Question

suppose $f(x) = x^2$. what is the graph of $g(x) = f(3x)$?

a. graph
b. graph
c. graph
d. graph

Explanation:

Step1: Find the function of g(x)

Given \( f(x)=x^{2} \), then \( g(x)=f(3x)=(3x)^{2}=9x^{2} \).

Step2: Analyze the transformation

The parent function \( f(x) = x^{2} \) is a parabola opening upwards with vertex at the origin. For \( g(x)=9x^{2} \), the coefficient 9 (greater than 1) causes a horizontal compression (since it's a coefficient inside the function, affecting the x - values) and a vertical stretch? Wait, no. Wait, the transformation for \( y = f(kx) \) when \( |k|>1 \) is a horizontal compression by a factor of \( \frac{1}{k} \). Here \( k = 3 \), so the graph of \( y=x^{2} \) is compressed horizontally by a factor of \( \frac{1}{3} \), making the parabola narrower. Also, \( g(x)=9x^{2} \) has a steeper slope (vertical stretch by a factor of 9) compared to \( f(x)=x^{2} \). Looking at the options, option A shows a narrower parabola (compressed horizontally and stretched vertically) compared to option B (which is the original \( y = x^{2} \) or a less compressed one). Option C is opening downwards (but \( g(x)=9x^{2} \) opens upwards), option D is also opening downwards. So the correct graph should be the narrower upward - opening parabola, which is option A.

Answer:

A. The graph with the narrower upward - opening parabola (as described in the analysis)