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Ferris wheel. The model for the height $h$ of a Ferris wheel car is $h=51+50 \sin 8 \pi t$where $t$ is measured in minutes. (The Ferris wheel has a radius of 50 feet.) This model yields a height of 51 feet when $t=0$ . Alter the model so that the height of the car is 1 foot when $t=0$ .

Sales The monthly sales $S$ (in thousands of units) of a seasonal product are modeled by$$S=58.3+32.5 \cos \frac{\pi t}{6}$$where $t$ is the time (in months), with $t=1$ corresponding to January. Use a graphing utility to graph the model for $S$ and determine the months when sales exceed $75,000$ units.

Pattern Recognition Use a graphing utility to compare the graph of$$f(x)=\frac{4}{\pi}\left(\sin \pi x+\frac{1}{3} \sin 3 \pi x\right)$$with the given graph. Try to improve the approximation byadding a term to $f(x) .$ Use a graphing utility to verify thatyour new approximation is better than the original. Can youfind other terms to add to make the approximation even better?What is the pattern? (Hint: Use sine terms.)

True or False? In Exercises $77-80$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$\begin{array}{l}{\text { A measurement of } 4 \text { radians corresponds to two complete }} \\ {\text { revolutions from the initial side to the terminal side of an angle. }}\end{array}$$

True or False? In Exercises $77-80$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Amplitude is always positive

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Think About It Sketch the graphs of$$f(x)=\sin x, \quad g(x)=|\sin x|, \quad \text { and } \quad h(x)=\sin (|x|)$$In general, how are the graphs of $|f(x)|$ and $f(|x|)$ related to the graph of $f ?$

(ANSWER NOT AVAILABLE)

No transcript available

Graph the given functions on a common screen. How are these graphs related?

$y=e^{x}, \quad y=e^{-x}, \quad y=8^{x}, \quad y=8^{-x}$

$y=2^{x}, \quad y=e^{x}, \quad y=5^{x}, \quad y=20^{x}$

$y=3^{x}, \quad y=10^{x}, \quad y=\left(\frac{1}{3}\right)^{x}, \quad y=\left(\frac{1}{10}\right)^{x}$

$y=0.9^{x}, \quad y=0.6^{x}, \quad y=0.3^{x}, \quad y=0.1^{x}$

Graphs of Two Functions Find all values of $x$ for which the graph of $f$ lies above the graph of $g$.$$f(x)=x^{2}+x ; \quad g(x)=\frac{2}{x}$$

Graph the three functions on a common screen. How are the graphs related?$$y=\sqrt{x}, \quad y=-\sqrt{x}, \quad y=\sqrt{x} \sin 5 \pi x$$

Graph functions $f$ and $g$ in the same rectangular coordinate system. Select integers from $-2$ to 2 , inclusive, for $x$. Then describe how the graph of g is related to the graph of $f .$ If applicable, use a graphing utility to confirm your hand-drawn graphs. $$f(x)=2^{x} \quad \text { and } \quad g(x)=2^{x+2}$$

Graphs of Two Functions Find all values of $x$ for which the graph of $f$ lies above the graph of $g$.$$f(x)=4 x ; \quad g(x)=\frac{1}{x}$$

Graph the three functions on a common screen. How are the graphs related?$$y=\frac{1}{1+x^{2}}, \quad y=-\frac{1}{1+x^{2}}, \quad y=\frac{\cos 2 \pi x}{1+x^{2}}$$

a. Draw graphs of the functions $$f(x)=x^{2}+x-6$$ and $$g(x)=\left|x^{2}+x-6\right|$$ How are the graphs of $f$ and $g$ related?b. Draw graphs of the functions $f(x)=x^{4}-6 x^{2}$ and $g(x)=\left|x^{4}-6 x^{2}\right| .$ How are the graphs of $f$ and $g$ related?c. In general, if $g(x)=|f(x)|$, how are the graphs of $f$ and $g$ related? Draw graphs to illustrate your answer.

