Problem 70
Question
Use a graphing utility to graph the function. Include two full periods. Be sure to choose an appropriate viewing window. $$ y=3 \cos \left(\frac{\pi x}{2}+\frac{\pi}{2}\right)-2 $$
Step-by-Step Solution
Verified Answer
The graph of the function \(y=3 \cos \left(\frac{\pi x}{2}+\frac{\pi}{2}\right)-2\) has an amplitude of 3, a period of 4, a phase shift of -1 units to the right, and a vertical shift of 2 units downward. Two full periods are shown from x = -5 to x = 3.
1Step 1: Understand the Function
The given function is in the form \(y=a \cos(bx+c)+d\), where 'a' is the amplitude, 'b' is related to the period of the function, 'c' is the phase shift, and 'd' is the vertical shift. For the given function \(y=3 \cos \left(\frac{\pi x}{2}+\frac{\pi}{2}\right)-2\), the amplitude is 3, the period is \(\frac{2\pi}{b}\), in this case \(\frac{2\pi}{\frac{\pi}{2}}=4\), the phase shift is \(-\frac{c}{b}\), here it is \(-\frac{\pi/2}{\pi/2}= -1\), and the vertical shift is -2.
2Step 2: Plot Key Points
The key points of a single period cosine curve are at x = 0, x = 1/4 period, x = 1/2 period, x = 3/4 period, x = period. Since the period is 4 and the phase shift is -1, the x values for key points are x = -1, x = -1+1, x = -1+2, x = -1+3, x = -1+4. Substitute these x into the function to get the corresponding y, then plot these points on the graph.
3Step 3: Draw the Function Curve
After plotting the key points, connect these points smoothly to create the cosine function curve. Remember that above and below these points, the curve will reach the maximum and minimum height that corresponds to the amplitude of 3.
4Step 4: Choose a Suitable Viewing Window
Since we need to graph two full periods, we need to adjust the viewing window of the graphing utility accordingly. In this case, for the x-axis from -1-4=-5 to -1+4*2=3 (the start of the first period to the end of the second period), and for the y-axis from -2-3=-5 to -2+3=1 (the vertical shift minus amplitude to the vertical shift plus amplitude).
Key Concepts
AmplitudePeriodPhase ShiftVertical Shift
Amplitude
Amplitude in trigonometric functions determines the height of the wave. For the function \(y = 3 \cos\left(\frac{\pi x}{2} + \frac{\pi}{2}\right) - 2\), the amplitude is represented by the constant \(a = 3\), which specifies how far the graph will stretch above and below its midline.
The midline of the graph is determined by the vertical shift, which we'll discuss later. Essentially, the amplitude shows how much the maximum and minimum values of the cosine wave differ from the midline:
The midline of the graph is determined by the vertical shift, which we'll discuss later. Essentially, the amplitude shows how much the maximum and minimum values of the cosine wave differ from the midline:
- A larger amplitude makes the peaks and valleys of the wave stand farther from the midline.
- A smaller amplitude results in peaks and valleys closer to the midline.
Period
The period of a trigonometric function is the horizontal length of one full cycle of the wave before it starts repeating. It's related to the speed at which the function oscillates. For the cosine function \(y = a \cos(bx + c) + d\), the period is given by \(\frac{2\pi}{b}\).
In our example function \(y = 3 \cos\left(\frac{\pi x}{2} + \frac{\pi}{2}\right) - 2\), \(b = \frac{\pi}{2}\), so the period is:\[\text{Period} = \frac{2\pi}{\frac{\pi}{2}} = 4\]This means that the entire wave pattern of the cosine function will complete one full cycle every 4 units along the x-axis. Understanding periods is essential for predicting when the wave pattern will repeat itself.
In our example function \(y = 3 \cos\left(\frac{\pi x}{2} + \frac{\pi}{2}\right) - 2\), \(b = \frac{\pi}{2}\), so the period is:\[\text{Period} = \frac{2\pi}{\frac{\pi}{2}} = 4\]This means that the entire wave pattern of the cosine function will complete one full cycle every 4 units along the x-axis. Understanding periods is essential for predicting when the wave pattern will repeat itself.
Phase Shift
Phase shift in trigonometric functions refers to the horizontal movement of the graph along the x-axis. For the function of the form \(y = a \cos(bx + c) + d\), the phase shift is calculated as \(-\frac{c}{b}\).
Applying this to our function \(y = 3 \cos\left(\frac{\pi x}{2} + \frac{\pi}{2}\right) - 2\), we find:\[\text{Phase Shift} = -\frac{\frac{\pi}{2}}{\frac{\pi}{2}} = -1\]This implies that the entire graph is shifted 1 unit to the left. The phase shift changes where the cosine wave starts repeating its cycle. Knowing the phase shift helps in plotting the key points correctly on the graph.
Applying this to our function \(y = 3 \cos\left(\frac{\pi x}{2} + \frac{\pi}{2}\right) - 2\), we find:\[\text{Phase Shift} = -\frac{\frac{\pi}{2}}{\frac{\pi}{2}} = -1\]This implies that the entire graph is shifted 1 unit to the left. The phase shift changes where the cosine wave starts repeating its cycle. Knowing the phase shift helps in plotting the key points correctly on the graph.
Vertical Shift
Vertical shift describes how far the entire graph of the function is moved up or down from the x-axis. It's especially important as it redefines the midline of the wave from which the amplitude is measured.
In an equation of the form \(y = a \cos(bx + c) + d\), the vertical shift is indicated by \(d\).
For our function \(y = 3 \cos\left(\frac{\pi x}{2} + \frac{\pi}{2}\right) - 2\), the vertical shift is -2, meaning the entire graph is moved 2 units downward.
This shift
In an equation of the form \(y = a \cos(bx + c) + d\), the vertical shift is indicated by \(d\).
For our function \(y = 3 \cos\left(\frac{\pi x}{2} + \frac{\pi}{2}\right) - 2\), the vertical shift is -2, meaning the entire graph is moved 2 units downward.
This shift
- lowers the horizontal midline of the amplitude to \(-2\),
- alters the graph so that peaks occur at 1 and valleys at -5,
- affects where the zero crossings of the wave happen.
Other exercises in this chapter
Problem 69
Write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example \(7 .)\) $$ \cos (\arcsin 2 x) $$
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Use a graphing utility to graph the two equations in the same viewing window. Use the graphs to determine whether the expressions are equivalent. Verify the res
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Convert the angle measure from degrees to radians. Round to three decimal places. $$ 345^{\circ} $$
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