Bot GPT 4-o mini To analyze the options and find the correct transformation of the given function \( f(x) = 2x^2 \), let's first describe the original function. The function is a parabola that opens upward, and its vertex is at the origin (0, 0). To make sense of the transformations given, we need to consider how each transformation affects the graph:
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\( f(x) + 1 \): This transformation shifts the graph of \( f(x) \) upward by 1 unit. So if the original graph was \( f(x) = 2x^2 \), then after the transformation, the new function would be \( f(x) + 1 = 2x^2 + 1 \). This means every point on the graph moves up by 1 unit.
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\( f(x) - 1 \): This transformation shifts the graph of \( f(x) \) downward by 1 unit. Thus, the new function would be \( f(x) - 1 = 2x^2 - 1 \). Every point on the graph moves down by 1 unit.
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\( 2f(x) \): This transformation vertically stretches the graph by multiplying the output of \( f(x) \) by 2. The new function would be \( 2f(x) = 2(2x^2) = 4x^2 \).
Now to determine which of these transformations corresponds to the provided points (0, 1), (-1, 3), (1, 3), (-2, 9), and (2, 9):
- For \( f(x) + 1 \) we get:
- \( f(0) + 1 = 2(0)^2 + 1 = 1 \) → point (0, 1)
- \( f(-1) + 1 = 2(-1)^2 + 1 = 2 + 1 = 3 \) → point (-1, 3)
- \( f(1) + 1 = 2(1)^2 + 1 = 2 + 1 = 3 \) → point (1, 3)
- \( f(-2) + 1 = 2(-2)^2 + 1 = 8 + 1 = 9 \) → point (-2, 9)
- \( f(2) + 1 = 2(2)^2 + 1 = 8 + 1 = 9 \) → point (2, 9)
Since all points match perfectly with the transformation \( f(x) + 1 \), this is the correct transformation that models the given points.
Thus, the correct answer is:
\( f(x) + 1 \).
1 answer