Graph the three functions on a common screen. How are the graphs related?$$y=x^{2}, \quad y=-x^{2}, \quad y=x^{2} \sin x$$

Graph functions $f$ and $g$ in the same rectangular coordinate system. Select integers from $-2$ to 2 , inclusive, for $x$. Then describe how the graph of g is related to the graph of $f .$ If applicable, use a graphing utility to confirm your hand-drawn graphs. $$f(x)=2^{x} \quad \text { and } \quad g(x)=2^{x-1}$$

Graph the three functions on a common screen. How are the graphs related?$$y=x, \quad y=-x, \quad y=x \cos x$$

Graphs of Two Functions Find all values of $x$ for which the graph of $f$ lies above the graph of $g$.$$f(x)=x^{2} ; \quad g(x)=3 x+10$$

Graphs of Two Functions Find all values of $x$ for which the graph of $f$ lies above the graph of $g$.$$f(x)=\frac{1}{x} ; \quad g(x)=\frac{1}{x-1}$$

The graph of a function f is illustrated. Use the graph of f as the first step toward graphing each of the following functions:$$\text { (a) } F(x)=f(x)+3 \quad \text { (b) } G(x)=f(x+2)$$$$\text { (c) } P(x)=-f(x) \quad \text { (d) } H(x)=f(x+1)-2$$$$\text { (e) } Q(x)=\frac{1}{2} f(x) \quad \text { (f) } g(x)=f(-x)$$$$(g) h(x)=f(2 x)$$

Consider $f(x)=\frac{-15 x^{2}+10 x}{5 x}$ and $g(x)=-3 x+2$.Graph $f(x)$ and $g(x)$ on a graphing calculator. How do the graphs appear?

Below is a graph of $f(x)=x^{2}+3 x-2$graph can't copya. Sketch a graph of $g(x)=(x-2)^{2}+3(x-2)-2$ and one of $h(x)=(x+2)^{2}+3(x+2)-2$ .b. How are the graphs of $g$ and $h$ related to the graph of $f ?$

Describing Transformations Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f .$$$\text { (a) } f(-x) \quad \text { (b) } 3 f(x)$$

Graph the pair of functions on the same set of coordinate axes and explain the differences between the two graphs.$$f(x)=2 \text { and } g(x)=2 x$$

A function $f$ is given. (a) Sketch the graph of $f .(b)$ Use the graph of $f$ to sketch the graph of $f^{-1} .(c)$ Find $f^{-1} .$$$f(x)=16-x^{2}, \quad x \geq 0$$

Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f$a. $f(-x)$b. $3 f(x)$

The figure shows the graphs of $y=2^{x}, y=e^{x}$$y=10^{x}, y=2^{-x}$$y=e^{-x},$ and $y=10^{-x}$Match each function with its graph. [The graphs are labeled (a) through (f).] Explain.

Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f$a. $-f(x)$b. $\frac{1}{3} f(x)$

Find all values of $x$ for which the graph of $f$ lies above the graph of $g$.$$f(x)=x^{2}+x ; \quad g(x)=\frac{2}{x}$$

Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f$a. $1-f(-x)$b. $2-\frac{1}{5} f(x)$

Graph the given functions, $f$ and $g,$ in the same rectangular coordinate system. Select integers for $x,$ starting with $-2$ and ending with $2 .$ Once you have obtained your graphs, describe how the graph of g is related to the graph of $f$$f(x)=3, g(x)=5$

Graph the following equations and explain why they are notgraphs of functions of $x .$$$|x|+|y|=1 \quad \text { b. }|x+y|=1$$

Describing Transformations Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f .$$$\text { (a) } \frac{1}{3} f(x-2)+5 \quad \text { (b) } 4 f(x+1)+3$$

$ y = e^x $ , $ y = e^{-x} $ , $ y = 8^x $ , $ y = 8^{-x} $

Describing Transformations Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f .$$$\text { (a) } y=-f(x)+5 \quad \text { (b) } y=3 f(x)-5$$

$ y = 2^x $ , $ y = e^x $ , $ y = 5^x $ , $ y =20^x $

(a) Draw the graphs of the functions$$\begin{array}{l}{f(x)=x^{2}+x-6} \\ {g(x)=\left|x^{2}+x-6\right|}\end{array}$$How are the graphs of $f$ and $g$ related?(b) Draw the graphs of the functions $f(x)=x^{4}-6 x^{2}$ and $g(x)=\left|x^{4}-6 x^{2}\right| .$ How are the graphs of $f$ and $g$ related?(c) In general, if $g(x)=|f(x)|$ , how are the graphs of $f$ and $g$ related? Draw graphs to illustrate your answer.

