Stewart Precalc 6e Section 5.3: Trigonometric Graphs

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Stewart Precalc 6e Section 5.3: Trigonometric Graphs 0/94
1 Concepts · Level 1
The trigonometric functions \(y = \sin x\) and \(y = \cos x\) have amplitude ___ and period ___. Sketch a graph of each function on the interval \([0, 2 \pi]\).
2 Concepts · Level 1
The trigonometric function \(y = 3 \sin 2x\) has amplitude ___ and period ___.
3 Skills - Graph the function · Level 1
Graph the function \(f(x) = 1 + \cos x\).
4 Skills - Graph the function · Level 1
Graph the function \(f(x) = 3 + \sin x\).
5 Skills - Graph the function · Level 1
Graph the function \(f(x) = -\sin x\).
6 Skills - Graph the function · Level 1
Graph the function \(f(x) = 2 - \cos x\).
7 Skills - Graph the function · Level 1
Graph the function \(f(x) = -2 + \sin x\).
8 Skills - Graph the function · Level 1
Graph the function \(f(x) = -1 + \cos x\).
9 Skills - Graph the function · Level 1
Graph the function \(g(x) = 3 \cos x\).
10 Skills - Graph the function · Level 1
Graph the function \(g(x) = 2 \sin x\).
11 Skills - Graph the function · Level 1
Graph the function \(g(x) = -\dfrac{1}{2} \sin x\).
12 Skills - Graph the function · Level 1
Graph the function \(g(x) = -\dfrac{2}{3} \cos x\).
13 Skills - Graph the function · Level 2
Graph the function \(g(x) = 3 + 3 \cos x\).
14 Skills - Graph the function · Level 2
Graph the function \(g(x) = 4 - 2 \sin x\).
15 Skills - Graph the function · Level 2
Graph the function \(h(x) = |\cos x|\).
16 Skills - Graph the function · Level 2
Graph the function \(h(x) = |\sin x|\).
17 Skills - Amplitude and period · Level 2
Find the amplitude and period of the function \(y = \cos 2x\), and sketch its graph.
18 Skills - Amplitude and period · Level 2
Find the amplitude and period of the function \(y = -\sin 2x\), and sketch its graph.
19 Skills - Amplitude and period · Level 2
Find the amplitude and period of the function \(y = -3 \sin 3x\), and sketch its graph.
20 Skills - Amplitude and period · Level 2
Find the amplitude and period of the function \(y = \dfrac{1}{2} \cos 4x\), and sketch its graph.
21 Skills - Amplitude and period · Level 2
Find the amplitude and period of the function \(y = 10 \sin\left(\dfrac{1}{2} x\right)\), and sketch its graph.
22 Skills - Amplitude and period · Level 2
Find the amplitude and period of the function \(y = 5 \cos\left(\dfrac{1}{4} x\right)\), and sketch its graph.
23 Skills - Amplitude and period · Level 2
Find the amplitude and period of the function \(y = -\dfrac{1}{3} \cos\left(\dfrac{1}{3} x\right)\), and sketch its graph.
24 Skills - Amplitude and period · Level 2
Find the amplitude and period of the function \(y = 4 \sin(-2x)\), and sketch its graph.
25 Skills - Amplitude and period · Level 2
Find the amplitude and period of the function \(y = -2 \sin 2 \pi x\), and sketch its graph.
26 Skills - Amplitude and period · Level 2
Find the amplitude and period of the function \(y = -3 \sin \pi x\), and sketch its graph.
27 Skills - Amplitude and period · Level 2
Find the amplitude and period of the function \(y = 1 + \dfrac{1}{2} \cos \pi x\), and sketch its graph.
28 Skills - Amplitude and period · Level 2
Find the amplitude and period of the function \(y = -2 + \cos 4 \pi x\), and sketch its graph.
29 Skills - Amplitude, period, phase shift · Level 2
Find the amplitude, period, and phase shift of \(y = \cos\left(x - \dfrac{\pi}{2}\right)\), and graph one complete period.
