To understand how the parent function \( f(x) = x^2 \) is transformed into \( y = f(3x) = (3x)^2 \), we need to consider how horizontal compression affects a function.
The transformation from \( f(x) \) to \( f(kx) \) where \( k > 1 \) represents a horizontal compression of the graph of the function by a factor of \( k \). In this case, with \( k = 3 \), the graph of the function \( y = (3x)^2 \) is compressed horizontally by a factor of 3.
Defining Features of \( y = f(3x) \):
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Vertex: The vertex of the parent function \( f(x) = x^2 \) is at the origin (0,0). This point remains unchanged in the transformation \( y = f(3x) \).
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Width: The graph of \( f(x) = x^2 \) has a certain width. The graph of \( y = (3x)^2 = 9x^2 \) will be narrower due to the horizontal compression. Specifically, it will rise more steeply compared to the original parabola.
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X-Intercepts: The x-intercepts for \( f(x) = x^2 \) are at x = 0. For the transformed function, the x-intercepts remain the same since both functions intersect the x-axis at the same point.
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Y-Intercept: The y-intercept remains the same at y = 0 for both functions.
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Domain and Range: The domain of both functions is all real numbers, \( (-\infty, \infty) \), and the range is \( [0, \infty) \) since both are parabolas opening upwards.
Visual Representation:
If you were to visually compare the graph of \( y = f(x) = x^2 \) with the graph of \( y = f(3x) \):
- Graph of \( y = x^2 \) would open upwards and be relatively wide.
- Graph of \( y = (3x)^2 \) would also open upwards but be more narrow, indicating a faster rise on both sides of the vertex compared to the parent function.
If you're looking to identify the correct graph from a set of graphs, look for the one that is a narrower version of the standard parabola \( y = x^2 \). This graph will illustrate the effect of the horizontal compression due to the factor of 3 in the function \( y = f(3x) \).