Questions: f(x) = sqrt(x), g(x) = sqrt(-x) + 8 reflect about the x-axis, then shift down 8 units reflect about the x-axis, then shift up 8 units reflect about the y-axis, then shift up 8 units reflect about the y-axis, then shift down 8 units shift down 8 units, then reflect about the y-axis

Solution Steps

To solve the problem, we need to apply transformations to the given functions \( f(x) = \sqrt{x} \) and \( g(x) = \sqrt{-x} + 8 \). The transformations include reflections about the x-axis and y-axis, and vertical shifts. For each transformation, we will modify the function accordingly:

1. Reflect about the x-axis, then shift down 8 units: Multiply the function by -1 and then subtract 8.
2. Reflect about the x-axis, then shift up 8 units: Multiply the function by -1 and then add 8.
3. Reflect about the y-axis, then shift up 8 units: Replace \( x \) with \(-x\) and then add 8.
4. Reflect about the y-axis, then shift down 8 units: Replace \( x \) with \(-x\) and then subtract 8.
5. Shift down 8 units, then reflect about the y-axis: Subtract 8 from the function and then replace \( x \) with \(-x\).

For the function \( f(x) = \sqrt{x} \): \[ f_1(x) = -\sqrt{x} - 8 \] For the function \( g(x) = \sqrt{-x} + 8 \): \[ g_1(x) = -\sqrt{-x} - 16 \]

For the function \( f(x) = \sqrt{x} \): \[ f_2(x) = 8 - \sqrt{x} \] For the function \( g(x) = \sqrt{-x} + 8 \): \[ g_2(x) = -\sqrt{-x} \]

For the function \( f(x) = \sqrt{x} \): \[ f_3(x) = \sqrt{-x} + 8 \] For the function \( g(x) = \sqrt{-x} + 8 \): \[ g_3(x) = \sqrt{x} + 16 \]

Final Answer