Which of the following correctly identifies the factor k for the vertical compression from the function f(x)=x3+2 to the function f(x)=1/4 x3+1/2 ?(1 point)

Question

Responses

k=2
k equals 2

k=0
k equals 0

k=14
k equals Start Fraction 1 over 4 End Fraction

k=4

Answer 1

To identify the factor \( k \) for the vertical compression from the function \( f(x) = x^3 + 2 \) to the function \( f(x) = \frac{1}{4} x^3 + \frac{1}{2} \), we need to analyze how the leading coefficient changes.

1. The original function is \( f(x) = x^3 + 2 \) (the leading term is \( x^3 \) with a coefficient of 1).
2. The new function is \( f(x) = \frac{1}{4} x^3 + \frac{1}{2} \) (the leading term now has a coefficient of \( \frac{1}{4} \)).

The leading coefficient has changed from 1 to \( \frac{1}{4} \).

The vertical compression factor \( k \) indicates how much the original function is compressed vertically. This can be determined by finding the ratio of the new leading coefficient to the original leading coefficient:

\[ k = \frac{\text{new coefficient}}{\text{original coefficient}} = \frac{\frac{1}{4}}{1} = \frac{1}{4} \]

Thus, the correct response is:

k equals \( \frac{1}{4} \).