Question

reflecting the cubic function quick check
given the function $f(x) = x^3$, which statement best describes the graph of $h(x) = -2f(x)$. (1 point)

• the graph will undergo both a reflection and a vertical stretch.
• the graph will undergo both a reflection and a vertical compression.
• the graph will undergo only a reflection.
• the graph will undergo only a vertical stretch.

Explanation:

Step1: Analyze the transformation of \( h(x) = -2f(x) \)

Given \( f(x) = x^3 \), then \( h(x)=-2x^3 \). For a function \( y = a\cdot f(x) \), if \( |a|> 1 \), it is a vertical stretch; if \( a<0 \), it is a reflection over the \( x \)-axis. Here, \( a = - 2 \), \( | - 2|=2>1 \) (so vertical stretch) and \( a=-2 < 0 \) (so reflection over \( x \)-axis).

Step2: Evaluate each option

• Option 1: "The graph will undergo both a reflection and a vertical stretch." Matches our analysis as \( a=-2 \) gives reflection (due to negative sign) and vertical stretch (due to \( |a| = 2>1 \)).
• Option 2: Vertical compression requires \( |a|<1 \), but \( | - 2| = 2>1 \), so incorrect.
• Option 3: There is also a vertical stretch, not only reflection, so incorrect.
• Option 4: There is also a reflection, not only vertical stretch, so incorrect.

Answer:

The graph will undergo both a reflection and a vertical stretch.