Question

scaling the cube root function quick check
what is the effect on the graph of $f(x) = \sqrt3{x}$ when $f(x)$ is replaced by $f(2x)$? (1 point)

• the graph stretches vertically by a factor of 1/2.
• the graph compresses vertically by a factor of 1/2.
• the graph stretches horizontally by a factor of 1/2.
• the graph compresses horizontally by a factor of 1/2.

Explanation:

Step1: Recall Horizontal Transformation Rule

For a function \( y = f(kx) \), if \( |k|>1 \), the graph of \( f(x) \) is horizontally compressed by a factor of \( \frac{1}{k} \); if \( 0<|k|<1 \), it's horizontally stretched by a factor of \( \frac{1}{k} \).
Here, the original function is \( f(x)=\sqrt[3]{x} \), and the new function is \( f(2x)=\sqrt[3]{2x} \). So \( k = 2 \), and \( |k|=2>1 \).

Step2: Determine the Transformation

Using the rule, since \( k = 2 \), the horizontal compression factor is \( \frac{1}{k}=\frac{1}{2} \). So the graph of \( f(x)=\sqrt[3]{x} \) compresses horizontally by a factor of \( \frac{1}{2} \) when replaced by \( f(2x) \).

Answer:

The graph compresses horizontally by a factor of 1/2. (The corresponding option: The graph compresses horizontally by a factor of 1/2.)