To determine which graph represents the function \( y = f(3x) \) where \( f(x) = x^2 \), we need to understand what it means to horizontally compress the function.
The function \( f(x) = x^2 \) is a parabola with its vertex at (0,0). The transformation \( f(3x) \) indicates a horizontal compression by a factor of 3. This means that for the new function, the same output value of \( y \) will occur at \( x \) values that are one-third of those in the parent function.
For \( f(x) = x^2 \), the points on the graph represent the following:
- The parent function has points like (1,1), (-1,1), (2,4), and (-2,4) since \( f(1)=1^2=1 \), and \( f(2)=2^2=4 \).
- When we apply the transformation \( f(3x) \), the points will change; for example:
- The point (1, 1) in the parent becomes \( (1/3, 1) \), so you should see a point at \( (1/3,1) \).
- \( (2,4) \) becomes \( (2/3, 4) \), and so forth.
Since we want to find the new values for specific \( x \) values, let's analyze the given options (assuming that the defining features are approximate):
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Graph #1 has dots at (-1, 9) and (1, 9):
- At \( x = 1/3 \), we would expect \( y = 1 \). This isn't present in the graph.
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Graph #2 has dots at (-3, 1) and (1, 3):
- At \( x = 1/3 \), the output is 1 (not matching the parent).
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Graph #3 has dots at (-1, 4) and (1, 4):
- At \( x = -1/3 \), \( y = 1 \) and \( y = 4 \) does not match.
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Graph #4 has dots at (-1, 1) and (1, 1):
- At \( x = -1/3 \), \( y = 1 = (-1/3) \).
Based on the horizontal compression we explored and the output values that would result, none of these precisely indicate points corresponding to the transformation of the parent function. However, the answer may vary based on interpreting the ranges and expected graph outputs. Since we're focusing on compression:
From the options given, Graph #4 (dots at (-1, 1) and (1, 1)) is the best candidate because it can closely match the transformation's shape along one of the axes even if values aren't completely resolved.
Keep in mind the possible conclusion relies heavily on defining what actually represents the correct mapping of the function under the transformation stated. For precise effects, consider drawing a graph or providing more context.