### Practice Test

For the following exercises, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline.

$f\left(x\right)=5\mathrm{cos}\phantom{\rule{0.3em}{0ex}}x$

$f\left(x\right)=\mathrm{sin}\left(3x\right)$

$f\left(x\right)=5\mathrm{sin}\left(3\left(x-\frac{\pi}{6}\right)\right)+4$

$f\left(x\right)=\mathrm{tan}\left(4x\right)$

$f\left(x\right)=\pi \mathrm{cos}\left(3x+\pi \right)$

$f\left(x\right)=\pi \mathrm{sec}\left(\frac{\pi}{2}x\right)$

For the following exercises, determine the amplitude, period, and midline of the graph, and then find a formula for the function.

Give in terms of a sine function.

Give in terms of a tangent function.

For the following exercises, find the amplitude, period, phase shift, and midline.

$y=8\mathrm{sin}\left(\frac{7\pi}{6}x+\frac{7\pi}{2}\right)+6$

The outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 68°F at midnight and the high and low temperatures during the day are 80°F and 56°F, respectively. Assuming $t$ is the number of hours since midnight, find a function for the temperature, $D,$ in terms of $t.$

Water is pumped into a storage bin and empties according to a periodic rate. The depth of the water is 3 feet at its lowest at 2:00 a.m. and 71 feet at its highest, which occurs every 5 hours. Write a cosine function that models the depth of the water as a function of time, and then graph the function for one period.

For the following exercises, find the period and horizontal shift of each function.

$n\left(x\right)=4\mathrm{csc}\left(\frac{5\pi}{3}x-\frac{20\pi}{3}\right)$

If $\mathrm{tan}\phantom{\rule{0.3em}{0ex}}x=3,$ find $\mathrm{tan}\left(-x\right).$

For the following exercises, graph the functions on the specified window and answer the questions.

Graph $m\left(x\right)=\mathrm{sin}\left(2x\right)+\mathrm{cos}\left(3x\right)$ on the viewing window $\left[-10,10\right]$ by $\left[-3,3\right].$ Approximate the graph’s period.

Graph $n\left(x\right)=0.02\mathrm{sin}\left(50\pi x\right)$ on the following domains in $x:$ $\left[0,1\right]$ and $\left[0,3\right].$ Suppose this function models sound waves. Why would these views look so different?

Graph $f\left(x\right)=\frac{\mathrm{sin}\phantom{\rule{0.3em}{0ex}}x}{x}$ on $\left[-0.5,0.5\right]$ and explain any observations.

For the following exercises, let $f\left(x\right)=\frac{3}{5}\mathrm{cos}\left(6x\right).$

What is the smallest possible value for $f\left(x\right)?$

For the following exercises, find and graph one period of the periodic function with the given amplitude, period, and phase shift.

Sine curve with amplitude 3, period $\frac{\pi}{3},$ and phase shift $\left(h,k\right)=\left(\frac{\pi}{4},2\right)$

Cosine curve with amplitude 2, period $\frac{\pi}{6},$ and phase shift $\left(h,k\right)=\left(-\frac{\pi}{4},3\right)$

For the following exercises, graph the function. Describe the graph and, wherever applicable, any periodic behavior, amplitude, asymptotes, or undefined points.

$f\left(x\right)=5\mathrm{cos}\left(3x\right)+4\mathrm{sin}\left(2x\right)$

For the following exercises, find the exact value.

${\mathrm{sin}}^{-1}\left(\frac{\sqrt{3}}{2}\right)$

${\mathrm{cos}}^{-1}\left(-\frac{\sqrt{3}}{2}\right)$

${\mathrm{cos}}^{-1}\left(\mathrm{tan}\left(\frac{7\pi}{4}\right)\right)$

${\mathrm{cos}}^{-1}\left(-0.4\right)$

For the following exercises, suppose $\mathrm{sin}\phantom{\rule{0.3em}{0ex}}t=\frac{x}{x+1}.$ Evaluate the following expressions.

$\mathrm{tan}\phantom{\rule{0.3em}{0ex}}t$

For the following exercises, determine whether the equation is true or false.

$\mathrm{arccos}\left(\mathrm{cos}\left(\frac{5\pi}{6}\right)\right)=\frac{5\pi}{6}$

The grade of a road is 7%. This means that for every horizontal distance of 100 feet on the road, the vertical rise is 7 feet. Find the angle the road makes with the horizontal in radians.