To compare the given functions [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = 3x^2 \)[/tex] and determine how their graphs relate to each other, let’s analyze the properties of these functions step-by-step.
1. Understand the Given Functions:
- The function [tex]\( f(x) = x^2 \)[/tex] is a standard quadratic function that forms a parabola opening upwards with its vertex at the origin (0, 0).
- The function [tex]\( g(x) = 3x^2 \)[/tex] is similar to [tex]\( f(x) \)[/tex] but has a coefficient of 3 multiplying the [tex]\( x^2 \)[/tex] term.
2. Effect of the Coefficient:
- When comparing [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], the presence of the coefficient 3 in [tex]\( g(x) = 3x^2 \)[/tex] affects the graph of the function.
- Specifically, multiplying the output of [tex]\( x^2 \)[/tex] by 3 results in making the function values larger by a factor of 3. For any input [tex]\( x \)[/tex]:
[tex]\[
f(x) = x^2 \quad \text{and} \quad g(x) = 3x^2
\][/tex]
3. Graphical Interpretation:
- The graph of [tex]\( f(x) = x^2 \)[/tex] is a standard parabola.
- For [tex]\( g(x) = 3x^2 \)[/tex], every point on the graph of [tex]\( f(x) \)[/tex] is multiplied by 3 in the vertical direction. For example:
- If [tex]\( x = 1 \)[/tex], then [tex]\( f(1) = 1 \)[/tex] and [tex]\( g(1) = 3 \)[/tex].
- If [tex]\( x = 2 \)[/tex], then [tex]\( f(2) = 4 \)[/tex] and [tex]\( g(2) = 12 \)[/tex].
- This scaling effect causes the graph of [tex]\( g(x) \)[/tex] to be "stretched" vertically compared to the graph of [tex]\( f(x) \)[/tex].
4. Determine the Correct Statement:
- With the explanation that [tex]\( g(x) = 3x^2 \)[/tex] makes the graph stretch vertically by a factor of 3, we can conclude:
- The graph of [tex]\( g(x) \)[/tex] is a vertically stretched version of the graph of [tex]\( f(x) \)[/tex].
Thus, the correct statement that best compares the graph of [tex]\( g(x) \)[/tex] with that of [tex]\( f(x) \)[/tex] is:
A. The graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] stretched vertically.
