How does the graph of $g(x)=(x-3)^3+4$ compare to the parent function $f(x)=x^3$?
A. $g(x)$ is shifted 3 units to the right and 4 units up.
B. $g(x)$ is shifted 4 units to the right and 3 units up.
C. $g(x)$ is shifted 3 units to the left and 4 units up.
D. $g(x)$ is shifted 4 units to the right and 3 units down.
Answer (2)
The parent function is f ( x ) = x 3 .
Replacing x with ( x − 3 ) shifts the graph 3 units to the right.
Adding 4 shifts the graph 4 units up.
Therefore, g ( x ) is shifted 3 units to the right and 4 units up: g ( x ) is shifted 3 units to the right and 4 units up.
Explanation
Understanding the Transformation We are given the parent function f ( x ) = x 3 and the transformed function g ( x ) = ( x − 3 ) 3 + 4 . We need to describe how the graph of g ( x ) compares to the graph of f ( x ) . This involves identifying any horizontal and vertical shifts.
Analyzing Horizontal Shift The function g ( x ) is obtained from f ( x ) by replacing x with ( x − 3 ) and adding 4. Replacing x with ( x − h ) results in a horizontal shift of h units. In this case, we have ( x − 3 ) , so the graph shifts 3 units to the right.
Analyzing Vertical Shift Adding a constant k to a function results in a vertical shift of k units. In this case, we are adding 4, so the graph shifts 4 units up.
Conclusion Combining the horizontal and vertical shifts, we can conclude that the graph of g ( x ) = ( x − 3 ) 3 + 4 is obtained from the graph of f ( x ) = x 3 by shifting it 3 units to the right and 4 units up.
Examples
Imagine you're designing a video game where you need to move a character along a path. The parent function f ( x ) = x 3 represents the initial path. If you want to shift the path 3 units to the right and 4 units up, you would use the transformed function g ( x ) = ( x − 3 ) 3 + 4 . This concept of shifting functions is crucial in game development for positioning objects and characters in the game world.
The graph of g ( x ) = ( x − 3 ) 3 + 4 is obtained from the parent function f ( x ) = x 3 by shifting it 3 units to the right and 4 units up. Therefore, the correct answer is A.
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