Question

how does the graph of $f(x) = \sqrt3{x}$ change when $f(x)$ is replaced by $af(x)$ if $-1 < a < 0$? (1 point) \
\bigcirc the graph is reflected across the x-axis and has been compressed vertically \
\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 y-axis and has been stretched vertically

Explanation:

Step1: Analyze Vertical Reflection

When we have a transformation \( y = af(x) \), if \( a < 0 \), the graph is reflected across the \( x \)-axis. Here, \( - 1< a<0 \), so there is a reflection across the \( x \)-axis.

Step2: Analyze Vertical Stretch/Compression

For the transformation \( y=af(x) \), if \( |a| < 1 \), the graph is vertically compressed, and if \( |a|>1 \), it is vertically stretched. Since \( - 1 < a<0 \), \( |a|=|a| \) and \( 0 < |a|<1 \), so the graph is vertically compressed.

Step3: Eliminate Other Options

• Option 2: Horizontal stretch is for transformations of the form \( y = f(bx) \) with \( 0 < b<1 \), not \( y = af(x) \), so this is wrong.
• Option 3: Reflection across \( y \)-axis is for \( y = f(-x) \), not \( y=af(x) \), and horizontal compression is for \( y = f(bx) \) with \( b > 1 \), so this is wrong.
• Option 4: Reflection across \( y \)-axis is incorrect (as explained) and vertical stretch is for \( |a|>1 \), but here \( |a| < 1 \), so this is wrong.

Answer:

A. The graph is reflected across the x - axis and has been compressed vertically