To describe how to graph the function [tex]\( g(x) = \sqrt[3]{x-5} + 7 \)[/tex] by transforming the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex], follow these steps:
1. Identify the parent function: The parent function for [tex]\( g(x) \)[/tex] is [tex]\( f(x) = \sqrt[3]{x} \)[/tex], which represents the cube root of [tex]\( x \)[/tex].
2. Analyze the transformation:
- The function [tex]\( g(x) \)[/tex] modifies [tex]\( f(x) \)[/tex] by altering the argument of the function and by adding a constant term.
- Specifically, [tex]\( g(x) \)[/tex] is written as [tex]\( g(x) = \sqrt[3]{x-5} + 7 \)[/tex].
3. Understand the transformations:
- The term [tex]\( x-5 \)[/tex] inside the cube root function shifts the graph horizontally.
- A subtraction of a constant inside the function (i.e., [tex]\( x-5 \)[/tex]) translates the graph to the right by the value of the constant, which is 5 units.
- The addition of 7 outside the function (i.e., [tex]\( +7 \)[/tex]) translates the graph vertically.
- Adding a constant to the function (i.e., [tex]\( +7 \)[/tex]) translates the graph upwards by the value of the constant, which is 7 units.
4. Conclusion:
- Based on the transformations identified, translating the parent function [tex]\( \sqrt[3]{x} \)[/tex] involves shifting it 5 units to the right (due to [tex]\( x-5 \)[/tex]) and 7 units up (due to [tex]\( +7 \)[/tex]).
Therefore, the correct description of how to graph [tex]\( g(x) = \sqrt[3]{x-5} + 7 \)[/tex] by transforming the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] is:
Translate the parent function 5 units to the right and 7 units up.
