Problem 34 Determine the amplitude of each ... [FREE SOLUTION]

Chapter 5: Problem 34

Determine the amplitude of each function. Then graph the function and $y=\cos x$ in the same rectangular coordinate system for $0 \leq x \leq 2 \pi$. $$y=-3 \cos x$$

Short Answer

Expert verified

The function $y=-3\cos x$ has an amplitude of 3. In its graph, compared to $y=\cos x$, the wave pattern is inverted and stretched vertically by a factor of 3.

Step by step solution

Determine the Amplitude of the Function

The amplitude of a function described by $y = A \cos x$, where $A$ is the coefficient of $\cos x$, is the absolute value of $A$. In the given function $y = -3 \cos x$, the amplitude is $|-3|$, which equals $3$.

Plot the Graph of the Given Function

Plotting the graph for $y = -3\cos x$ involves recognizing key points, which can be derived from the basic cosine function $y=\cos x$. The only difference here is that each point's y-coordinate from $y=\cos x$ needs to be multiplied by -3 (the coefficient of $\cos x$). For instance, when $x=0$ in $y=\cos x$, $y=1$. In $y=-3\cos x$, this point becomes (0,-3). Similarly, we can get points like $(\pi/2,0), (\pi,3), (3\pi/2,0), (2\pi,-3)$ and use them to draw the graph of $y=-3\cos x$. The wave pattern should appear inverted (due to the negative sign) and stretched vertically (due to 3) compared to the basic cosine function.

Graph the Function $y=\cos x$

The function $y=\cos x$ has key points at (0,1), $(\pi/2,0), (\pi,-1), (3\pi/2,0), (2\pi,1)$. Plot these points and join them in a wave pattern to graph the function, ensuring that it completes one cycle in the interval $0 \leq x \leq 2\pi$.

Compare the Graphs

After graphing both functions, you can clearly see that both graphs display the wave pattern characteristic of cosine functions. However, compared to $y=\cos x$, the graph of $y=-3\cos x$ is inverted (due to the -3 factor) and stretched vertically by a factor of 3. Both graphs make one complete cycle within the range 0 to $2\pi$.

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91影视

Determine the amplitude of each function. Then graph the function and \(y=\cos x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\). $$y=-3 \cos x$$

Short Answer

Step by step solution

Determine the Amplitude of the Function

Plot the Graph of the Given Function

Graph the Function \(y=\cos x\)

Compare the Graphs

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