Answer :
To graph the function \( g(x) = \sqrt[3]{x-5} + 7 \) by transforming the parent function, we need to understand how transformations affect the graph of a function. The parent function in this case is \( f(x) = \sqrt[3]{x} \), which is the cube root function.
The transformations that occur are:
1. Horizontal Translation (Shift Right or Left):
- If we replace \( x \) with \( x - k \) in the function \( f(x) \), this translates the graph horizontally.
- In \( g(x) = \sqrt[3]{x-5} + 7 \), the expression inside the cube root is \( x - 5 \). This means that the graph of \( \sqrt[3]{x} \) is translated 5 units to the right.
2. Vertical Translation (Shift Up or Down):
- If we add or subtract a constant \( k \) from the function \( f(x) \), this translates the graph vertically.
- In \( g(x) = \sqrt[3]{x-5} + 7 \), the term \( + 7 \) outside the cube root function indicates a vertical translation. This means that the graph of \( \sqrt[3]{x} \) is translated 7 units up.
Thus, combining these two transformations:
- The graph of \( \sqrt[3]{x} \) is translated 5 units to the right.
- The graph is then translated 7 units up.
Therefore, the correct description of how to graph \( g(x) = \sqrt[3]{x-5} + 7 \) by transforming the parent function is:
Translate the parent function 5 units to the right and 7 units up.
The transformations that occur are:
1. Horizontal Translation (Shift Right or Left):
- If we replace \( x \) with \( x - k \) in the function \( f(x) \), this translates the graph horizontally.
- In \( g(x) = \sqrt[3]{x-5} + 7 \), the expression inside the cube root is \( x - 5 \). This means that the graph of \( \sqrt[3]{x} \) is translated 5 units to the right.
2. Vertical Translation (Shift Up or Down):
- If we add or subtract a constant \( k \) from the function \( f(x) \), this translates the graph vertically.
- In \( g(x) = \sqrt[3]{x-5} + 7 \), the term \( + 7 \) outside the cube root function indicates a vertical translation. This means that the graph of \( \sqrt[3]{x} \) is translated 7 units up.
Thus, combining these two transformations:
- The graph of \( \sqrt[3]{x} \) is translated 5 units to the right.
- The graph is then translated 7 units up.
Therefore, the correct description of how to graph \( g(x) = \sqrt[3]{x-5} + 7 \) by transforming the parent function is:
Translate the parent function 5 units to the right and 7 units up.
