To analyze the transformation from the parent function \( f(x) = \sqrt{x} \) to the function \( g(x) = \sqrt{x} - 5 \), we can identify the changes in the graph step by step:
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Parent Function: The graph of \( f(x) = \sqrt{x} \) is a curve that starts at the origin (0,0) and rises to the right, increasing slowly. It is defined for \( x \geq 0 \).
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Transformation Description:
- The function \( g(x) = \sqrt{x} - 5 \) involves a vertical translation of the parent function.
- Specifically, the "-5" indicates that the graph of \( f(x) = \sqrt{x} \) is shifted downwards by 5 units.
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Impact on the Graph:
- The entire graph of \( f(x) \) is moved down 5 units. This means that every point (x, y) on the graph of \( f(x) \) will move to (x, y - 5) on the graph of \( g(x) \).
- For example, the point (0, 0) on the graph of \( f(x) \) will move to (0, -5) on the graph of \( g(x) \).
- The new graph will still have its shape similar to the square root curve, but it will be lower on the y-axis.
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Domain and Range:
- The domain of \( g(x) \) remains the same as that of \( f(x) \): \( x \geq 0 \).
- The range of \( g(x) \) shifts down as well: originally, the range of \( f(x) \) was \( [0, \infty) \), and after the transformation, the range of \( g(x) \) is \( [-5, \infty) \).
In summary, the transformation from \( f(x) = \sqrt{x} \) to \( g(x) = \sqrt{x} - 5 \) is a vertical downward shift of 5 units.
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