Question

cubic & cube root functions unit online practice
complete this assessment to review what youve learned. it will not count toward your grade.
the function $f(x) = x^3$ is compressed vertically, and the resulting function is $g(x) = af(x)$. identify the value of
a if an ordered pair on the graph of $f(x)$, which is $(-3, -27)$, becomes $(-3, -9)$ after the vertical compression.
(1 point)
2
3
$\frac{1}{3}$
$\frac{1}{2}$

Explanation:

Step1: Recall the function transformation

We know that \( g(x)=af(x) \), and \( f(x)=x^3 \), so \( g(x)=a x^3 \). An ordered pair \((x,y)\) on \( f(x) \) will transform to \((x, a y)\) on \( g(x) \) because \( g(x)=a f(x) \), so if \((x,y)\) is on \( f(x) \), then \( y = f(x) \) and \( g(x)=a y \).

Step2: Substitute the ordered pairs

We have the original ordered pair \((-3, -27)\) on \( f(x) \), so \( x=-3 \), \( f(-3)=(-3)^3=-27 \). After transformation, the ordered pair is \((-3, -9)\) on \( g(x) \). Since \( g(-3)=a f(-3) \), substitute \( f(-3)=-27 \) and \( g(-3)=-9 \) into the equation:
\( -9=a\times(-27) \)

Step3: Solve for \( a \)

To find \( a \), we can solve the equation \( -9 = -27a \). Divide both sides of the equation by \(-27\):
\( a=\frac{-9}{-27}=\frac{1}{3} \)

Answer:

\(\frac{1}{3}\) (corresponding to the option with \(\frac{1}{3}\))