To compare the functions [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = (2x)^2 \)[/tex], let's analyze their properties and transformations.
1. Find the equation of [tex]\( g(x) \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
g(x) = (2x)^2
\][/tex]
Simplify this expression:
[tex]\[
(2x)^2 = 4x^2
\][/tex]
So, [tex]\( g(x) = 4x^2 \)[/tex].
2. Compare [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- [tex]\( f(x) = x^2 \)[/tex]
- [tex]\( g(x) = 4x^2 \)[/tex]
Notice that [tex]\( g(x) \)[/tex] = 4 times [tex]\( f(x) \)[/tex]:
[tex]\[
g(x) = 4 \cdot f(x)
\][/tex]
3. Interpret the transformation:
The factor of 4 indicates a vertical stretch. In other words, each value of [tex]\( g(x) \)[/tex] is 4 times the corresponding value of [tex]\( f(x) \)[/tex], meaning the graph of [tex]\( g(x) \)[/tex] is stretched vertically by a factor.
4. Evaluate the given statements:
A. The graph of [tex]\( g(x) \)[/tex] is horizontally stretched by a factor of 2.
- This would change the input [tex]\( x \)[/tex] in the function, not the output.
B. The graph of [tex]\( g(x) \)[/tex] is shifted 2 units to the right.
- A horizontal shift to the right would be represented by [tex]\( f(x) = (x - 2)^2 \)[/tex].
C. The graph of [tex]\( g(x) \)[/tex] is horizontally compressed by a factor of 2.
- This affects the input [tex]\( x \)[/tex], changing it to [tex]\( f(2x) \)[/tex].
D. The graph of [tex]\( g(x) \)[/tex] is vertically stretched by a factor of 2.
- This suggests that each value of [tex]\( g(x) \)[/tex] would be twice the value of [tex]\( f(x) \)[/tex].
Although the correct transformation is a vertical stretch, it is by a factor of 4. However, given the provided multiple-choice options and the closest correct statement associated with a vertical stretch, the best choice is:
D. The graph of [tex]\( g(x) \)[/tex] is vertically stretched by a factor of 2.
Therefore, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
