Graph the functions for one period. In each case, specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.

(a) $y=2 \sin x$

(b) $y=-\sin 2 x$

Adnan G.

Numerade Educator

Graph the functions for one period. In each case, specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.

(a) $y=3 \sin x$

(b) $y=\sin 3 x$

Adnan G.

Numerade Educator

Graph the functions for one period. In each case, specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.

(a) $y=\cos 2 x$

(b) $y=2 \cos 2 x$

Adnan G.

Numerade Educator

Graph the functions for one period. In each case, specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.

(a) $y=\cos (x / 2)$

(b) $y=-\frac{1}{2} \cos (x / 2)$

Adnan G.

Numerade Educator

Graph the functions for one period. In each case, specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.

(a) $y=3 \sin (\pi x / 2)$

(b) $y=-3 \sin (\pi x / 2)$

Adnan G.

Numerade Educator

Graph the functions for one period. In each case, specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.

(a) $y=2 \sin \pi x$

(b) $y=-2 \sin \pi x$

Adnan G.

Numerade Educator

Graph the functions for one period. In each case, specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.

(a) $y=\cos 2 \pi x$

(b) $y=-4 \cos 2 \pi x$

Adnan G.

Numerade Educator

Graph the functions for one period. In each case, specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.

(a) $y=-2 \cos (x / 4)$

(b) $y=-2 \cos (\pi x / 4)$

Adnan G.

Numerade Educator

et the viewing rectangle so that it extends from 0 to $2 \pi$ in the $x$ -direction and from $-4$ to 4 in the $y$ -direction. On the same set of axes, graph the four functions $y=\sin x$ $y=2 \sin x, y=3 \sin x,$ and $y=4 \sin x .$ What is the amplitude in each case? What is the period?

Adnan G.

Numerade Educator

(a)Without using a graphing utility, specify the amplitude and the period for each of the following four functions:

$y=\cos x, y=2 \cos x, y=3 \cos x,$ and $y=4 \cos x$

(b) Check your answers in part (a) by graphing the four functions. (Use the viewing rectangle specified in Exercise 1.)

Adnan G.

Numerade Educator

(a)Without using a graphing utility, specify the amplitude and the period for $y=2 \sin \pi x$ and for $y=\sin 2 \pi x$

(b) Check your answers in part (a) by graphing the two functions. (Use a viewing rectangle that extends from 0 to 2 in the $x$ -direction and from $-2$ to 2 in the $y$ -direction.

Adnan G.

Numerade Educator

Graph the function for one period. Specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.

$$y=1+\sin 2 x$$

Adnan G.

Numerade Educator

Graph the function for one period. Specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.

$$y=\sin (x / 2)-2$$

Adnan G.

Numerade Educator

Graph the function for one period. Specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.

$$y=1-\cos (\pi x / 3)$$

Adnan G.

Numerade Educator

Graph the function for one period. Specify the amplitude, period, $x$ -intercepts, and interval(s) on which the function is increasing.

$$y=-2-2 \cos 3 \pi x$$

Adnan G.

Numerade Educator

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.

$$f(x)=\sin \left(x-\frac{\pi}{6}\right)$$

Adnan G.

Numerade Educator

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.

$$g(x)=\cos \left(x+\frac{\pi}{3}\right)$$

Adnan G.

Numerade Educator

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.

$$F(x)=-\cos \left(x+\frac{\pi}{4}\right)$$

Adnan G.

Numerade Educator

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.

$$G(x)=-\sin (x+2)$$

Adnan G.

Numerade Educator

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.

$$y=\sin \left(2 x-\frac{\pi}{2}\right)$$

Adnan G.

Numerade Educator

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.

$$y=\sin \left(3 x+\frac{\pi}{2}\right)$$

Adnan G.

Numerade Educator

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.

$$y=\cos (2 x-\pi)$$

Adnan G.

Numerade Educator

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.

$$y=\cos \left(x-\frac{\pi}{2}\right)$$

Adnan G.

