Question
Which option correctly describes the transformation from f(x)=x3
to g(x)=(6x)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
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
a horizontal compression by a factor of start fraction 1 over 6 end fraction
to g(x)=(6x)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
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
a horizontal compression by a factor of start fraction 1 over 6 end fraction
Answers
GPT-4o mini
To analyze the transformation from \( f(x) = x^3 \) to \( g(x) = (6x)^3 \), we can rewrite \( g(x) \):
\[
g(x) = (6x)^3 = 6^3 \cdot x^3 = 216x^3
\]
However, the key part to analyze for the transformation is the \( (6x) \) inside the cubic function. Specifically, the transformation can be understood by substituting \( x \) in \( f(x) \):
\[
g(x) = f(6x)
\]
This means that we are replacing \( x \) with \( 6x \), which represents a horizontal compression. In general, if we have \( f(kx) \), where \( k > 1 \), it compresses the graph horizontally by a factor of \( k \).
Since \( k = 6 \) in this case, the transformation is a horizontal compression by a factor of 6.
Therefore, the correct option is:
**a horizontal compression by a factor of 6.**
\[
g(x) = (6x)^3 = 6^3 \cdot x^3 = 216x^3
\]
However, the key part to analyze for the transformation is the \( (6x) \) inside the cubic function. Specifically, the transformation can be understood by substituting \( x \) in \( f(x) \):
\[
g(x) = f(6x)
\]
This means that we are replacing \( x \) with \( 6x \), which represents a horizontal compression. In general, if we have \( f(kx) \), where \( k > 1 \), it compresses the graph horizontally by a factor of \( k \).
Since \( k = 6 \) in this case, the transformation is a horizontal compression by a factor of 6.
Therefore, the correct option is:
**a horizontal compression by a factor of 6.**