Problem 28 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

In each of the following cases, use the graphing utility of your calculator to draw the graphs of the given pair of functions \(f(x)\) and \(g(x)\) on the same screen. Describe the relationship between the graphs of \(f(x)\) and \(g(x)\). a. \(f(x)=\sin x\) and \(g(x)=2 \sin x\) b. \(f(x)=\cos x\) and \(g(x)=2 \cos 2 x\) c. \(f(x)=\sin x\) and \(g(x)=\sin \left(x+\frac{\pi}{2}\right)\) d. \(f(x)=\cos x\) and \(g(x)=2+\cos x\)

Step-by-Step Solution

Answer

In part (a), the amplitude of g(x) is twice that of f(x). In part (b), the amplitude is doubled and the frequency is doubled. In part (c), g(x) is shifted left by \( \frac{\pi}{2}\). In part (d), g(x) is shifted up by 2 units.

Step 1 - Graphing the functions for part (a)

Plot the graphs of the functions \(f(x) = \sin x\) and \(g(x) = 2 \sin x\) using a graphing utility on the same screen. Observe the amplitude of the sine waves.

Step 2 - Describing the relationship between graphs in part (a)

The graph of \(g(x) = 2 \sin x\) has an amplitude that is twice that of the graph of \(f(x) = \sin x\). This means that the peaks and troughs of \(g(x)\) are twice as high and twice as low as those of \(f(x)\).

Step 3 - Graphing the functions for part (b)

Plot the graphs of the functions \(f(x) = \cos x\) and \(g(x) = 2 \cos 2x\) using a graphing utility on the same screen. Observe the frequency and amplitude.

Step 4 - Describing the relationship between graphs in part (b)

The graph of \(g(x) = 2 \cos 2x\) has an amplitude that is twice that of \(f(x) = \cos x\) and a frequency that is twice that of \(f(x)\). This results in \(g(x)\) completing two cycles for each cycle of \(f(x)\) and the peaks being twice as high.

Step 5 - Graphing the functions for part (c)

Plot the graphs of the functions \(f(x) = \sin x\) and \(g(x) = \sin \left(x + \frac{\pi}{2}\right)\) using a graphing utility on the same screen. Observe the horizontal shift.

Step 6 - Describing the relationship between graphs in part (c)

The graph of \(g(x) = \sin \left(x + \frac{\pi}{2}\right)\) is a horizontal shift of the graph of \(f(x) = \sin x\) by \(\frac{\pi}{2}\) units to the left.

Step 7 - Graphing the functions for part (d)

Plot the graphs of the functions \(f(x) = \cos x\) and \(g(x) = 2 + \cos x\) using a graphing utility on the same screen. Observe the vertical shift.

Step 8 - Describing the relationship between graphs in part (d)

The graph of \(g(x) = 2 + \cos x\) is a vertical shift of the graph of \(f(x) = \cos x\) by 2 units upwards.

Key Concepts

Amplitude

In trigonometric functions, the amplitude refers to the maximum absolute value of the function. It dictates how high and low the graph will go.
For example, when you look at the functions in part (a) of the exercise, you compare \(f(x) = \sin x\) with \(g(x) = 2 \sin x\). Here, \(f(x)\) has an amplitude of 1 while \(g(x)\) has an amplitude of 2.
This means that the peaks of \(g(x)\) will be twice as high and the troughs twice as low compared to \(f(x)\). The mathematical expression of amplitude in this case is simply the coefficient in front of the trigonometric function, which dictates the stretching of the graph vertically.

Frequency

The frequency of a trigonometric function concerns how many cycles it completes within a certain interval, typically within \(2\pi\). It defines the periodic nature of the graph.
In part (b) of the exercise, the frequency of \(f(x) = \cos x\) is compared with \(g(x) = 2 \cos 2x\). The function \(f(x)\) completes one cycle over \(2\pi\), while \(g(x)\) completes two cycles over the same interval. This increased frequency makes \(g(x)\) compress horizontally, leading to its graph being 'squished' so it completes more cycles in the same amount of space.
In mathematical terms, the frequency is directly related to the coefficient inside the trigonometric function (the number multiplied by the variable). A higher coefficient results in more cycles. So while \(f(x)\) has a frequency represented by 1 (one cycle per \(2\pi\)), \(g(x)\) has a frequency of 2 (two cycles per \(2\pi\)).

Horizontal Shift

A horizontal shift occurs when the entire graph of a trigonometric function moves left or right along the x-axis.
Consider part (c) of the exercise: \(f(x) = \sin x\) and \(g(x) = \sin(x + \pi/2)\). Here, \(g(x)\) is a horizontal shift of \(f(x)\) by \(\pi/2\) units to the left.
Essentially, every point on the graph of \(f(x)\) is moved \(\pi/2\) units leftward to produce the graph of \(g(x)\). To generalize, horizontal shifts are determined by the value added to or subtracted from the variable inside the function. For example, \(f(x) = \sin(x - c)\) would be a shift to the right by \(c\) units, while \(f(x) = \sin(x + c)\) would be a shift to the left by \(c\) units.

Vertical Shift

A vertical shift occurs when the entire graph of a trigonometric function moves up or down along the y-axis.
Analyzing part (d) of the exercise, \(f(x) = \cos x\) and \(g(x) = 2 + \cos x\), shows an example of vertical shift. Here, \(g(x)\) is a vertical shift of \(f(x)\) by 2 units upwards.
Each point on the graph of \(f(x)\) is elevated by 2 units to produce \(g(x)\). Vertical shifts are denoted by constants added to the function outside of the trigonometric expression. Thus, for \(f(x) = \cos x + c\), the entire graph will shift up by \(c\) units. Conversely, if it is \(f(x) = \cos x - c\), then the shift is downwards by \(c\) units.