To analyze the transformations that occurred from the graph of the parent function \( f(x) = x^2 \) to the given function \( f(x) = \frac{1}{2}(x + 7)^2 + 4 \), we can break it down into parts by looking at how it differs from the parent function.
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Parent function: \( f(x) = x^2 \)
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Horizontal Shift:
- The term \( (x + 7)^2 \) indicates a shift to the left by 7 units. The standard transformation for horizontal shifts is that if \( g(x) = (x - h)^2 \), then the graph shifts right by \( h \) units. Here, because of the plus sign, it shifts left.
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Vertical Stretch/Compression:
- The coefficient \( \frac{1}{2} \) indicates a vertical compression (not a stretch). A factor less than 1 compresses the graph vertically.
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Vertical Shift:
- The \( + 4 \) at the end indicates a shift up by 4 units.
Putting this together, we can summarize the transformations:
- Left 7 units
- Vertical compression by a factor of \( \frac{1}{2} \)
- Up 4 units
Therefore, the transformations that occurred are:
- Left 7 units
- Up 4 units
- Vertical compression (not listed in your options, but important to note)
The following transformations are not applicable:
- Up 7 units
- Down 7 units
- Down 4 units
- Right 7 units
- Right 4 units
- Vertical stretch
- Reflection x-axis
- Reflection y-axis
So, the correct answers are that the graph of \( f(x) \) is shifted left 7 units and up 4 units, and it experiences a vertical compression.
1 answer