Problem 33 - 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 \(0 \leq x \leq 2 \pi\). $$y=-2 \cos x$$

Step-by-Step Solution

Answer

The amplitude of the function \(y = -2 \cos x\) is 2. When graphed over the domain \(0 \leq x \leq 2 \pi\), it oscillates between -2 and 2, and is a vertically stretched and reflected version of the graph of \(y = \cos x\), which oscillates between -1 and 1.

Step 1: Determine the amplitude

From the given function \(y = -2 \cos x\), we can determine the amplitude by taking the absolute value of the coefficient of the cosine function. In this case, the amplitude is \(|-2| = 2\).

Step 2: Graph the function

To graph \(y = -2 \cos x\), first mark out the intervals of the graph on the x-axis with the domain \(0 \leq x \leq 2 \pi\) (0 to approximately 6.28). For \(y = -2 \cos x\), plot the corresponding y-values. Start at \(y = -2\) at \(x = 0\) as cosine of 0 is 1 (1 multiplied by -2 is -2). The cosine function achieves its minimum at \(\pi\) (multiplied by -2 gives 2), and returns to 1 at \(2\pi\) (multiplied by -2 gives -2). Your graph should oscillate between -2 and 2, starting and ending at -2 over this domain.

Step 3: Graph \(y = \cos x\)

The graph of \(y = \cos x\) will look very similar to the previous graph, but will have an amplitude of 1 instead of 2. Begin at \(y = 1\) at \(x = 0\), reaches a minimum at \(y = -1\) at \(x = \pi\), and returns to 1 at \(x = 2\pi\). The graph will oscillate between these two points over the given domain.

Step 4: Analyze the graphs

Comparing the two graphs, it can be seen that the graph of \(y = -2 \cos x\) is a vertically stretched version of \(y = \cos x\), and it is reflected over the x-axis. The amplitude is the absolute value of the maximum or minimum point on each graph from the x-axis, confirming our earlier calculation that the amplitude of \(y = -2 \cos x\) is 2, whereas the amplitude of \(y = \cos x\) is 1.