To interpret the function \( f(x) = 4x^2 \), we note that this is a transformation of the parent function \( y = x^2 \).
The key features to consider include:
- Shape of the Graph: The graph maintains the parabolic shape of the parent function \( y = x^2 \).
- Vertical Stretch: The coefficient \( 4 \) in \( f(x) = 4x^2 \) indicates a vertical stretch of the graph by a factor of \( 4 \). This means for any \( x \), the \( y \)-value of \( f(x) \) is four times that of the parent function. Consequently, the graph will be "narrower" than the parent function because it rises more steeply.
- Intercepts: While the x-intercept (which is at the origin (0,0) for both functions) remains unchanged, the changes in the y-values make it clear that we are looking at a transformation–specifically vertical scaling.
Now, analyzing the provided options:
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a: This statement talks about changes in the x-intercept, which remain the same (both functions have the same x-intercept).
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b: This statement discusses changes in the y-intercept. The y-intercept of the parent function is \( 0 \), and for \( f(x) \), the y-intercept is also \( 0 \). Thus, there's no vertical shift.
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c: This focuses on changes in the x-coordinates, which is not applicable here; the x-coordinates do not change in a vertical stretch.
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d: This option correctly identifies that the vertical stretch affects the y-coordinates. Specifically, each y-coordinate of the points on the graph of the parent function \( y = x^2 \) is multiplied by \( 4 \) in the transformed graph \( f(x) = 4x^2 \).
Given these analyses, the correct interpretation is:
d. this is a vertical stretch of the parent function y = x^2 due to the changes in the y-coordinates of each point.
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