Graph the given functions, $f$ and $g,$ in the same rectangular coordinate system. Select integers for $x,$ starting with $-2$ and ending with $2 .$ Once you have obtained your graphs, describe how the graph of g is related to the graph of $f$$f(x)=-2 x, g(x)=-2 x+3$

Graph functions $f$ and $g$ in the same rectangular coordinate system. Select integers from $-2$ to 2 , inclusive, for $x$. Then describe how the graph of g is related to the graph of $f .$ If applicable, use a graphing utility to confirm your hand-drawn graphs. $$f(x)=3^{x} \text { and } g(x)=3^{-x}$$

Graph functions $f$ and $g$ in the same rectangular coordinate system. Select integers from $-2$ to 2 , inclusive, for $x$. Then describe how the graph of g is related to the graph of $f .$ If applicable, use a graphing utility to confirm your hand-drawn graphs. $$f(x)=2^{x} \text { and } g(x)=2^{x}+2$$

Graph functions $f$ and $g$ in the same rectangular coordinate system. Select integers from $-2$ to 2 , inclusive, for $x$. Then describe how the graph of g is related to the graph of $f .$ If applicable, use a graphing utility to confirm your hand-drawn graphs. $$f(x)=3^{x} \text { and } g(x)=-3^{x}$$

Graph the three functions on a common screen. How are the graphs related?$$y=\cos 3 \pi x, \quad y=-\cos 3 \pi x, \quad y=\cos 3 \pi x \cos 21 \pi x$$

Suppose that the graph of a function $f$ is known. Explain how the graph of $y=4 f(x)$ differs from the graph of $y=f(4 x) .$

Suppose that the graph of a function $f$ is known. Explain how the graph of $y=4 f(x)$ differs from the graph of $y=f(4 x)$

Show that $f$ and $g$ are inverse functions algebraically. Use a graphing utility to graph $f$ and $g$ in the same viewing window. Describe the relationship between the graphs.$$f(x)=\sqrt{x-4} ; \quad g(x)=x^{2}+4, \quad x \geq 0$$

Explain how the graph of $g$ is obtained from the graph of $f$a. $f(x)=x^{3}, \quad g(x)=(x-4)^{3}$b. $f(x)=x^{3}, \quad g(x)=x^{3}-4$

Use the graphs to find a $\delta>0$ such that$$|f(x)-L|<\varepsilon \quad \text { whenever } \quad 0<|x-c|<\delta$$CANT COPY THE GRAPH

Use the graphs to find a $\delta>0$ such that$$|f(x)-L|<\varepsilon \quad \text { whenever } \quad 0<|x-c|<\delta$$ CANT COPY THE GRAPH

The figure shows the graphs of $ y = 2^x $, $ y = e^x $, $ y = 10^x $, $ y = 2^{-x} $, $ y = e^{-x} $, and $ y = 10^{-x} $. Match each function with its graph. [The graphs are labeled (a) through (f).] Explain your reasoning.

Explain how each graph is obtained from the graph of $ y = f(x) $.

(a) $ y = f(x) + 8 $(b) $ y = f (x + 8) $(c) $ y = 8f(x) $(d) $ y = f(8x) $(e) $ y = -f(x) - 1 $(f) $ y = 8f (\frac{1}{8}x) $

Graph the three functions on a common screen. How are the graphs related?$$y=\sin 2 \pi x, \quad y=-\sin 2 \pi x, \quad y=\sin 2 \pi x \sin 10 \pi x$$

Graph the given functions, $f$ and $g,$ in the same rectangular coordinate system. Select integers for $x,$ starting with $-2$ and ending with $2 .$ Once you have obtained your graphs, describe how the graph of g is related to the graph of $f$$f(x)=x^{3}, g(x)=x^{3}+2$

Describing Transformations Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f .$$$\text { (a) } y=f(4 x) \quad \text { (b) } y=f\left(\frac{1}{4} x\right)$$

Graph the following equations and explain why they are not graphs of functions of $x .$a. $|x|+|y|=1 \quad$ b. $|x+y|=1$

In Exercises 101 - 104, sketch the graphs of $ f $ and $ g $ and describe the relationship between the graphs of $ f $ and $ g $. What is the relationship between the functions $ f $ and $ g $?

