To derive the graph of \( g(x) = \frac{1}{2}(x - 4)^3 + 5 \) from the parent function \( f(x) = x^2 \), we will go through a series of transformations. However, it seems there might be some confusion as \( g(x) \) is based on a cubic function while \( f(x) \) is a quadratic function. Assuming you meant to relate \( g(x) \) to a cubic parent function like \( f(x) = x^3 \), we can detail the appropriate transformations.
Here’s the step-by-step breakdown of the transformations from \( f(x) = x^3 \) to \( g(x) = \frac{1}{2}(x - 4)^3 + 5 \):
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Horizontal Shift:
- Start with \( f(x) = x^3 \).
- To shift the graph to the right by 4 units, replace \( x \) with \( x - 4 \): \[ f(x) = (x - 4)^3 \]
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Vertical Compression:
- To compress the graph vertically by a factor of \( \frac{1}{2} \), multiply the entire function by \( \frac{1}{2} \): \[ f(x) = \frac{1}{2}(x - 4)^3 \]
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Vertical Shift:
- Finally, to shift the graph up by 5 units, add 5 to the function: \[ g(x) = \frac{1}{2}(x - 4)^3 + 5 \]
Thus, the order of transformations from the graph of \( f(x) = x^3 \) to the graph of \( g(x) \) is as follows:
- Shift right by 4 units.
- Compress vertically by a factor of \( \frac{1}{2} \).
- Shift up by 5 units.
If you intended to relate \( g(x) \) to a quadratic function, please clarify, and I can adjust the explanation accordingly!
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