Let's compare the functions [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = (4x)^2 \)[/tex].
1. Start by analyzing the written functions.
- [tex]\( f(x) = x^2 \)[/tex] is a basic quadratic function, which represents a standard parabola opening upwards with its vertex at the origin (0, 0).
- [tex]\( g(x) = (4x)^2 \)[/tex] can be rewritten as [tex]\( g(x) = 16x^2 \)[/tex].
2. Next, look at the transformation that [tex]\( g(x) \)[/tex] represents.
We can rewrite [tex]\( g(x) \)[/tex] to identify the coefficient inside the function:
[tex]\[
g(x) = (4x)^2
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
g(x) = 16x^2
\][/tex]
3. Recognizing the transformation involves understanding how the term [tex]\( (4x) \)[/tex] affects the graph. The factor inside the function (here, the 4) changes the graph's horizontal scaling:
- If we replace [tex]\( x \)[/tex] with [tex]\( cx \)[/tex] in a function [tex]\( f(x) \)[/tex], where [tex]\( c>1 \)[/tex]:
- It horizontally compresses the graph by a factor of [tex]\( c \)[/tex].
- So, in [tex]\( g(x) = (4x)^2 \)[/tex]:
- The [tex]\( 4 \)[/tex] inside the parentheses denotes a horizontal compression by a factor of 4.
4. Finally, examine the given choices to find the correct description of [tex]\( g(x) \)[/tex] with respect to [tex]\( f(x) \)[/tex]:
- A. The graph of [tex]\( g(x) \)[/tex] is horizontally compressed by a factor of 4.
- B. The graph of [tex]\( g(x) \)[/tex] is horizontally stretched by a factor of 4.
- C. The graph of [tex]\( g(x) \)[/tex] is shifted 4 units to the right.
- D. The graph of [tex]\( g(x) \)[/tex] is vertically stretched by a factor of 4.
Since we determined [tex]\( g(x) = (4x)^2 \)[/tex] involves a horizontal compression by a factor of 4, the correct statement is:
A. The graph of [tex]\( g(x) \)[/tex] is horizontally compressed by a factor of 4.