30 Skills - Amplitude, period, phase shift · Level 2
Find the amplitude, period, and phase shift of \(y = 2 \sin\left(x - \dfrac{\pi}{3}\right)\), and graph one complete period.
31 Skills - Amplitude, period, phase shift · Level 2
Find the amplitude, period, and phase shift of \(y = -2 \sin\left(x - \dfrac{\pi}{6}\right)\), and graph one complete period.
32 Skills - Amplitude, period, phase shift · Level 2
Find the amplitude, period, and phase shift of \(y = 3 \cos\left(x + \dfrac{\pi}{4}\right)\), and graph one complete period.
33 Skills - Amplitude, period, phase shift · Level 3
Find the amplitude, period, and phase shift of \(y = -4 \sin 2\left(x + \dfrac{\pi}{2}\right)\), and graph one complete period.
34 Skills - Amplitude, period, phase shift · Level 3
Find the amplitude, period, and phase shift of \(y = \sin\left(\dfrac{1}{2}\right)\left(x + \dfrac{\pi}{4}\right)\), and graph one complete period.
35 Skills - Amplitude, period, phase shift · Level 3
Find the amplitude, period, and phase shift of \(y = 5 \cos\left(3x - \dfrac{\pi}{4}\right)\), and graph one complete period.
36 Skills - Amplitude, period, phase shift · Level 3
Find the amplitude, period, and phase shift of \(y = 2 \sin(\left(\dfrac{2}{3}\right) x - \dfrac{\pi}{6})\), and graph one complete period.
37 Skills - Amplitude, period, phase shift · Level 3
Find the amplitude, period, and phase shift of \(y = \dfrac{1}{2} - \dfrac{1}{2} \cos\left(2x - \dfrac{\pi}{3}\right)\), and graph one complete period.
38 Skills - Amplitude, period, phase shift · Level 3
Find the amplitude, period, and phase shift of \(y = 1 + \cos\left(3x + \dfrac{\pi}{2}\right)\), and graph one complete period.
39 Skills - Amplitude, period, phase shift · Level 3
Find the amplitude, period, and phase shift of \(y = 3 \cos \pi\left(x + \dfrac{1}{2}\right)\), and graph one complete period.
40 Skills - Amplitude, period, phase shift · Level 3
Find the amplitude, period, and phase shift of \(y = 3 + 2 \sin 3(x + 1)\), and graph one complete period.
41 Skills - Amplitude, period, phase shift · Level 3
Find the amplitude, period, and phase shift of \(y = \sin(\pi + 3x)\), and graph one complete period.
42 Skills - Amplitude, period, phase shift · Level 3
Find the amplitude, period, and phase shift of \(y = \cos\left(\dfrac{\pi}{2} - x\right)\), and graph one complete period.
43 Skills - Equation from graph · Level 3
The graph of one complete period of a sine or cosine curve is given. (a) Find the amplitude, period, and phase shift. (b) Write an equation that represents the curve in the form \(y = a \sin k(x - b)\) or \(y = a \cos k(x - b)\).
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44 Skills - Equation from graph · Level 3
The graph of one complete period of a sine or cosine curve is given. (a) Find the amplitude, period, and phase shift. (b) Write an equation that represents the curve in the form \(y = a \sin k(x - b)\) or \(y = a \cos k(x - b)\).
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45 Skills - Equation from graph · Level 3
The graph of one complete period of a sine or cosine curve is given. (a) Find the amplitude, period, and phase shift. (b) Write an equation that represents the curve in the form \(y = a \sin k(x - b)\) or \(y = a \cos k(x - b)\).
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46 Skills - Equation from graph · Level 3
The graph of one complete period of a sine or cosine curve is given. (a) Find the amplitude, period, and phase shift. (b) Write an equation that represents the curve in the form \(y = a \sin k(x - b)\) or \(y = a \cos k(x - b)\).
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47 Skills - Equation from graph · Level 3
The graph of one complete period of a sine or cosine curve is given. (a) Find the amplitude, period, and phase shift. (b) Write an equation that represents the curve in the form \(y = a \sin k(x - b)\) or \(y = a \cos k(x - b)\).
