To describe the transformations between the function \( y = \sqrt{x} \) and \( y = -(\sqrt{x} + 1) + 5 \), we can break down the transformations step by step.
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Starting function: \[ y = \sqrt{x} \]
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First transformation: Adding 1 to \( \sqrt{x} \) inside the parentheses: \[ y = \sqrt{x} + 1 \] This shifts the graph of \( y = \sqrt{x} \) upward by 1 unit.
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Second transformation: Negating the entire expression: \[ y = -(\sqrt{x} + 1) \] This reflects the graph across the x-axis, turning it upside down.
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Third transformation: Adding 5 after negating: \[ y = -(\sqrt{x} + 1) + 5 \] This can be simplified further to: \[ y = -\sqrt{x} - 1 + 5 = -\sqrt{x} + 4 \] The addition of 5 shifts the graph upward by 5 units.
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Final transformation: Together, the transformations from \( y = \sqrt{x} \) to \( y = -(\sqrt{x} + 1) + 5 \) result in:
- Shift up by 1 unit (to get \( y = \sqrt{x} + 1 \))
- Reflection across the x-axis (to get \( y = -(\sqrt{x} + 1) \))
- Shift up by 5 units (to apply the final form).
Therefore, the complete transformation sequence is:
- Shift the graph of \( y = \sqrt{x} \) up by 1 unit.
- Reflect across the x-axis.
- Finally, shift up by 5 units to produce the final function \( y = -\sqrt{x} + 4 \).
Summary:
The transformation from \( y = \sqrt{x} \) to \( y = -(\sqrt{x} + 1) + 5 \) involves a shift up by 1, reflection across the x-axis, and a subsequent shift up by 5.
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