $T(x)$ represent the volume in cubic inches of a box with square base of side length $x$ inches and height 84 inches. Which of the following is the correct formula for $T(x)$ in terms of $V(x)$ ?

A) $T(m)=12 \cdot V(144 x)$
B) $T(m)=V(12 x)$
C) $T(m)=12 V(x)$
D) $T(m)=144 V(x)$

Question 57

The domain of $f(x)$ is $-16 \leq x \leq 16$ and the range is $0 \leq f(x) \leq 11$ . What is the domain and range of $g(x)=f(-4 x)$ ?

Question 58

Describe the effect of the transformation $f\left(\frac{x}{7}\right)+6$ on the graph of $f(x)$ .

Question 59

Let $V=f(m)$ be the value of an investment $m$ months after the account was open. Write an expression that represents the value of the investment increased by $25 \%$ .

Question 60

Which of the following describes the effect of the transformation $g(x)=9 f\left(\frac{x}{9}\right)$ on the graph of the function $f$ ?

A) The graph is stretched vertically by a factor of 9 and stretched horizontally by a factor of 9.
B) The graph is stretched vertically by a factor of 9 and shrunk horizontally by a factor of $\frac{1}{9}$ .
C) The graph is shrunk vertically by a factor of $\frac{1}{9}$ and stretched horizontally by a factor of 9 .
D) The graph is shrunk vertically by a factor of $\frac{1}{9}$ and shrunk horizontally by a factor of $\frac{1}{9}$ .

Question 61

Which of the following describes the effect of the transformation $g(x)=\frac{f(7 x)}{7}$ on the graph of the function $f$ ?

A) The graph is stretched vertically by a factor of 7 and stretched horizontally by a factor of 7.
B) The graph is stretched vertically by a factor of 7 and compressed horizontally by a factor of $\frac{1}{7}$ .
C) The graph is compressed vertically by a factor of $\frac{1}{7}$ and stretched horizontally by a factor of 7 .
D) The graph is compressed vertically by a factor of $\frac{1}{7}$ and compressed horizontally by a factor of $\frac{1}{7}$ .

Question 62

The point $(-4,5)$ lies on the graph of $f$ . What point must lie on the graph of $3 f\left(\frac{x}{5}\right)$ ?

Question 63

The point $(2,-8)$ lies on the graph of $f$ . If the graph of $f$ is compressed vertically by a factor of $\frac{1}{7}$ and stretched horizontally by a factor of 3 , what point must lie on the transformed graph?

Question 64

The point $(4,-2)$ lies on the graph of $f$ . If the graph of $f$ is stretched vertically by a factor of 3 and compressed horizontally by a factor of $\frac{1}{5}$ , what point must lie on the transformed graph?

Question 65

The point $(-4,4)$ lies on the graph of $f$ . If the graph of $f$ is compressed vertically by a factor of $\frac{1}{7}$ and compressed horizontally by a factor of $\frac{1}{11}$ , what point must lie on the transformed graph?

Question 66

Educational psychologists suggest that a child's intellectual capacity, $I$ , can be represented as a function of time, $t$ , in months, as follows:
$I=f(t)=a\left(1-e^{-b t}\right)$
Where $a>0, b>0$ are constants. What is the effect on the graph of $I$ of increasing the value of $a$ ?

A) The graph gets closer to the asymptote less quickly.
B) The asymptote moves to the right.
C) The graph gets closer to the asymptote more quickly.
D) The asymptote moves up.

Question 67

Let $g(x)=\frac{1}{x^{2}}$ . Which of the following formulas transforms the graph of $g$ by shifting it 5 units to the right, reflecting it about the $x$ -axis, and then shifting it up 3 units?

