Which describes how to graph $g(x)=\sqrt[3]{x-5}+7$ by transforming the parent function?
A. Translate the parent function 5 units to the left and 7 units up.
B. Translate the parent function 5 units to the right and 7 units up.
C. Translate the parent function 5 units down and 7 units to the right.
D. Translate the parent function 5 units up and 7 units to the right.

The function g ( x ) = 3 x − 5 ​ + 7 is a transformation of the parent function f ( x ) = 3 x ​ .
The term ( x − 5 ) indicates a horizontal translation of 5 units to the right.
The term + 7 indicates a vertical translation of 7 units up.
Therefore, the graph of g ( x ) is obtained by translating the graph of f ( x ) 5 units to the right and 7 units up. Translate the parent function 5 units to the right and 7 units up. ​

Explanation

Analyze the function Let's analyze the given function g ( x ) = 3 x − 5 ​ + 7 and determine how it relates to the parent function f ( x ) = 3 x ​ . We need to identify the transformations applied to the parent function to obtain the given function.

Identify the transformations The function g ( x ) involves two transformations of the parent function f ( x ) = 3 x ​ .

Horizontal Translation: The term ( x − 5 ) inside the cube root indicates a horizontal translation. Specifically, since we are subtracting 5 from x , the graph shifts 5 units to the right .

Vertical Translation: The term + 7 outside the cube root indicates a vertical translation. Since we are adding 7, the graph shifts 7 units up .


Therefore, the graph of g ( x ) = 3 x − 5 ​ + 7 is obtained by translating the graph of f ( x ) = 3 x ​ by 5 units to the right and 7 units up.

State the final transformation Based on our analysis, the correct description of the transformation is:

Translate the parent function 5 units to the right and 7 units up.
Examples
Understanding transformations of functions is crucial in many fields. For example, in physics, understanding how graphs of motion change with shifts in time or position helps analyze the movement of objects. In economics, shifting demand or supply curves can model the impact of taxes or subsidies. In computer graphics, transformations are used to manipulate objects in 3D space. Suppose you are designing a game where an object's trajectory is defined by the cube root function. By applying horizontal and vertical translations, you can easily reposition the object's path on the screen, making it start at a different point or reach a different height. This allows for dynamic and interactive game design.

Answered by GinnyAnswer | 2025-07-08