Question

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)=14x3+12
?(1 point)
Responses

k=0
k equals 0

k=4
k equals 4

k=2
k equals 2

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

Answer 1

To identify the factor \( k \) for the vertical compression from the function \( f(x) = x^3 + 2 \) to the function \( f(x) = 14x^3 + 12 \), we need to compare the coefficients of the terms in the two functions that impact their vertical stretch or compression.

1. The original function is \( f(x) = x^3 + 2 \).
2. The transformed function is \( f(x) = 14x^3 + 12 \).

The term that has a coefficient affecting the vertical stretch/compression is the coefficient of \( x^3 \).

In \( f(x) = x^3 + 2 \), the coefficient of \( x^3 \) is \( 1 \).
In \( f(x) = 14x^3 + 12 \), the coefficient of \( x^3 \) is \( 14 \).

Since a vertical stretch can be interpreted as multiplying by a factor, in this case, the transition from \( 1 \) (in the original function) to \( 14 \) (in the new function) indicates a vertical stretch rather than a compression.

Thus, if we consider the "compression" factor \( k \) as the ratio \( \frac{\text{new coefficient}}{\text{original coefficient}} \):

\[
k = \frac{14}{1} = 14
\]

However, since the problem states that we are looking for a vertical compression, there is no vertical compression in this example; instead, there is a vertical stretch by a factor of 14.

If we wanted to think about how the function changes under the inverse of compression, we might say \( k \) is focused on the reciprocal impact of compression. But in this scenario, the answer directly indicated is still \( k = 14 \).

Therefore, the correct response would be:

**k = 14**