Question

Which option correctly describes the transformation from f(x)=x3
𝑓
(
𝑥
)
=
𝑥
3
to g(x)=(6x)3
𝑔
(
𝑥
)
=
(
6
𝑥
)
3
?(1 point)
Responses

a horizontal stretch by a factor of 6
a horizontal stretch by a factor of 6

a horizontal stretch by a factor of 16
1
6
a horizontal stretch by a factor of start fraction 1 over 6 end fraction

a horizontal compression by a factor of 6
a horizontal compression by a factor of 6

a horizontal compression by a factor of 16

Answer 1

To analyze the transformation from \( f(x) = x^3 \) to \( g(x) = (6x)^3 \), we can first rewrite \( g(x) \):

\[
g(x) = (6x)^3 = 6^3 \cdot x^3 = 216x^3
\]

However, to understand the horizontal transformation, we can focus specifically on how \( g(x) \) can be expressed in relation to \( f(x) \):

The expression \( g(x) = (6x)^3 \) can be thought of as:

\[
g(x) = f(6x)
\]

This indicates that we have replaced \( x \) with \( 6x \) in the function \( f(x) \).

Now, when you replace \( x \) with \( kx \) (where \( k > 1 \)), it results in a horizontal compression. Specifically, replacing \( x \) with \( 6x \) compresses the graph horizontally by a factor of \( \frac{1}{6} \).

Thus, the correct option is:
**a horizontal compression by a factor of 6.**