How does the graph of [tex]g(x)=-(x-2)^4[/tex] compare to the parent function of [tex]f(x)=x^4[/tex]?

A. [tex]g(x)[/tex] is shifted 2 units to the right and reflected over the [tex]x[/tex]-axis.
B. [tex]g(x)[/tex] is shifted 2 units to the left and reflected over the [tex]x[/tex]-axis.
C. [tex]g(x)[/tex] is shifted 2 units to the right and 1 unit up.
D. [tex]g(x)[/tex] is shifted 2 units to the right and 1 unit down.

The function g ( x ) = − ( x − 2 ) 4 is a transformation of the parent function f ( x ) = x 4 .
The term ( x − 2 ) indicates a horizontal shift of 2 units to the right.
The negative sign indicates a reflection over the x-axis.
Therefore, g ( x ) is shifted 2 units to the right and reflected over the x -axis. g ( x ) is shifted 2 units to the right and reflected over the x -axis. ​

Explanation

Analyze the Problem We are given two functions: the parent function f ( x ) = x 4 and the transformed function g ( x ) = − ( x − 2 ) 4 . We need to describe how the graph of g ( x ) compares to the graph of f ( x ) . This involves identifying the transformations applied to f ( x ) to obtain g ( x ) .

Identify the Transformations The function g ( x ) has two transformations applied to the parent function f ( x ) = x 4 :

Horizontal Shift: The term ( x − 2 ) inside the function indicates a horizontal shift. Specifically, it shifts the graph 2 units to the right. This is because replacing x with ( x − 2 ) causes the graph to move in the positive x-direction.

Reflection: The negative sign in front of the function, − ( x − 2 ) 4 , indicates a reflection over the x-axis. This is because the y-values of the transformed function are the negative of the y-values of the function ( x − 2 ) 4 .

Describe the Combined Transformations Combining these transformations, we can say that the graph of g ( x ) = − ( x − 2 ) 4 is obtained by shifting the graph of f ( x ) = x 4 two units to the right and reflecting it over the x-axis.

Select the Correct Option Now, let's compare our result with the given options:



g ( x ) is shifted 2 units to the right and reflected over the x -axis.
g ( x ) is shifted 2 units to the left and reflected over the x -axis.
g ( x ) is shifted 2 units to the right and 1 unit up.
g ( x ) is shifted 2 units to the right and 1 unit down.

The first option matches our analysis.

Final Answer The graph of g ( x ) = − ( x − 2 ) 4 is shifted 2 units to the right and reflected over the x -axis.

Examples
Understanding transformations of functions is crucial in many fields. For example, in physics, understanding how equations transform can help predict the trajectory of a projectile. If f ( x ) represents the initial height of a projectile at a given horizontal distance x , then g ( x ) = − f ( x − 2 ) might represent the trajectory if the projectile is launched 2 units further and the trajectory is inverted due to gravity. Similarly, in signal processing, transformations are used to shift and reflect signals for analysis and manipulation. For instance, if f ( t ) represents a sound wave, g ( t ) = − f ( t − T ) could represent the same sound wave played T seconds later and inverted.

Answered by GinnyAnswer | 2025-07-03

The graph of g ( x ) = − ( x − 2 ) 4 is obtained by shifting the parent function f ( x ) = x 4 2 units to the right and reflecting it over the x-axis. The correct answer is A.
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Answered by Anonymous | 2025-07-04