Jay Abramson

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.

1.

$f (x) = 0.5 \sin x$

2.

$f (x) = 5 \cos x$

3.

$f (x) = 5 \sin x$

4.

$f (x) = \sin (3 x)$

5.

$f (x) = - \cos (x + \frac{π}{3}) + 1$

6.

$f (x) = 5 \sin (3 (x - \frac{π}{6})) + 4$

7.

$f (x) = 3 \cos (\frac{1}{3} x - \frac{5 π}{6})$

8.

$f (x) = \tan (4 x)$

9.

$f (x) = - 2 \tan (x - \frac{7 π}{6}) + 2$

10.

$f (x) = π \cos (3 x + π)$

11.

$f (x) = 5 \csc (3 x)$

12.

$f (x) = π \sec (\frac{π}{2} x)$

13.

$f (x) = 2 \csc (x + \frac{π}{4}) - 3$

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

14.

Give in terms of a sine function.

A graph of two periods of a sine function, graphed from -2 to 2. Range is [-6,-2], period is 2, and amplitude is 2.

15.

Give in terms of a sine function.

A graph of two periods of a sine function, graphed over -2 to 2. Range is [-2,2], period is 2, and amplitude is 2.

16.

Give in terms of a tangent function.

A graph of two periods of a tangent function, graphed over -3pi/4 to 5pi/4. Vertical asymptotes at x=-pi/4, 3pi/4. Period is pi.

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

17.

$y = \sin (\frac{π}{6} x + π) - 3$

18.

$y = 8 \sin (\frac{7 π}{6} x + \frac{7 π}{2}) + 6$

19.

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 $t .$

20.

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.

21.

$g (x) = 3 \tan (6 x + 42)$

22.

$n (x) = 4 \csc (\frac{5 π}{3} x - \frac{20 π}{3})$

23.

Write the equation for the graph in Figure 1 in terms of the secant function and give the period and phase shift.

A graph of 2 periods of a secant function, graphed over -2 to 2. The period is 2 and there is no phase shift.

Figure 1

24.

If $\tan x = 3,$ find $\tan (- x) .$

25.

If $\sec x = 4,$ find $\sec (- x) .$

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

26.

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

27.

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

28.

Graph $f (x) = \frac{\sin x}{x}$ on $[- 0.5, 0.5]$ and explain any observations.

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

29.

What is the largest possible value for $f (x) ?$

30.

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

31.

Where is the function increasing on the interval $[0, 2 π] ?$

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

32.

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

33.

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

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

34.

$f (x) = 5 \cos (3 x) + 4 \sin (2 x)$

35.

$f (x) = e^{\sin t}$

For the following exercises, find the exact value.

36.

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

37.

$\tan^{- 1} (\sqrt{3})$

38.

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

39.

$\cos^{- 1} (\sin (π))$

40.

$\cos^{- 1} (\tan (\frac{7 π}{4}))$

41.

$\cos (\sin^{- 1} (1 - 2 x))$

42.

$\cos^{- 1} (- 0.4)$

43.

$\cos (\tan^{- 1} (x^{2}))$

For the following exercises, suppose $\sin t = \frac{x}{x + 1} .$ Evaluate the following expressions.

44.

$\tan t$

45.

$\csc t$

46.

Given Figure 2, find the measure of angle $θ$ to three decimal places. Answer in radians.

An illustration of a right triangle with angle theta. Opposite the angle theta is a side with length 12, adjacent to the angle theta is a side with length 19.

Figure 2

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

47.

$\arcsin (\sin (\frac{5 π}{6})) = \frac{5 π}{6}$

48.

$\arccos (\cos (\frac{5 π}{6})) = \frac{5 π}{6}$

49.

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.