Problem 34 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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 1: 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 \(3\).

Step 2: 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.

Step 3: 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 \(0 \leq x \leq 2\pi\).

Step 4: 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 \(2\pi\).