1. **Problem Statement:** We are given the function $f(x) = \sqrt[5]{x}$ and its transformations: - $f(x) = \sqrt[5]{x} + 6$ - $f(x) = \sqrt[5]{x} - 6$ - $f(x) = \sqrt[5]{50x}$ - $f(x) = \sqrt[5]{\frac{x}{50}}$ We want to understand these transformations and how they affect the graph of the original function. 2. **Original Function:** The function $f(x) = \sqrt[5]{x} = x^{\frac{1}{5}}$ is the fifth root function. It is defined for all real $x$ and is an odd function, meaning it is symmetric about the origin. 3. **Transformations Explained:** - $\sqrt[5]{x} + 6$: This shifts the graph of $\sqrt[5]{x}$ vertically upward by 6 units. - $\sqrt[5]{x} - 6$: This shifts the graph vertically downward by 6 units. - $\sqrt[5]{50x}$: This compresses the graph horizontally by a factor of $\frac{1}{50}$ because multiplying $x$ by 50 inside the function makes the graph change faster. - $\sqrt[5]{\frac{x}{50}}$: This stretches the graph horizontally by a factor of 50 because dividing $x$ by 50 inside the function makes the graph change slower. 4. **Summary of Effects:** - Vertical shifts add or subtract a constant outside the function. - Horizontal stretches/compressions multiply $x$ inside the function by a factor. 5. **Final Answer:** The graphs are: - $y = \sqrt[5]{x}$ (original) - $y = \sqrt[5]{x} + 6$ - $y = \sqrt[5]{x} - 6$ - $y = \sqrt[5]{50x}$ - $y = \sqrt[5]{\frac{x}{50}}$ Each transformation modifies the original graph as explained above.