Problem 58 - Precalculus Essentials

Question

Use a vertical shift to graph one period of the function. $$y=2 \cos \frac{1}{2} x+1$$

Step-by-Step Solution

Answer

The function \(y = 2 \cos \frac{1}{2}x + 1\) has a vertical shift of 1 unit up, an amplitude of 2, and a period of \(4 \pi\). So, when graphing, the center line of the cosine function will be at y=1, the maximum and minimum y-values will be y=3 and y=-1 respectively, and one complete cycle of the function will occur between x=0 and x=\(4 \pi \).

Step 1: Identify the vertical shift

As we can see, a constant of 1 is being added to the output of the cosine function. That means, this function has a vertical shift of 1 unit upwards.

Step 2: Identify the Amplitude

The coefficient before the cosine function, 2, is the amplitude which the magnitude of the highest or lowest point from the horizontal axis. This means the function is stretched to 2 units above and below the center line.

Step 3: Identify the period of the function

Given that the coefficient within the cosine function is \( \frac{1}{2} \). The period of cosine function is \(2 \pi \), so the period of this function will be \(2 \pi \) divided by \( \frac{1}{2} \), equal to \(4 \pi \).

Step 4: Graph the Function

When graphing the function, you would start from the vertical shift, which is at 1 on the y-axis. The maximum height of the wave would be 1 unit above this (at y=3) due to the amplitude of 2, and the minimum height of the wave would be 1 unit below this (at y=-1). For the x-values, since the period of the wave is \(4 \pi \), one complete cycle of the waveform would occur between x=0 and x=\(4 \pi \).