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48 Skills - Equation from graph · Level 3
The graph of one complete period of a sine or cosine curve is given. (a) Find the amplitude, period, and phase shift. (b) Write an equation that represents the curve in the form \(y = a \sin k(x - b)\) or \(y = a \cos k(x - b)\).
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49 Skills - Equation from graph · Level 3
The graph of one complete period of a sine or cosine curve is given. (a) Find the amplitude, period, and phase shift. (b) Write an equation that represents the curve in the form \(y = a \sin k(x - b)\) or \(y = a \cos k(x - b)\).
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50 Skills - Equation from graph · Level 3
The graph of one complete period of a sine or cosine curve is given. (a) Find the amplitude, period, and phase shift. (b) Write an equation that represents the curve in the form \(y = a \sin k(x - b)\) or \(y = a \cos k(x - b)\).
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51 Skills - Viewing rectangle · Level 3
Determine an appropriate viewing rectangle for \(f(x) = \cos 100 x\), and use it to draw the graph.
52 Skills - Viewing rectangle · Level 3
Determine an appropriate viewing rectangle for \(f(x) = 3 \sin 120 x\), and use it to draw the graph.
53 Skills - Viewing rectangle · Level 3
Determine an appropriate viewing rectangle for \(f(x) = \sin\left(\dfrac{x}{40}\right)\), and use it to draw the graph.
54 Skills - Viewing rectangle · Level 3
Determine an appropriate viewing rectangle for \(f(x) = \cos\left(\dfrac{x}{80}\right)\), and use it to draw the graph.
55 Skills - Viewing rectangle · Level 3
Determine an appropriate viewing rectangle for \(y = \tan 25 x\), and use it to draw the graph.
56 Skills - Viewing rectangle · Level 3
Determine an appropriate viewing rectangle for \(y = \csc 40 x\), and use it to draw the graph.
57 Skills - Viewing rectangle · Level 3
Determine an appropriate viewing rectangle for \(y = \sin^2 20 x\), and use it to draw the graph.
58 Skills - Viewing rectangle · Level 3
Determine an appropriate viewing rectangle for \(y = \sqrt{\tan 10 \pi x}\), and use it to draw the graph.
59 Skills - Graphical addition · Level 2
Graph \(f\), \(g\), and \(f + g\) on a common screen to illustrate graphical addition: \(f(x) = x\), \(g(x) = \sin x\).
60 Skills - Graphical addition · Level 2
Graph \(f\), \(g\), and \(f + g\) on a common screen to illustrate graphical addition: \(f(x) = \sin x\), \(g(x) = \sin 2 x\).
61 Skills - Variable amplitude · Level 3
Graph the three functions on a common screen: \(y = x^2\), \(y = -x^2\), \(y = x^2 \sin x\). How are the graphs related?
62 Skills - Variable amplitude · Level 3
Graph the three functions on a common screen: \(y = x\), \(y = -x\), \(y = x \cos x\). How are the graphs related?
63 Skills - Variable amplitude · Level 3
Graph the three functions on a common screen: \(y = \sqrt{x}\), \(y = -\sqrt{x}\), \(y = \sqrt{x} \sin 5 \pi x\) (for \(x \geq 0\)). How are the graphs related?
64 Skills - Variable amplitude · Level 3
Graph the three functions on a common screen: \(y = 1/(1+x^2)\), \(y = -1/(1+x^2)\), \(y = \cos\dfrac{2 \pi x}{1+x^2}\). How are the graphs related?
65 Skills - Variable amplitude · Level 3
Graph the three functions on a common screen: \(y = \cos 3 \pi x\), \(y = -\cos 3 \pi x\), \(y = \cos 3 \pi x \cos 21 \pi x\). How are the graphs related?
66 Skills - Variable amplitude · Level 3
Graph the three functions on a common screen: \(y = \sin 2 \pi x\), \(y = -\sin 2 \pi x\), \(y = \sin 2 \pi x \sin 10 \pi x\). How are the graphs related?
