Question

how does the graph of $f(x) = \sqrt3{x}$ change when $f(x)$ is replaced by $f(bx)$ when $b$ is between 0 and -1? (1 point) \
\bigcirc the graph is reflected over the $y$-axis and stretches horizontally. \
\bigcirc the graph is reflected over the $x$-axis and compresses horizontally. \
\bigcirc the graph is reflected over the $y$-axis and stretches vertically. \
\bigcirc the graph is reflected over the $x$-axis and compresses vertically.

Explanation:

Step1: Analyze reflection

For a function \( y = f(x) \), replacing \( x \) with \( -x \) reflects over the \( y \)-axis. Here, \( b \) is negative (between 0 and -1), so \( f(bx)=f(-|b|x) \), which is a reflection over the \( y \)-axis.

Step2: Analyze horizontal stretch/compression

The transformation \( f(kx) \) for \( 0 < |k| < 1 \) is a horizontal stretch (since \( k=\frac{1}{a} \) where \( a > 1 \) for stretch). Here, \( |b| \) is between 0 and 1 (since \( b\in(0, - 1)\)), so \( |b|=\frac{1}{a} \) with \( a>1 \), so it's a horizontal stretch.

Answer:

The graph is reflected over the \( y \)-axis and stretches horizontally. (The first option: The graph is reflected over the \( y \)-axis and stretches horizontally)