$ f(x) = 8^x $, $ g(x) = \log_8 x $

Family of Functions A family of functions is given. (a) Draw graphs of the family for $c=1,2,3,$ and $4 .$ (b) How are the graphs in part (a) related?$$f(x)=\log (c x)$$

Graphing Transformations The graph of a function $f$ is given. Sketch the graphs of the following transformations of $f .$$$\text { (a) } y=f(3 x) \quad \text { (b) } y=f\left(\frac{1}{3} x\right)$$

Explain how the graph of $f$ can be obtained from the graph of $y=\frac{1}{x}$ or $y=\frac{1}{x^{2}}$ Draw a sketch of the graph of $f$ by hand. Then generate an accurate depiction of the graph with a graphing calculator. Finally, give the domain and range.$$f(x)=-\frac{2}{x^{2}}$$

$5-14$ . Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f .$ (a) $y=3-2 f(x) \quad$ (b) $y=2-f(-x)$

Graph functions $f$ and $g$ in the same rectangular coordinate system. Select integers from $-2$ to 2 , inclusive, for $x$. Then describe how the graph of g is related to the graph of $f .$ If applicable, use a graphing utility to confirm your hand-drawn graphs.$$f(x)=2^{x} \text { and } g(x)=2^{x+1}$$

Graph functions $f$ and $g$ in the same rectangular coordinate system. Select integers from $-2$ to 2 , inclusive, for $x$. Then describe how the graph of g is related to the graph of $f .$ If applicable, use a graphing utility to confirm your hand-drawn graphs. $$f(x)=2^{x} \text { and } g(x)=2^{x}+1$$

$37-40$ Graphs of Two Functions Find all values of $x$ forwhich the graph of $f$ lies above the graph of $g .$$$f(x)=x^{2} ; \quad g(x)=3 x+10$$

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## Recommended Questions

Graph the given functions on a common screen. How are these graphs related?

$y=e^{x}, \quad y=e^{-x}, \quad y=8^{x}, \quad y=8^{-x}$

Graph the given functions on a common screen. How are these graphs related?

$y=2^{x}, \quad y=e^{x}, \quad y=5^{x}, \quad y=20^{x}$

Graph the given functions on a common screen. How are these graphs related?

$y=3^{x}, \quad y=10^{x}, \quad y=\left(\frac{1}{3}\right)^{x}, \quad y=\left(\frac{1}{10}\right)^{x}$

Graph the given functions on a common screen. How are these graphs related?

$y=0.9^{x}, \quad y=0.6^{x}, \quad y=0.3^{x}, \quad y=0.1^{x}$

Graphs of Two Functions Find all values of $x$ for which the graph of $f$ lies above the graph of $g$.

$$

f(x)=x^{2}+x ; \quad g(x)=\frac{2}{x}

$$

Graph the three functions on a common screen. How are the graphs related?

$$

y=\sqrt{x}, \quad y=-\sqrt{x}, \quad y=\sqrt{x} \sin 5 \pi x

$$

Graph functions $f$ and $g$ in the same rectangular coordinate system. Select integers from $-2$ to 2 , inclusive, for $x$. Then describe how the graph of g is related to the graph of $f .$ If applicable, use a graphing utility to confirm your hand-drawn graphs.

$$f(x)=2^{x} \quad \text { and } \quad g(x)=2^{x+2}$$

Graphs of Two Functions Find all values of $x$ for which the graph of $f$ lies above the graph of $g$.

$$

f(x)=4 x ; \quad g(x)=\frac{1}{x}

$$

Graph the three functions on a common screen. How are the graphs related?

$$

y=\frac{1}{1+x^{2}}, \quad y=-\frac{1}{1+x^{2}}, \quad y=\frac{\cos 2 \pi x}{1+x^{2}}

$$

a. Draw graphs of the functions $$f(x)=x^{2}+x-6$$ and $$g(x)=\left|x^{2}+x-6\right|$$ How are the graphs of $f$ and $g$ related?

b. Draw graphs of the functions $f(x)=x^{4}-6 x^{2}$ and $g(x)=\left|x^{4}-6 x^{2}\right| .$ How are the graphs of $f$ and $g$ related?

c. In general, if $g(x)=|f(x)|$, how are the graphs of $f$ and $g$ related? Draw graphs to illustrate your answer.