67 Skills - Max and min · Level 3
Find the maximum and minimum values of the function \(y = \sin x + \sin 2x\).
68 Skills - Max and min · Level 3
Find the maximum and minimum values of \(y = x - 2 \sin x\), \(0 \leq x \leq 2 \pi\).
69 Skills - Max and min · Level 3
Find the maximum and minimum values of \(y = 2 \sin x + \sin^2 x\).
70 Skills - Max and min · Level 3
Find the maximum and minimum values of \(y = \cos x / (2 + \sin x)\).
71 Skills - Solutions in interval · Level 2
Find all solutions of the equation \(\cos x = 0.4\) that lie in the interval \([0, \pi]\). State each answer correct to two decimal places.
72 Skills - Solutions in interval · Level 2
Find all solutions of the equation \(\tan x = 2\) that lie in the interval \([0, \pi]\). State each answer correct to two decimal places.
73 Skills - Solutions in interval · Level 2
Find all solutions of the equation \(\csc x = 3\) that lie in the interval \([0, \pi]\). State each answer correct to two decimal places.
74 Skills - Solutions in interval · Level 3
Find all solutions of the equation \(\cos x = x\) that lie in the interval \([0, \pi]\). State each answer correct to two decimal places.
75 Skills - Function analysis · Level 3
For \(f(x) = (1 - \cos x)/x\): (a) Is \(f\) even, odd, or neither? (b) Find the \(x\)-intercepts of the graph of \(f\). (c) Graph \(f\) in an appropriate viewing rectangle. (d) Describe the behavior of the function as \(x \rightarrow \pm \infty\). (e) Notice that \(f(x)\) is not defined when \(x = 0\). What happens as \(x\) approaches \(0\)?
76 Skills - Function analysis · Level 3
For \(f(x) = \sin 4 x / (2 x)\): (a) Is \(f\) even, odd, or neither? (b) Find the \(x\)-intercepts of the graph of \(f\). (c) Graph \(f\) in an appropriate viewing rectangle. (d) Describe the behavior of the function as \(x \rightarrow \pm \infty\). (e) Notice that \(f(x)\) is not defined when \(x = 0\). What happens as \(x\) approaches \(0\)?
77 Applications - Height of a wave · Level 2
As a wave passes by an offshore piling, the height of the water is modeled by the function \(h(t) = 3 \cos(\left(\dfrac{\pi}{10}\right) t)\), where \(h(t)\) is the height in feet above mean sea level at time \(t\) seconds. (a) Find the period of the wave. (b) Find the wave height, that is, the vertical distance between the trough and the crest of the wave.
78 Applications - Sound vibrations · Level 2
A tuning fork is struck, producing a pure tone as its tines vibrate. The vibrations are modeled by the function \(v(t) = 0.7 \sin(880 \pi t)\), where \(v(t)\) is the displacement of the tines in millimeters at time \(t\) seconds. (a) Find the period of the vibration. (b) Find the frequency of the vibration, that is, the number of times the fork vibrates per second. (c) Graph the function \(v\).
79 Applications - Blood pressure · Level 3
A certain person's blood pressure is modeled by \(p(t) = 115 + 25 \sin(160 \pi t)\), where \(p(t)\) is pressure in mmHg at time \(t\) minutes. (a) Find the period of \(p\). (b) Find the number of heartbeats per minute. (c) Graph the function \(p\). (d) Find the blood pressure reading. How does this compare to normal blood pressure?
80 Applications - Variable stars · Level 2
The brightness of R Leonis is modeled by \(b(t) = 7.9 - 2.1 \cos(\left(\dfrac{\pi}{156}\right) t)\), where \(t\) is measured in days. (a) Find the period of R Leonis. (b) Find the maximum and minimum brightness. (c) Graph the function \(b\).
