To compare the graph of [tex]\( g(x) \)[/tex] with [tex]\( f(x) \)[/tex], let's analyze the two functions in detail.
First, consider the function [tex]\( f(x) = x^2 \)[/tex].
Now, let's look at the function [tex]\( g(x) \)[/tex].
[tex]\[ g(x) = (3x)^2 \][/tex]
We can simplify the expression for [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = (3x)^2 = 9x^2 \][/tex]
Now, let's compare [tex]\( g(x) = 9x^2 \)[/tex] with [tex]\( f(x) = x^2 \)[/tex].
1. Vertical Stretch or Compression:
- The function [tex]\( g(x) = 9x^2 \)[/tex] has a coefficient of 9 multiplying [tex]\( x^2 \)[/tex]. This indicates that the graph of [tex]\( f(x) = x^2 \)[/tex] is vertically stretched by a factor of 9 in [tex]\( g(x) \)[/tex]. This means the values of [tex]\( g(x) \)[/tex] will be 9 times higher than the corresponding values of [tex]\( f(x) \)[/tex] for the same [tex]\( x \)[/tex].
2. Horizontal Stretch or Compression:
- To determine the horizontal transformation, let's consider the argument [tex]\( 3x \)[/tex] inside the function. The term [tex]\( 3x \)[/tex] means we are scaling the input [tex]\( x \)[/tex] by a factor of 3. In general, if we have [tex]\( f(ax) \)[/tex], the graph is horizontally compressed by a factor of [tex]\( \frac{1}{a} \)[/tex]. Therefore, [tex]\( (3x)^2 \)[/tex] results in a horizontal compression by a factor of [tex]\( \frac{1}{3} \)[/tex].
3. Shifting:
- There is no horizontal or vertical shift present in the function [tex]\( g(x) \)[/tex].
Considering all the transformations, we can conclude that the graph of [tex]\( g(x) \)[/tex] is horizontally compressed by a factor of 3 and vertically stretched by a factor of 9, but since the question does not offer the option for vertical stretch by a factor of 9, we focus on the horizontal transformation.
Thus, the best statement that compares the graph of [tex]\( g(x) \)[/tex] with the graph of [tex]\( f(x) \)[/tex] is:
D. The graph of [tex]\( g(x) \)[/tex] is horizontally compressed by a factor of 3.