Graph the three functions on a common screen. How are the graphs related?

$$

y=x^{2}, \quad y=-x^{2}, \quad y=x^{2} \sin x

$$

Graph functions $f$ and $g$ in the same rectangular coordinate system. Select integers from $-2$ to 2 , inclusive, for $x$. Then describe how the graph of g is related to the graph of $f .$ If applicable, use a graphing utility to confirm your hand-drawn graphs.

$$f(x)=2^{x} \quad \text { and } \quad g(x)=2^{x-1}$$

Graph the three functions on a common screen. How are the graphs related?

$$

y=x, \quad y=-x, \quad y=x \cos x

$$

Graphs of Two Functions Find all values of $x$ for which the graph of $f$ lies above the graph of $g$.

$$

f(x)=x^{2} ; \quad g(x)=3 x+10

$$

Graphs of Two Functions Find all values of $x$ for which the graph of $f$ lies above the graph of $g$.

$$

f(x)=\frac{1}{x} ; \quad g(x)=\frac{1}{x-1}

$$

The graph of a function f is illustrated. Use the graph of f as the first step toward graphing each of the following functions:

$$

\text { (a) } F(x)=f(x)+3 \quad \text { (b) } G(x)=f(x+2)

$$

$$

\text { (c) } P(x)=-f(x) \quad \text { (d) } H(x)=f(x+1)-2

$$

$$

\text { (e) } Q(x)=\frac{1}{2} f(x) \quad \text { (f) } g(x)=f(-x)

$$

$$

(g) h(x)=f(2 x)

$$

The graph of a function f is illustrated. Use the graph of f as the first step toward graphing each of the following functions:

$$

\text { (a) } F(x)=f(x)+3 \quad \text { (b) } G(x)=f(x+2)

$$

$$

\text { (c) } P(x)=-f(x) \quad \text { (d) } H(x)=f(x+1)-2

$$

$$

\text { (e) } Q(x)=\frac{1}{2} f(x) \quad \text { (f) } g(x)=f(-x)

$$

$$

(g) h(x)=f(2 x)

$$

The graph of a function f is illustrated. Use the graph of f as the first step toward graphing each of the following functions:

$$

\text { (a) } F(x)=f(x)+3 \quad \text { (b) } G(x)=f(x+2)

$$

$$

\text { (c) } P(x)=-f(x) \quad \text { (d) } H(x)=f(x+1)-2

$$

$$

\text { (e) } Q(x)=\frac{1}{2} f(x) \quad \text { (f) } g(x)=f(-x)

$$

$$

(g) h(x)=f(2 x)

$$

Consider $f(x)=\frac{-15 x^{2}+10 x}{5 x}$ and $g(x)=-3 x+2$.

Graph $f(x)$ and $g(x)$ on a graphing calculator. How do the graphs appear?

Below is a graph of $f(x)=x^{2}+3 x-2$

graph can't copy

a. Sketch a graph of $g(x)=(x-2)^{2}+3(x-2)-2$ and one of $h(x)=(x+2)^{2}+3(x+2)-2$ .

b. How are the graphs of $g$ and $h$ related to the graph of $f ?$

Describing Transformations Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f .$

$$

\text { (a) } f(-x) \quad \text { (b) } 3 f(x)

$$

Graph the pair of functions on the same set of coordinate axes and explain the differences between the two graphs.

$$f(x)=2 \text { and } g(x)=2 x$$

A function $f$ is given. (a) Sketch the graph of $f .(b)$ Use the graph of $f$ to sketch the graph of $f^{-1} .(c)$ Find $f^{-1} .$

$$

f(x)=16-x^{2}, \quad x \geq 0

$$

Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f$

a. $f(-x)$

b. $3 f(x)$

The figure shows the graphs of $y=2^{x}, y=e^{x}$

$y=10^{x}, y=2^{-x}$

$y=e^{-x},$ and $y=10^{-x}$

Match each function with its graph. [The graphs are labeled (a) through (f).] Explain.

Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f$

a. $-f(x)$

b. $\frac{1}{3} f(x)$

Find all values of $x$ for which the graph of $f$ lies above the graph of $g$.

$$f(x)=x^{2}+x ; \quad g(x)=\frac{2}{x}$$

Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f$

a. $1-f(-x)$

b. $2-\frac{1}{5} f(x)$

Graph the given functions, $f$ and $g,$ in the same rectangular coordinate system. Select integers for $x,$ starting with $-2$ and ending with $2 .$ Once you have obtained your graphs, describe how the graph of g is related to the graph of $f$

$f(x)=3, g(x)=5$

Graph the following equations and explain why they are not

graphs of functions of $x .$

$$

|x|+|y|=1 \quad \text { b. }|x+y|=1

$$

Describing Transformations Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f .$

$$

\text { (a) } \frac{1}{3} f(x-2)+5 \quad \text { (b) } 4 f(x+1)+3

$$

Graph the given functions on a common screen. How are these graphs related?

$ y = e^x $ , $ y = e^{-x} $ , $ y = 8^x $ , $ y = 8^{-x} $

Describing Transformations Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f .$

$$

\text { (a) } y=-f(x)+5 \quad \text { (b) } y=3 f(x)-5

$$

Graph the given functions on a common screen. How are these graphs related?

$ y = 2^x $ , $ y = e^x $ , $ y = 5^x $ , $ y =20^x $

(a) Draw the graphs of the functions

$$\begin{array}{l}{f(x)=x^{2}+x-6} \\ {g(x)=\left|x^{2}+x-6\right|}\end{array}$$

How are the graphs of $f$ and $g$ related?

(b) Draw the graphs of the functions $f(x)=x^{4}-6 x^{2}$ and $g(x)=\left|x^{4}-6 x^{2}\right| .$ How are the graphs of $f$ and $g$ related?

(c) In general, if $g(x)=|f(x)|$ , how are the graphs of $f$ and $g$ related? Draw graphs to illustrate your answer.

Graph the given functions, $f$ and $g,$ in the same rectangular coordinate system. Select integers for $x,$ starting with $-2$ and ending with $2 .$ Once you have obtained your graphs, describe how the graph of g is related to the graph of $f$

$f(x)=-2 x, g(x)=-2 x+3$

Graph functions $f$ and $g$ in the same rectangular coordinate system. Select integers from $-2$ to 2 , inclusive, for $x$. Then describe how the graph of g is related to the graph of $f .$ If applicable, use a graphing utility to confirm your hand-drawn graphs.

$$f(x)=3^{x} \text { and } g(x)=3^{-x}$$

Graph functions $f$ and $g$ in the same rectangular coordinate system. Select integers from $-2$ to 2 , inclusive, for $x$. Then describe how the graph of g is related to the graph of $f .$ If applicable, use a graphing utility to confirm your hand-drawn graphs.

$$f(x)=2^{x} \text { and } g(x)=2^{x}+2$$

Graph functions $f$ and $g$ in the same rectangular coordinate system. Select integers from $-2$ to 2 , inclusive, for $x$. Then describe how the graph of g is related to the graph of $f .$ If applicable, use a graphing utility to confirm your hand-drawn graphs.

$$f(x)=3^{x} \text { and } g(x)=-3^{x}$$

Graph the three functions on a common screen. How are the graphs related?

$$

y=\cos 3 \pi x, \quad y=-\cos 3 \pi x, \quad y=\cos 3 \pi x \cos 21 \pi x

$$

Suppose that the graph of a function $f$ is known. Explain how the graph of $y=4 f(x)$ differs from the graph of $y=f(4 x) .$

Suppose that the graph of a function $f$ is known. Explain how the graph of $y=4 f(x)$ differs from the graph of $y=f(4 x)$

Show that $f$ and $g$ are inverse functions algebraically. Use a graphing utility to graph $f$ and $g$ in the same viewing window. Describe the relationship between the graphs.