Numerade Educator

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.

$$y=3 \sin \left(\frac{1}{2} x+\frac{\pi}{6}\right)$$

Adnan G.

Numerade Educator

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.

$$y=-2 \sin (\pi x+\pi)$$

Adnan G.

Numerade Educator

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.

$$y=4 \cos \left(3 x-\frac{\pi}{4}\right)$$

Adnan G.

Numerade Educator

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.

$$y=\cos (x+1)$$

Adnan G.

Numerade Educator

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.

$$y=\frac{1}{2} \sin \left(\frac{\pi x}{2}-\pi^{2}\right)$$

Adnan G.

Numerade Educator

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.

$$y=\cos \left(2 x-\frac{\pi}{3}\right)+1$$

Adnan G.

Numerade Educator

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.

$$y=1-\cos \left(2 x-\frac{\pi}{3}\right)$$

Adnan G.

Numerade Educator

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.

$$y=3 \cos \left(\frac{2 x}{3}+\frac{\pi}{6}\right)$$

Adnan G.

Numerade Educator

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.

(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]

(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.

(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).

$$y=-2.5 \cos (3 x+4)$$

Adnan G.

Numerade Educator

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.

(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]

(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.

(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).

$$y=-2.5 \cos (3 \pi x+4)$$

Adnan G.

Numerade Educator

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.

(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]

(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.

(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).

$$y=-2.5 \cos \left(\frac{1}{3} x+4\right)$$

Adnan G.

Numerade Educator

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.

(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]

(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.

(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).

$$y=-2.5 \cos \left(\frac{1}{3} \pi x+4\right)$$

Adnan G.

Numerade Educator

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.

(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]

(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.

(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).

$$y=\sin (0.5 x-0.75)$$

Adnan G.

Numerade Educator

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.

(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]

(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.

(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).

$$y=\sin (0.5 x+0.75)$$

Adnan G.

Numerade Educator

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.

(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]

(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.

(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).

$$y=0.02 \cos (100 \pi x-4 \pi)$$

Adnan G.

Numerade Educator

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.

(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]

(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.

(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).

$$y=0.02 \cos (0.01 \pi x-4 \pi)$$

Adnan G.

Numerade Educator

Determine whether the equation for the graph has the form $y=A \sin B x$ or $y=A \cos B x(\text { with } B>0)$ and then find the values of $A$ and $B$.

(GRAPH CANT COPY)

Adnan G.

Numerade Educator

(GRAPH CANT COPY)

Adnan G.

Numerade Educator

(GRAPH CANT COPY)

Adnan G.

Numerade Educator

(GRAPH CANT COPY)

Adnan G.

Numerade Educator

(GRAPH CANT COPY)

Adnan G.

Numerade Educator

(GRAPH CANT COPY)

Adnan G.

Numerade Educator

You are given a table of average monthly temperatures and a scatter plot based on the data. Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures. The data in these exercises, as well as in Exercise $51,$ are from the Global Historical Climatology Network, and can be accessed on the Internet through the following website created by Robert Hoare.

(GRAPH CANT COPY)

Average Monthly Temperatures for Phoenix, Arizona (based on daily maximums)

$$\begin{array}{lc}\hline \text { Month } & \text { Temperature ('F) } \\\hline \text { Jan. } & 65.5 \\

\text { Feb. } & 70.2 \\\text { Mar. } & 75.2 \\\text { Apr. } & 84.6 \\\text { May } & 93.2 \\\text { June } & 102.7 \\\text { July } & 105.1 \\\text { Aug. } & 103.1 \\\text { Sept. } & 98.6 \\\text { Oct. } & 88.2 \\

\text { Nov. } & 74.7 \\\text { Dec. } & 66.4 \\\hline\end{array}$$

Adnan G.

Numerade Educator

You are given a table of average monthly temperatures and a scatter plot based on the data. Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures. The data in these exercises, as well as in Exercise $51,$ are from the Global Historical Climatology Network, and can be accessed on the Internet through the following website created by Robert Hoare.

