To determine which graph illustrates the transformation from the parent function \( f(x) = x^2 \) to the transformed function \( y = f(3x) \), we need to understand the effect of the transformation on the graph.
The function \( y = f(3x) \) represents a horizontal compression of the parent function. Specifically, it compresses the graph of \( f(x) \) horizontally by a factor of \( \frac{1}{3} \). This means that the points on the graph will get closer to the y-axis.
For the parent function \( f(x) = x^2 \):
- The original key points are (0, 0), (1, 1), and (-1, 1).
- After applying the transformation, the new key points will be (0, 0), \( \left(\frac{1}{3}, 1\right) \), and \( \left(-\frac{1}{3}, 1\right) \).
Considering this, the potential candidates for the graph showing \( y = f(3x) \) should display a narrower parabola. Given the choices:
- Graph A: x-axis ranges from -5 to 6, indicating a significant range but may not show enough detail at the peak around the transformed key points.
- Graph B: x-axis ranges from -10 to 10, which is very broad and may not show the effect of compression strongly.
- Graph C: x-axis ranges from -5 to 5, which might allow us to see the points more clearly.
- Graph D: This is identical to Graph C, just noting that both have the same range.
Since both Graph C and D have the same x-axis restrictions and show a horizontally compressed parabola, any could fit the description, but as per the response designations, they represent similar graphed behaviours, displaying a narrower parabola.
Without more specific points or images, both Graph C and D would effectively show \( y = f(3x) \), assuming they visually represent the characteristics of a compressed parabola. However, typically only one option will be designated as the answer.
If forced to choose from the provided options, I would suggest C or D; both are acceptable. If only one option must be chosen based on your context, please select one based on other depicted characteristics or available output not described here.
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