Question

scaling the cubic function quick check
the graph of $g(x) = x^3 + 1$ needs to be horizontally compressed by a factor of $\frac{1}{12}$. what should the value of $b$ be in the new function $h(x) = (bx)^3 + 1$? (1 point)
\bigcirc 12
\bigcirc $-\frac{1}{12}$
\bigcirc $-12$
\bigcirc $\frac{1}{12}$

Explanation:

Step1: Recall horizontal compression rule

For a function \( y = f(x) \), a horizontal compression by a factor of \( \frac{1}{k} \) (where \( k>0 \)) is given by \( y = f(kx) \).

Step2: Apply to the given function

The original function is \( g(x)=x^{3}+1 \). We need to horizontally compress it by a factor of \( \frac{1}{12} \). Using the horizontal compression rule, the compressed function should be \( g(12x)=(12x)^{3}+1 \). Comparing this with \( h(x)=(bx)^{3}+1 \), we can see that \( b = 12 \). Also, since horizontal compression (or stretch) by a positive factor doesn't involve reflection (so \( b \) is positive here as we are compressing, not reflecting), we choose \( b = 12 \).

Answer:

12