Problem 31 - 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 function \(y = 2 \cos(x)\) is 2. When compared on the graph, we observe that \(y = 2 \cos(x)\) reaches up to 2 and down to -2 whereas \(y = \cos(x)\) only reaches up to 1 and down to -1.

Step 1: Find the Amplitude

The amplitude is determined from the coefficient of the cosine function. In this case, the function is \(y = 2 \cos(x)\), making the amplitude 2.

Step 2: Sketch the Function

Plot the function \(y = 2 \cos(x)\) from \(x = 0\) to \(x = 2 \pi\). Due to the amplitude being twice the default value, the graph will reach a maximum at \(y = 2\) and a minimum at \(y = -2\). Also, the cosine function starts at its maximum point.

Step 3: Compare with y = cos(x)

Next, plot the function \(y = \cos(x)\) on the same graph for comparison. This function has an amplitude of 1. It starts at its maximum value, decreases to its minimum and then rises back to its maximum in the given range.