Question

which of the following correctly identifies the factor $k$ for the vertical compression from the function $f(x)=x^3 + 2$ to the function $f(x)=\frac{1}{4}x^3+\frac{1}{2}$? (1 point)○ $k=0$○ $k=\frac{1}{4}$○ $k=2$○ $k=4$

Explanation:

Step1: Recall vertical compression rule

For a function $y=f(x)$, a vertical compression by factor $k$ (where $0

Step2: Apply rule to given functions

Start with original $f(x)=x^3+2$. A vertical compression gives $k(x^3+2)=kx^3+2k$.

Step3: Match to transformed function

Set equal to $\frac{1}{4}x^3+\frac{1}{2}$:
$kx^3+2k = \frac{1}{4}x^3+\frac{1}{2}$
Equate coefficients: $k=\frac{1}{4}$, and verify $2k=2\times\frac{1}{4}=\frac{1}{2}$, which matches.

Answer:

B. $k=\frac{1}{4}$