81 Discovery - Compositions · Level 3
This exercise explores the effect of the inner function \(g\) on a composite function \(y = f(g(x))\). (a) Graph the function \(y = \sin \sqrt{x}\) using the viewing rectangle \([0, 400]\) by \([-1.5, 1.5]\). In what ways does this graph differ from the graph of the sine function? (b) Graph the function \(y = \sin(x^2)\) using the viewing rectangle \([-5, 5]\) by \([-1.5, 1.5]\). In what ways does this graph differ from the graph of the sine function?
82 Discovery - Periodic functions I · Level 3
Recall that a function \(f\) is periodic if there is a positive number \(p\) such that \(f(t + p) = f(t)\) for every \(t\), and the least such \(p\) (if it exists) is the period of \(f\). The graph of a function of period \(p\) looks the same on each interval of length \(p\), so we can easily determine the period from the graph. Determine whether each of the four functions whose graphs are shown is periodic; if it is periodic, find the period.
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83 Discovery - Periodic functions II · Level 3
Use a graphing device to graph the following functions. From the graph, determine whether the function is periodic; if it is periodic, find the period. (a) \(y = |\sin x|\). (b) \(y = \sin |x|\). (c) \(y = 2^{\cos x}\). (d) \(y = x - \lfloor x \rfloor\). (e) \(y = \cos(\sin x)\). (f) \(y = \cos(x^2)\).
84 Discovery - Sinusoidal curves · Level 3
The graph of \(y = \sin x\) is the same as the graph of \(y = \cos x\) shifted to the right \(\dfrac{\pi}{2}\) units. So the sine curve \(y = \sin x\) is also at the same time a cosine curve: \(y = \cos\left(x - \dfrac{\pi}{2}\right)\). In fact, any sine curve is also a cosine curve with a different phase shift, and any cosine curve is also a sine curve. Sine and cosine curves are collectively referred to as sinusoidal. For the curve whose graph is shown, find all possible ways of expressing it as a sine curve \(y = a \sin(x - b)\) or as a cosine curve \(y = a \cos(x - b)\). Explain why you think you have found all possible choices for \(a\) and \(b\) in each case.
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85 Example - Cosine curves (transformations) · Level 2
Sketch the graph of each function.
(a) \(f(x) = 2 + \cos x\)
(b) \(g(x) = -\cos x\)

Enter your answer directly below each part above.

86 Example - Stretching a Cosine Curve · Level 2
Find the amplitude of \(y = -3 \cos x\), and sketch its graph.
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87 Example - Amplitude and Period · Level 2
Find the amplitude and period of each function, and sketch its graph.
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(a) \(y = 4 \cos(3 x)\)
(b) \(y = -2 \sin(\left(\dfrac{1}{2}\right) x)\)

Enter your answer directly below each part above.

88 Example - A Shifted Sine Curve · Level 3
Find the amplitude, period, and phase shift of \(y = 3 \sin(2\left(x - \dfrac{\pi}{4}\right))\), and graph one complete period.
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89 Example - A Shifted Cosine Curve · Level 3
Find the amplitude, period, and phase shift of \(y = \dfrac{3}{4} \cos(2 x + (2 \pi)/3)\) and graph one complete period.
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90 Example - Choosing the Viewing Rectangle · Level 3
Graph the function \(f(x) = \sin(50 x)\) in an appropriate viewing rectangle.
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91 Example - Sum of Sine and Cosine Curves · Level 2
Graph \(f(x) = 2 \cos x\), \(g(x) = \sin 2x\), and \(h(x) = 2 \cos x + \sin 2x\) on a common screen to illustrate the method of graphical addition.
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92 Example - Cosine Curve with Variable Amplitude · Level 3
Graph the functions \(y = x^2\), \(y = -x^2\), and \(y = x^2 \cos 6 \pi x\) on a common screen. Comment on and explain the relationship among the graphs.
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93 Example - Cosine Curve with Variable Amplitude · Level 3
Graph the function \(f(x) = \cos 2 \pi x \cos 16 \pi x\).
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94 Example - Sine Curve with Decaying Amplitude · Level 3
The function \(f(x) = \sin(x)/x\) is important in calculus. Graph this function and comment on its behavior when \(x\) is close to \(0\).
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