Problem 3 - Precalculus Essentials

Question

Determine the amplitude of each function. Then graph the function and \(y=\sin x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\). $$y=\frac{1}{3} \sin x$$

Step-by-Step Solution

Answer

The amplitude of the function \(y=\frac{1}{3} \sin x\) is \(\frac{1}{3}\), which means that its graph will reach peak and dip of \(\frac{1}{3}\) only, compared to the graph of \(y=\sin x\).

Step 1: Observing the function

Observe that given function is \(y=\frac{1}{3} \sin x\). This is a sine function where coefficient of \(\sin x\) is \(\frac{1}{3}\). Recall that for any function \(y=a \sin x\), the value of \(a\) is the amplitude.

Step 2: Finding the amplitude

The amplitude of the given function is \(\frac{1}{3}\).

Step 3: Comparing to normal sine function

Now, compare it with the function \(y = \sin x\). For \(y = \sin x\), the amplitude is 1, as there is no coefficient in front of the \(\sin x\). This means that the given function will reach only a third of the height of the \(y = \sin x\) at its peak and dip.

Step 4: Drawing the graph

Plot the graphs of \(y=\frac{1}{3} \sin x\) and \(y= \sin x\) for the range \(0 \leq x \leq 2 \pi\). Note that both graphs will fluctuate between their amplitude values and demonstrate standard sine wave behavior. The wave of \(y=\frac{1}{3} \sin x\) will be shorter due to its lower amplitude.