Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. (Include two full periods.) $$\begin{array}{l} f(x)=4 \sin x \\ g(x)=4 \sin x-1 \end{array}$$

Step-by-Step Solution

Answer

The graph of \(f(x)=4 \sin x\) has an amplitude of 4 and no vertical shift, while the graph of \(g(x)=4 \sin x-1\) also has an amplitude of 4 but with a vertical shift of -1 unit. Therefore, the graph of \(g(x)\) is similar to \(f(x)\) but shifted down by 1 unit.

Step 1: Understanding the functions

First, let's analyze the functions: the function \(f(x)=4 \sin x\) is a sinusoidal function with an amplitude of 4. This means the highest and lowest points on the graph will be 4 and -4 respectively. The function \(g(x)=4 \sin x-1\) is similar to \(f(x)\) but it has a vertical shift of -1 unit. That means the whole graph will be shifted down by one unit compared to \(f(x)\).

Step 2: Plot the function \(f(x)=4 \sin x\)

The graph of \(f(x)=4 \sin x\) starts at the origin (0,0). It reaches its peak at \(\pi/2, 4\), falls back to zero at \(\pi, 0\), goes to its minimum at \(3\pi/2, -4\), and finally comes back to zero at \(2\pi, 0\). This is one period of the function, and this pattern will repeat itself for another period.

Step 3: Plot the function \(g(x)=4 \sin x-1\)

Next, to plot \(g(x)=4 \sin x-1\), we use the same pattern as \(f(x)\), but we shift whole graph down by 1 unit. So, the graph starts at (0,-1), goes to its peak at \(\pi/2, 3\), returns to -1 at \(\pi, -1\), reaches its minimum value at \(3\pi/2, -5\), and finally returns to -1 at \(2\pi, -1\). This pattern repeats for the next period.

Key Concepts

Amplitude of a Sine Function

The amplitude of a sine function is a measure of how