Question

Describe the graph of the function f(x)=\sin \left((2)/(3) x\right) as a transformation of its parent function.
A. The graph is a vertical stretch by a factor of (2)/(3).
B. The graph is a horizontal compression by a factor of (2)/(3).
C. The graph is a horizontal stretch by a factor of (2)/(3).
D. The graph is a vertical compression by a factor of (2)/(3).

Answer 1

To analyze the transformation of the function \( f(x) = \sin\left(\frac{2}{3} x\right) \) compared to its parent function \( g(x) = \sin(x) \), we should focus on the coefficient of \( x \) inside the sine function.

The transformation involves the argument of the sine function. In this case, we see that:

1. The sine function \( g(x) = \sin(x) \) has a period of \( 2\pi \).
2. The function \( f(x) = \sin\left(\frac{2}{3} x\right) \) has an argument that scales \( x \) by \( \frac{2}{3} \).

The period of the function \( f(x) \) can be determined by the formula for the period of a sine function, which is given by:

\[
\text{Period} = \frac{2\pi}{\text{coefficient of } x}
\]

In this case, the coefficient of \( x \) in \( \frac{2}{3} x \) is \( \frac{2}{3} \). So, the period of \( f(x) \) is:

\[
\text{Period} = \frac{2\pi}{\frac{2}{3}} = 2\pi \cdot \frac{3}{2} = 3\pi
\]

Since \( \frac{2}{3} < 1 \), it indicates that the graph of \( f(x) = \sin\left(\frac{2}{3} x\right) \) undergoes a horizontal stretch. Specifically, because the input \( x \) is multiplied by \( \frac{2}{3} \), the function takes longer to complete one period compared to the parent function.

Thus, the correct characterization of the transformation is:

**C. The graph is a horizontal stretch by a factor of \( \frac{3}{2} \) (the reciprocal of \( \frac{2}{3} \)).**

However, since the given options do not specify the correct factor for horizontal stretch directly, and you are asked to select one from the provided options only, option C is the most relevant in the context of horizontal stretch transformation.