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Chapter 4: Problem 35

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. (Include two full periods.) $$\begin{aligned} &f(x)=-\frac{1}{2} \sin \frac{x}{2}\\\ &g(x)=3-\frac{1}{2} \sin \frac{x}{2} \end{aligned}$$

Short Answer

Expert verified
The graph of \(f(x) = -\frac{1}{2}\sin\frac{x}{2}\) is a reflection of the sine function squashed towards the x-axis and that of \(g(x) = 3 -\frac{1}{2}\sin\frac{x}{2}\) is the graph of \(f(x)\) shifted vertically upwards by 3 units. Both graphs have a period of \(4\pi\), so they need to be plotted from \(x = 0\) to \(x = 8\pi\) to include two full periods.

Step by step solution

01

Analyze the functions

Begin by looking at the formulas for \(f(x)\) and \(g(x)\). The first function, \(f(x)\), is \(-\frac{1}{2}\sin\frac{x}{2}\) which is the sine function scaled by a factor of -1/2 and horizontally stretched by a factor of 2. This results in a vertically squashed reflection of the sine curve. The second function, \(g(x)\), is \(3 -\frac{1}{2}\sin\frac{x}{2}\). This is the first function with the addition of 3 which results in a vertical shift of the graph of 3 units upwards.
02

Plot the graphs

Start by producing the \(y = \sin x\) plot. Since \(-\frac{1}{2}\sin\frac{x}{2}\) has been horizontally stretched by a factor of 2 and vertically scaled -1/2, the resultant graph will be a reflection of the sine curve squashed towards the x-axis. For plotting \(g(x) = 3 -\frac{1}{2}\sin\frac{x}{2}\), add 3 to every y-coordinate of the \(f(x) = -\frac{1}{2}\sin\frac{x}{2}\) function, which will vertically shift the graph of \(f(x)\) up by 3 units to produce the graph of \(g(x)\).
03

Include two full periods

The period of the functions is \(4\pi\) (2 times the normal period of \(\pi\), since there is a horizontal stretch by a factor of 2). Therefore, the graph should extend from \(x = 0\) to \(x = 8\pi\) to include two full periods of the sine function.

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Most popular questions from this chapter

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) As \(x \rightarrow 0^{+},\) the value of \(f(x) \rightarrow\) (b) As \(x \rightarrow 0^{-},\) the value of \(f(x) \rightarrow\) (c) As \(x \rightarrow \pi^{+},\) the value of \(f(x) \rightarrow\) (d) As \(x \rightarrow \pi^{-},\) the value of \(f(x) \rightarrow\) $$f(x)=\csc x$$

Determine whether the statement is true or false. Justify your answer. You can obtain the graph of \(y=\sec x\) on a calculator by graphing a translation of the reciprocal of \(y=\sin x\)

Sketch a graph of the function and compare the graph of \(g\) with the graph of \(f(x)=\arcsin x\). $$g(x)=\arcsin (x-1)$$

Determine whether the statement is true or false. Justify your answer. $$\sin \frac{5 \pi}{6}=\frac{1}{2} \quad \rightarrow \quad \arcsin \frac{1}{2}=\frac{5 \pi}{6}$$

Fill in the blank. If not possible, state the reason. $$\text { As } x \rightarrow-\infty, \text { the value of } \arctan x \rightarrow\text { _____ } .$$
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