The quadratic parent function is given by \( f(x) = x^2 \). When we apply transformations to create \( g(x) = f(x - 3) + 5 \), we can analyze the transformations as follows:
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Horizontal Shift: The term \( (x - 3) \) indicates a horizontal shift to the right by 3 units. This moves the vertex from its original position.
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Vertical Shift: The term \( +5 \) indicates a vertical shift upwards by 5 units.
The vertex of the parent function \( f(x) = x^2 \) is at the point \( (0, 0) \).
Applying the transformations:
- After shifting right by 3 units, the new x-coordinate of the vertex becomes \( 0 + 3 = 3 \).
- After shifting up by 5 units, the new y-coordinate of the vertex becomes \( 0 + 5 = 5 \).
Thus, the vertex of the graph of \( g(x) \) is at the point \( (3, 5) \).
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