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Chapter 4: Problem 56

Use a vertical shift to graph one period of the function. $$y=\cos x+3$$

Short Answer

Expert verified
The function \(y = \cos x + 3\) is graphed as a cosine wave, oscillating between 2 and 4 over one period from x = 0 to x = \(2\pi\).

Step by step solution

01

Identify the Basic Function

The basic function is \(\cos x\). The graph of the cosine function, \(\cos x\), is a wave that oscillates between -1 and 1 over a period of \(2\pi\). The peak of the wave is at \((0,1)\) and its minimum is at \((\pi,-1)\). This is the base pattern upon which modifications are applied.
02

Apply the Vertical Shift

The given function is \(y=\cos x+3\), which is the cosine function vertically shifted 3 units upwards. To apply the vertical shift, every y-coordinate of the original cosine function will be increased by 3. This means the peak of the wave now becomes \((0,1+3)=(0,4)\) and the trough of the wave becomes \((\pi,-1+3)=(\pi,2)\).
03

Draw the Graph

With this information, one period of the function can be graphed. The graph should show a wave pattern from x = 0 to x = \(2\pi\), oscillating between 2 and 4, with the highest point at \(y = 4\) when \(x = 0\) and the lowest point at \(y = 2\) when \(x = \pi\) or \(x = 2\pi\). The graph will show a complete wave pattern for one period which is \(2\pi\).

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After using the four-step procedure to graph \(y=-\cot \left(x+\frac{\pi}{4}\right),\) I checked my graph by verifying it was the graph of \(y=\cot x\) shifted left \(\frac{\pi}{4}\) unit and reflected about the \(x\) -axis.

Write the equation for a cosecant function satisfying the given conditions. $$\text { period: } 3 \pi ; \text { range: }(-\infty,-2] \cup[2, \infty)$$

Use a graphing utility to graph $$ y=\sin x+\frac{\sin 2 x}{2}+\frac{\sin 3 x}{3}+\frac{\sin 4 x}{4} $$ in a \(\left[-2 \pi, 2 \pi, \frac{\pi}{2}\right]\) by [-2,2,1] viewing rectangle. How do these waves compare to the smooth rolling waves of the basic sine curve?

Use a graphing utility to graph each pair of functions in the same viewing rectangle. Use a viewing rectangle that shows the graphs for at least two periods. $$y=-3.5 \cos \left(\pi x-\frac{\pi}{6}\right) \text { and } y=-3.5 \sec \left(\pi x-\frac{\pi}{6}\right)$$

will help you prepare for the material covered in the next section. $$\text { Simplify: } \frac{-\frac{3 \pi}{4}+\frac{\pi}{4}}{2}$$
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