Question

reflecting the cube root function quick check
what is the effect on the graph of $f(x) = \sqrt3{x}$ when $f(x)$ is replaced by $-2f(x)$? (1 point)
\bigcirc the graph is reflected across the $x$-axis and has been stretched horizontally.
\bigcirc the graph is reflected across the $y$-axis and has been compressed horizontally.
\bigcirc the graph is reflected across the $x$-axis and has been stretched vertically.
\bigcirc the graph is reflected across the $y$-axis and has been compressed vertically.

Explanation:

Step1: Analyze reflection

For a function \( y = f(x) \), replacing \( f(x) \) with \( -f(x) \) reflects the graph across the \( x \)-axis. Here we have \( -2f(x) \), so the negative sign means reflection across \( x \)-axis.

Step2: Analyze vertical stretch/compression

For a function \( y = f(x) \), replacing \( f(x) \) with \( a f(x) \) (where \( |a|>1 \)) vertically stretches the graph by a factor of \( |a| \). Here \( a=-2 \), \( | - 2|=2>1 \), so it's a vertical stretch by factor 2.

Answer:

The graph is reflected across the \( x \)-axis and has been stretched vertically. (The corresponding option: The graph is reflected across the \( x \)-axis and has been stretched vertically.)