(GRAPH CANT COPY)

Average Monthly Temperatures for Bangor, Maine

$$\begin{array}{lc}\hline \text { Month } & \text { Temperature ('F) } \\\hline \text { Jan. } & 18.0 \\

\text { Feb. } & 20.8 \\\text { Mar. } & 30.7 \\\text { Apr. } & 42.6 \\\text { May } & 54.1 \\

\text { June } & 62.8 \\\text { July } & 68.5 \\\text { Aug. } & 67.3 \\\text { Sept. } & 58.1 \\

\text { Oct. } & 47.1 \\\text { Nov. } & 36.9 \\\text { Dec. } & 23.7 \\\hline\end{array}$$

Adnan G.

Numerade Educator

You are given a table of average monthly temperatures and a scatter plot based on the data. Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures. The data in these exercises, as well as in Exercise $51,$ are from the Global Historical Climatology Network, and can be accessed on the Internet through the following website created by Robert Hoare.

(GRAPH CANT COPY)

Average Monthly Temperatures for Cape Town, South Africa

$$\begin{array}{lc}\hline \text { Month } & \text { Temperature }\left(^{\circ} \mathrm{F}\right) \\

\hline \text { Jan. } & 69.8 \\\text { Feb. } & 70.0 \\\text { Mar. } & 67.8 \\\text { Apr. } & 63.1 \\

\text { May } & 58.8 \\\text { June } & 55.6 \\\text { July } & 54.3 \\\text { Aug. } & 55.4 \\

\text { Sept. } & 57.7 \\\text { Oct. } & 61.2 \\\text { Nov. } & 64.8 \\\text { Dec. } & 67.8 \\

\hline\end{array}$$

Adnan G.

Numerade Educator

You are given a table of average monthly temperatures and a scatter plot based on the data. Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures. The data in these exercises, as well as in Exercise $51,$ are from the Global Historical Climatology Network, and can be accessed on the Internet through the following website created by Robert Hoare.

(GRAPH CANT COPY)

Average Monthly Temperatures for Beira, Mozambique

$$\begin{array}{lc}\hline \text { Month } & \text { Temperature }\left(^{\circ} \mathrm{F}\right) \\

\hline \text { Jan. } & 81.3 \\\text { Feb. } & 81.3 \\\text { Mar. } & 80.1 \\\text { Apr. } & 77.5 \\

\text { May } & 73.0 \\\text { June } & 69.4 \\\text { July } & 68.7 \\\text { Aug. } & 70.2 \\

\text { Sept. } & 73.4 \\\text { Oct. } & 76.6 \\\text { Nov. } & 79.0 \\\text { Dec. } & 80.4 \\\hline\end{array}$$

Adnan G.

Numerade Educator

You are given a table of average monthly temperatures and a scatter plot based on the data. Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures. The data in these exercises, as well as in Exercise $51,$ are from the Global Historical Climatology Network, and can be accessed on the Internet through the following website created by Robert Hoare.

(GRAPH CANT COPY)

The following two scatter plots and table display average temperature data for two locations with very different climates: Dar es Salaam is south of the Equator, in Tanzania, on the Indian Ocean; Tiksi is in Russia, far north of the Arctic Circle, on the Arctic Ocean.

Average Monthly Temperatures for Dar es Salaam, Tanzania, and Tiksi, Russia

(TABLE CANT COPY)

(a) Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures for Dar es Salaam.

(b) Follow part (a) for Tiksi.

(c) By looking at the scatter plot for temperatures in Dar es Salaam, say if the average rate of change of temperature over the interval from January through July is positive or negative.

(d) Use the table of values to compute the average rate of change of temperature in Dar es Salaam over the interval from January through July. Be sure to include units as part of your answer, and check that the sign is consistent with your response in part (c).

(e) By looking at the scatter plot for temperatures in Tiksi, say if the average rate of change of temperature over the interval from January through July is positive or negative.

