Question

reflecting the cubic function quick check
given the function $f(x) = x^3$, analyze the changes in the graph to determine which statement best describes the graph of $g(x) = -\frac{1}{2}f(x)$. (1 point)

• the graph will undergo a reflection due to the negative sign, and a vertical stretch because $|a| > 1$.
• the graph will only undergo a reflection due to the negative sign.
• the graph will undergo a vertical stretch because $0 < |a| < 1$.
• the graph will undergo a reflection due to the negative sign, and a vertical compression because $0 < |a| < 1$.

Explanation:

Step1: Recall Transformations

For a function \( y = a f(x) \), the coefficient \( a \) affects vertical transformations. A negative \( a \) reflects over the \( x \)-axis. If \( |a|>1 \), it's a vertical stretch; if \( 0<|a|<1 \), it's a vertical compression.

Step2: Analyze \( g(x)=-\frac{1}{2}f(x) \)

Here, \( a = -\frac{1}{2} \). The negative sign means reflection over \( x \)-axis. \( |a|=\frac{1}{2} \), and \( 0 < \frac{1}{2} < 1 \), so it's a vertical compression (since \( |a| \) is between 0 and 1).

Step3: Evaluate Options

• First option: \( |a|=\frac{1}{2}<1 \), not stretch. Wrong.
• Second option: Ignores the \( \frac{1}{2} \) (compression). Wrong.
• Third option: Ignores the reflection. Wrong.
• Fourth option: Reflection (negative) and compression (\( 0<|a|<1 \)). Correct.

Answer:

The graph will undergo a reflection due to the negative sign, and a vertical compression because \( 0 < |a| < 1 \)