$$f(x)=\sqrt{x-4} ; \quad g(x)=x^{2}+4, \quad x \geq 0$$

Explain how the graph of $g$ is obtained from the graph of $f$

a. $f(x)=x^{3}, \quad g(x)=(x-4)^{3}$

b. $f(x)=x^{3}, \quad g(x)=x^{3}-4$

Use the graphs to find a $\delta>0$ such that

$$|f(x)-L|<\varepsilon \quad \text { whenever } \quad 0<|x-c|<\delta$$

CANT COPY THE GRAPH

Use the graphs to find a $\delta>0$ such that

$$|f(x)-L|<\varepsilon \quad \text { whenever } \quad 0<|x-c|<\delta$$

CANT COPY THE GRAPH

Use the graphs to find a $\delta>0$ such that

$$|f(x)-L|<\varepsilon \quad \text { whenever } \quad 0<|x-c|<\delta$$

CANT COPY THE GRAPH

$$|f(x)-L|<\varepsilon \quad \text { whenever } \quad 0<|x-c|<\delta$$

CANT COPY THE GRAPH

$$|f(x)-L|<\varepsilon \quad \text { whenever } \quad 0<|x-c|<\delta$$

CANT COPY THE GRAPH

$$|f(x)-L|<\varepsilon \quad \text { whenever } \quad 0<|x-c|<\delta$$

CANT COPY THE GRAPH

$$|f(x)-L|<\varepsilon \quad \text { whenever } \quad 0<|x-c|<\delta$$

CANT COPY THE GRAPH

$$|f(x)-L|<\varepsilon \quad \text { whenever } \quad 0<|x-c|<\delta$$

CANT COPY THE GRAPH

The figure shows the graphs of $ y = 2^x $, $ y = e^x $, $ y = 10^x $, $ y = 2^{-x} $, $ y = e^{-x} $, and $ y = 10^{-x} $. Match each function with its graph. [The graphs are labeled (a) through (f).] Explain your reasoning.

Explain how each graph is obtained from the graph of $ y = f(x) $.

(a) $ y = f(x) + 8 $

(b) $ y = f (x + 8) $

(c) $ y = 8f(x) $

(d) $ y = f(8x) $

(e) $ y = -f(x) - 1 $

(f) $ y = 8f (\frac{1}{8}x) $

Graph the three functions on a common screen. How are the graphs related?

$$

y=\sin 2 \pi x, \quad y=-\sin 2 \pi x, \quad y=\sin 2 \pi x \sin 10 \pi x

$$

Graph the given functions, $f$ and $g,$ in the same rectangular coordinate system. Select integers for $x,$ starting with $-2$ and ending with $2 .$ Once you have obtained your graphs, describe how the graph of g is related to the graph of $f$

$f(x)=x^{3}, g(x)=x^{3}+2$

Describing Transformations Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f .$

$$

\text { (a) } y=f(4 x) \quad \text { (b) } y=f\left(\frac{1}{4} x\right)

$$

Graph the following equations and explain why they are not graphs of functions of $x .$

a. $|x|+|y|=1 \quad$ b. $|x+y|=1$

In Exercises 101 - 104, sketch the graphs of $ f $ and $ g $ and describe the relationship between the graphs of $ f $ and $ g $. What is the relationship between the functions $ f $ and $ g $?

$ f(x) = 8^x $, $ g(x) = \log_8 x $

Family of Functions A family of functions is given. (a) Draw graphs of the family for $c=1,2,3,$ and $4 .$ (b) How are the graphs in part (a) related?

$$

f(x)=\log (c x)

$$

Graphing Transformations The graph of a function $f$ is given. Sketch the graphs of the following transformations of $f .$

$$

\text { (a) } y=f(3 x) \quad \text { (b) } y=f\left(\frac{1}{3} x\right)

$$

Explain how the graph of $f$ can be obtained from the graph of $y=\frac{1}{x}$ or $y=\frac{1}{x^{2}}$ Draw a sketch of the graph of $f$ by hand. Then generate an accurate depiction of the graph with a graphing calculator. Finally, give the domain and range.

$$f(x)=-\frac{2}{x^{2}}$$

$5-14$ . Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f .$

(a) $y=3-2 f(x) \quad$ (b) $y=2-f(-x)$

Graph functions $f$ and $g$ in the same rectangular coordinate system. Select integers from $-2$ to 2 , inclusive, for $x$. Then describe how the graph of g is related to the graph of $f .$ If applicable, use a graphing utility to confirm your hand-drawn graphs.

$$f(x)=2^{x} \text { and } g(x)=2^{x+1}$$

Graph functions $f$ and $g$ in the same rectangular coordinate system. Select integers from $-2$ to 2 , inclusive, for $x$. Then describe how the graph of g is related to the graph of $f .$ If applicable, use a graphing utility to confirm your hand-drawn graphs.

$$f(x)=2^{x} \text { and } g(x)=2^{x}+1$$

$37-40$ Graphs of Two Functions Find all values of $x$ for

which the graph of $f$ lies above the graph of $g .$

$$f(x)=x^{2} ; \quad g(x)=3 x+10$$