Problem 55 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The given function \(y=\cos x - 3\) is a vertical shift of the cosine function, where the entire graph of the basic cosine function is moved 3 units downwards.

Step 1: Understand the Basic Cosine Function

The basic cosine function \(y=\cos x\) has a period of \(2 \pi\), and wave amplitude of 1. It reaches its maximum value at \(x=0\) (and multiples of \(2 \pi\)) and minimum values at \(x=\pi\) (and multiples of \(2 \pi\)).

Step 2: Understanding the Impact of -3 on the Cosine Function

The -3 at the end of the function \(y=\cos x - 3\) indicates a vertical shift of the cosine function downwards by 3 units on the y-axis. The wave amplitude remains the same as the basic cosine function, at 1. So, the existing minimum and maximum values for the function will be shifted down by 3 units.

Step 3: Graphing the Function

To graph this function, first draw the basic cosine function. Then, translate the graph 3 units downwards. The maximum of the function will now be at \(y=1-3=-2\), and the minimum of the function will be at \(y=-1-3=-4\).