Problem 56 - Precalculus Essentials

Question

Use a vertical shift to graph one period of the function. $$y=\cos x+3$$

Step-by-Step Solution

Answer

The function \(y = \cos x + 3\) is graphed as a cosine wave, oscillating between 2 and 4 over one period from x = 0 to x = \(2\pi\).

Step 1: Identify the Basic Function

The basic function is \(\cos x\). The graph of the cosine function, \(\cos x\), is a wave that oscillates between -1 and 1 over a period of \(2\pi\). The peak of the wave is at \((0,1)\) and its minimum is at \((\pi,-1)\). This is the base pattern upon which modifications are applied.

Step 2: Apply the Vertical Shift

The given function is \(y=\cos x+3\), which is the cosine function vertically shifted 3 units upwards. To apply the vertical shift, every y-coordinate of the original cosine function will be increased by 3. This means the peak of the wave now becomes \((0,1+3)=(0,4)\) and the trough of the wave becomes \((\pi,-1+3)=(\pi,2)\).

Step 3: Draw the Graph

With this information, one period of the function can be graphed. The graph should show a wave pattern from x = 0 to x = \(2\pi\), oscillating between 2 and 4, with the highest point at \(y = 4\) when \(x = 0\) and the lowest point at \(y = 2\) when \(x = \pi\) or \(x = 2\pi\). The graph will show a complete wave pattern for one period which is \(2\pi\).