Question

suppose $f(x) = x^2$ and $g(x) = 4x^2$. which statement best compares the graph of $g(x)$ with the graph of $f(x)$?

a. the graph of $g(x)$ is the graph of $f(x)$ vertically stretched by a factor of 4.

b. the graph of $g(x)$ is the graph of $f(x)$ vertically compressed by a factor of 4.

c. the graph of $g(x)$ is the graph of $f(x)$ shifted 4 units left.

d. the graph of $g(x)$ is the graph of $f(x)$ horizontally stretched by a factor of 4.

Explanation:

Step1: Recall Vertical Stretch Rule

For a function \( y = a f(x) \), if \( |a|>1 \), the graph of \( f(x) \) is vertically stretched by a factor of \( |a| \).
Here, \( f(x)=x^2 \) and \( g(x) = 4x^2=4f(x) \), with \( a = 4>1 \).

Step2: Analyze Each Option

• Option A: Matches the vertical stretch rule (factor 4).
• Option B: Vertical compression occurs when \( 0<|a|<1 \), not here.
• Option C: Shifting left would be \( f(x + 4) \), not \( 4f(x) \).
• Option D: Horizontal stretch is for \( f(bx) \) with \( 0<|b|<1 \), not this form.

Answer:

A. The graph of \( g(x) \) is the graph of \( f(x) \) vertically stretched by a factor of 4.