Question

given the function $f(x) = x^3$, which of the following functions will have a graph that undergoes a reflection and a vertical stretch? (1 point)

$\bigcirc\\ g(x) = -\frac{1}{8}f(x)$

$\bigcirc\\ g(x) = -\frac{1}{2}f(x)$

$\bigcirc\\ g(x) = 7f(x)$

$\bigcirc\\ g(x) = -\frac{7}{2}f(x)$

Explanation:

Step1: Recall Transformations

For a function \( y = a f(x) \), \( |a| > 1 \) means vertical stretch, \( |a| < 1 \) means vertical compression, and a negative \( a \) means reflection over x - axis.

Step2: Analyze Each Option

• Option 1: \( g(x)=-\frac{1}{8}f(x) \). \( |-\frac{1}{8}|=\frac{1}{8}<1 \) (compression), and reflection. Not stretch.
• Option 2: \( g(x)=-\frac{1}{2}f(x) \). \( |-\frac{1}{2}|=\frac{1}{2}<1 \) (compression), and reflection. Not stretch.
• Option 3: \( g(x) = 7f(x) \). \( 7>1 \) (stretch), but no reflection (positive \( a \)).
• Option 4: \( g(x)=-\frac{7}{2}f(x) \). \( |-\frac{7}{2}|=\frac{7}{2}>1 \) (stretch) and \( a = -\frac{7}{2}<0 \) (reflection).

Answer:

\( g(x)=-\frac{7}{2}f(x) \) (the fourth option)