(f) Use the table of values to compute the average rate of change of temperature in Tiksi over the interval from January through July. As before, be sure to include units as part of your answer, and check that the sign is consistent with your response in part (e).

Adnan G.

Numerade Educator

The following scatter plot displays average monthly temperatures for Death Valley, California. (The averages were obtained using daily maximums.) Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures. (No table is given; you will need to rely on the scatter plot and make estimates.)

(TABLE CANT COPY)

Adnan G.

Numerade Educator

In Section 9.2 you'll see the identity $\sin ^{2} x=\frac{1}{2}-\frac{1}{2} \cos 2 x$. Use this identity to graph the function $y=\sin ^{2} x$ for one period.

Adnan G.

Numerade Educator

In Section 9.2 you'll see the identity $\cos ^{2} x=\frac{1}{2}+\frac{1}{2} \cos 2 x$. Use this identity to graph the function $y=\cos ^{2} x$ for one period.

Adnan G.

Numerade Educator

In Section 9.2 we derive the identity $\sin 2 x=2 \sin x \cos x$. Use this to graph $y=\sin x \cos x$ for one period.

Adnan G.

Numerade Educator

In Section 9.2 we derive the identity $\cos 2 x=\cos ^{2} x-\sin ^{2} x .$ Use this to graph $y=\cos ^{2} x-\sin ^{2} x$ for one period.

Adnan G.

Numerade Educator

In Example 2 we showed that the period of $y=\cos 3 x$ is $2 \pi / 3 .$ Use the same method to show that the period of $y=A \cos B x,$ with $B>0,$ is $2 \pi / B$

Adnan G.

Numerade Educator

Adapt the method of Example 6 to determine the amplitude, period, and phase shift for the given function. Graph the function over one period indicating the $x$ -intercepts and the coordinates of the highest and lowest points on the graph.

(a) $y=3 \cos (\pi-2 x)$

(b) $y=\frac{5}{2} \sin (-4 \pi x-\pi)$

Adnan G.

Numerade Educator

( Let $F(x)=\sin x, G(x)=x^{2},$ and $H(x)=x^{3} .$ Which, if any, of the following four composite functions have graphs that do not go below the $x$ -axis? First, try to answer without using a graphing utility, then use the graphing utility to check yourself. (You will learn more this way than if you were to draw the graphs immediately.)

$$\begin{array}{ll}y=G(F(x)) & y=F(G(x)) \\y=F(H(x)) & y=H(F(x))\end{array}$$

Adnan G.

Numerade Educator

(a) Graph the two functions $y=\sin x$ and $y=\sin (\sin x)$ in the standard viewing rectangle. Then for a closer look, switch to a viewing rectangle extending from 0 to $2 \pi$ in the $x$ -direction and from $-1$ to 1 in the $y$ -direction. Compare the two graphs; write out your observations in complete sentences.

(b) Use the graphing utility to estimate the amplitude of the function $y=\sin (\sin x)$

(c) Using your knowledge of the sine function, explain why the amplitude of the function $y=\sin (\sin x)$ is the number sin $1 .$ Then evaluate $\sin 1$ and use the result to check your approximation in part (b).

Adnan G.

Numerade Educator

Let $f(x)=e^{x / 20}(\sin x)$

(a) Graph the function $f$ using a viewing rectangle that extends from $-5$ to 5 in both the $x$ - and the $y$ -directions. Note that the resulting graph resembles a sine curve.

(b) Change the viewing rectangle so that $x$ extends from 0 to 50 and $y$ extends from $-10$ to $10 .$ Describe what you see. Is the function periodic?

(c) Add the graphs of the two functions $y=e^{x / 20}$ and $y=-e^{x / 20}$ to your picture in part (b). Describe what you see.

Adnan G.

Numerade Educator

For this exercise, use the standard viewing rectangle.

(a) Graph the function $y=\ln \left(\sin ^{2} x\right)$

(b) Graph the function $y=\ln (1-\cos x)+\ln (1+\cos x)$

(c) Explain why the two graphs are identical.

Adnan G.

Numerade Educator