Question

the graph of $f(x)$ is shown. which graph represents $g(x)=f(2x)$?

Explanation:

Step1: Identify horizontal transformation rule

For $g(x)=f(kx)$ where $k>1$, the graph of $f(x)$ is horizontally compressed by a factor of $\frac{1}{k}$. Here $k=2$, so compress by $\frac{1}{2}$.

Step2: Map key points of $f(x)$

Original $f(x)$ has x-intercepts/vertices at $x=-2, 0, 2$. Apply compression: $x \to \frac{x}{2}$, so new x-values are $-1, 0, 1$.

Step3: Match to option

Find the graph with vertices at $x=-1, 0, 1$ (narrower than original, same y-values).

Answer:

The third option (the graph with a narrow "W" shape, vertices at $x=-1, 0, 1$ with the same vertical range as $f(x)$)