Question

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

Answer 1

To analyze the function \( f(x) = \sin\left(\frac{3}{2} x\right) \) and how it transforms from its parent function \( g(x) = \sin(x) \), we need to look at the effect of the coefficient inside the sine function.

In general, the function \( f(x) = \sin(kx) \) where \( k \) is a constant, will result in a horizontal transformation of the graph:

1. If \( k > 1 \), the graph is horizontally compressed by a factor of \( \frac{1}{k} \).
2. If \( 0 < k < 1 \), the graph is horizontally stretched by a factor of \( \frac{1}{k} \).

In this case, we have \( k = \frac{3}{2} \) which is greater than 1. Therefore, the graph of \( f(x) = \sin\left(\frac{3}{2} x\right) \) is horizontally compressed by a factor of \( \frac{1}{\frac{3}{2}} = \frac{2}{3} \).

Thus, the correct transformation description for the graph of \( f(x) = \sin\left(\frac{3}{2} x\right) \) is:

**B. The graph is a horizontal compression by a factor of \( \frac{3}{2} \).**