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)

Question

Responses

k=4
k equals 4

k=2
k equals 2

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

k=0

Answer 1

To determine the factor \( k \) for the vertical compression from the function \( f(x) = x^3 + 2 \) to the function \( f(x) = 14x^3 + 12 \), we can focus on the cubic term which has the main effect for vertical scaling.

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

We can see that the leading term \( x^3 \) in the original function has been changed to \( 14x^3 \) in the new function.

Vertical Compression Factor:

To find \( k \):

• The function is scaled vertically by multiplying the output of \( f(x) \) by \( k \).

The coefficient of \( x^3 \) in the new function is 14 and in the original function, it is 1 (since we can write it as \( 1x^3 \)).

Thus:

• If we denote the original function's output as \( y = f(x) = x^3 + 2 \), a vertical compression would imply that the new function is obtained by multiplying the original function \( f(x) \) outputs by a factor \( k \) less than 1.

With that said, the multiplication factor that transforms the leading coefficient from 1 to 14 is not a compression but an expansion.

Evaluating Options:

Given that we started with \( k \cdot (x^3 + 2) \) and got \( 14x^3 + 12 \):

Thus, \( k \) is not a compression; in fact, this is an expansion.

Final Conclusion:

Among the options given:

• The correct answer associated with vertical scaling from 1 to 14 must highlight that this is actually not a compression but rather an expansion.

The options provided do not offer a valid answer if we are strictly looking at vertical compression. However, the closest interpretation in concept is that \( k \) is effectively representing how far away the function moves away from the original.

Given that, none of the answers directly contribute to a vertical compression factor in the conventional sense leading to sufficient clarity in reasoning.

However, if revised strictly for "scaling," the conclusion can't meet the compression requirement but can ascertain \( k = 14 \) as applied vertical stretching as a note, despite suggesting otherwise in context.

So, the only hinting option that aligns with this transformation includes:

k = 14

Even though referring to it as compression is misleading, it channels larger "stretching via coefficient," reinforcing our understanding of function properties in cubic transformations, as thus.