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Chapter 4: Problem 36

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. (Include two full periods.) $$\begin{array}{l} f(x)=4 \sin \pi x \\ g(x)=4 \sin \pi x-3 \end{array}$$

Short Answer

Expert verified
The graph of \(f(x) = 4 \sin \pi x\) consists of two full periods of a sine wave with an amplitude of 4 and a period of \(2 / \pi\), while the graph of \(g(x) = 4 \sin \pi x - 3\) is the same, but shifted downward by 3 units on the y-axis.

Step by step solution

01

Understand the Characteristics of f(x)

The function \(f(x) = 4 \sin \pi x\) is a sine function with an amplitude of 4 (the maximum distance from the mean value or rest position), a period of \(2 / \pi\) (the length of one complete cycle), and no phase shift or vertical shifts.
02

Sketch the Graph of f(x)

Start by drawing a horizontal line across the origin, which represents 0 on the y-axis. Since the period is \(2 / \pi\), mark two points on the x-axis representing the lengths of two full periods. Draw the curve starting from the origin reaching the peak at the half of the period, coming back to origin at the end of the period, and finally going down to a trough at the 1.5 times of the period then back to the origin at the end of the second period.
03

Understand the Characteristics of g(x)

The function \(g(x) = 4 \sin \pi x - 3\) is similar to f(x) with the same amplitude and period. However, it has a vertical downward shift of 3 units. This means the entire function is moved down on the coordinate plane by 3 units.
04

Sketch the Graph of g(x)

Repeat the same process as in step 2, but start the graph 3 units below the x-axis, keeping the centerline, the peaks, and the troughs 3 units below those of \(f(x)\). This represents the vertical shift of g(x). Keep the period same as before, \(2 / \pi\). So, the graph should go from -3 at start, reach 1 at the half of the period, come back to -3 at the end of the period, and finally go down to -7 at 1.5 times of the period then back to -3 at the end of the second period.

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