A) $3-\frac{1}{(x-5)^{2}}$
B) $-\frac{1}{3+(x-5)^{2}}$
C) $\frac{1}{3-(x-5)^{2}}$
D) $3-\frac{1}{(x)^{2}-5}$

Question 68

Let $g(x)=\frac{1}{x^{2}}$ . The domain of the formula that transforms the graph of $g$ by shifting it 8 units to the right, reflecting it about the $x$ -axis, and then shifting it up 4 units is all real numbers except $x=$ ------------

Question 69

Let $g(x)=\frac{1}{x^{2}}$ . Let $h(x)$ be the function that transforms the graph of $g$ by shifting it 5 units to the right, reflecting it about the $x$ -axis, and then shifting it up 7 units. Is $h(x)$ even, odd, or neither?

Question 70

A cup of coffee is initially at $150^{\circ} \mathrm{F}$ , which is $80^{\circ} \mathrm{F}$ above room temperature. The difference between the coffee's temperature and the room's temperature decreases at the hourly rate of $80 \%$ . Which of the following formulas gives $T(t)$ , the coffee's temperature after $t$ hours have elapsed?

A) $T(t)=150+70(0.2)^{t}$
B) $T(t)=80+70(0.2)^{t}$
C) $T(t)=150+80(0.2)^{t}$
D) $T(t)=70+80(0.2)^{t}$

Question 71

Below is a graph of the function $f(x)$ . If you were to sketch a graph of $y=-f(x+3)+2$ , what point would the point $(-4,2)$ correspond to?

Question 72

Below is a graph of the function $f(x)$ . If you were to sketch a graph of $y=f(-x)+2$ , what point would the point $(-2,1)$ correspond to?

Question 73

Given the graph of $y=f(x)$ below, find a possible formula for $f(x)$ .

A) $f(x)=-(x-5)^{2}$
B) $f(x)=\log (x-4)$
C) $f(x)=e^{x-5}-1$
D) $f(x)=x^{2}-25$

Question 74

One of these graphs shows a transformation of an exponential function, and one is a transformation of a log function. The remaining graph is not a transformation of either an exponential or log function. Which graph is the transformation of a log function?

A)

B)

Question 75

The graph of $g(x)$ contains the point $(1,5)$ . What is the corresponding point on the graph of $y=-4 g(x)+4$ ?

Question 76

The first figure is the graph of $f(x)$ . Which of the other figures (a)-(f) show the graph of $f(0.5 x)$ ?

A)
B)
C)
D)
E)
F)

Question 77

The graph of $f(x)$ is shown in the first figure. Which of the other graphs (a)-(d) correspond to the formula $-f(x)-2$ ?

A)
B)

C)

D)

Question 78

Let $P(n)$ be the profit made by a water park in terms of the number of people, $n$ , visiting the park each day. If $n_{0}$ people attended the park yesterday, what is meant by the expression $P\left(n_{0}\right)+10$ ?

A) Yesterday's profit if an additional 10 people had visited
B) 10 percent of yesterday's profit
C) Yesterday's profit if 10 times as many people had visited
D) Yesterday's profit plus an additional $\$ 10$

Question 79

The point $(a, b)$ lies on the graph of $y=f(x)$ . If the graph is reflected across the $y$ -axis and compressed toward the $y$ -axis by a factor of $d$ (with $d<1$ ), and then translated upward 4 units, what are the new coordinates for the point $(a, b)$ ?

A) $(d(a+4), 4+b)$
B) $(-d a+4, b+4)$
C) $(d a, b+4)$
D) $(-d a, b+4)$

Question 80

Let $f(x)=e^{x}$ . Which of the following describe how the function $g(x)=4 e^{6(x-3)}$ transforms the graph of $f$ ?

A) The graph of $f$ is compressed horizontally by a factor of $\frac{1}{6}$ , then shifted to the right 3 units and stretched vertically by a factor of 4 .
B) The graph of $f$ is shifted to the right 3 units, then compressed horizontally by a factor of $\frac{1}{6}$ and stretched vertically by a factor of 4 .
C) The graph of $f$ is compressed vertically by a factor of $4 e^{-18}$ , and compressed horizontally by a factor of $\frac{1}{6}$ .
D) The graph of $f$ is stretched vertically by a factor of 4 , and compressed horizontally by a factor of $\frac{1}{6}$ .