Question

how does the graph of $f(x) = \sqrt3{x}$ change when $f(x)$ is replaced by $af(x)$ when $a$ is positive and greater than 1? (1 point)

\bigcirc the graph vertically compresses by a scale factor $a$.

\bigcirc the graph shifts to the right $a$ units.

\bigcirc the graph vertically stretches by a scale factor $a$.

\bigcirc the graph shifts to the left $a$ units.

Explanation:

Step1: Recall Vertical Stretch/Compression Rules

For a function \( y = f(x) \), when we have \( y = af(x) \):

• If \( a>1 \), the graph vertically stretches by a scale factor of \( a \).
• If \( 0 < a < 1 \), the graph vertically compresses by a scale factor of \( a \).

Step2: Analyze the Given Transformation

Here, the original function is \( f(x)=\sqrt[3]{x} \), and we are transforming it to \( af(x) \) where \( a>1 \) (positive and greater than 1).

• The option about shifting right/left is incorrect because horizontal shifts are of the form \( f(x - h) \), not \( af(x) \).
• Since \( a>1 \), it's a vertical stretch, not compression.

Answer:

The graph vertically stretches by a scale factor \( a \). (Corresponding option: The graph vertically stretches by a scale factor \( a \).)