To obtain the graph of [tex]y=(x-8)^2[/tex], shift the graph of [tex]y=x^2[/tex].

The problem involves understanding how to shift the graph of a function.
Replacing x with ( x − h ) in y = f ( x ) shifts the graph horizontally.
If 0"> h > 0 , the shift is to the right by h units.
Therefore, to obtain the graph of y = ( x − 8 ) 2 , shift the graph of y = x 2 to the right by 8 units. right by 8 units ​ .

Explanation

Understanding the Problem We are given two quadratic functions, y = x 2 and y = ( x − 8 ) 2 . The problem asks us to describe how to transform the graph of y = x 2 to obtain the graph of y = ( x − 8 ) 2 . This involves understanding horizontal shifts of functions.

Understanding Horizontal Shifts Recall that if we have a function y = f ( x ) , replacing x with ( x − h ) results in a horizontal shift. If 0"> h > 0 , the graph shifts to the right by h units. If h < 0 , the graph shifts to the left by ∣ h ∣ units.

Applying the Shift In our case, we have y = ( x − 8 ) 2 . Comparing this to y = f ( x − h ) , we see that f ( x ) = x 2 and h = 8 . Since 0"> h = 8 > 0 , the graph of y = x 2 is shifted to the right by 8 units to obtain the graph of y = ( x − 8 ) 2 .

Conclusion Therefore, to obtain the graph of y = ( x − 8 ) 2 , we shift the graph of y = x 2 to the right by 8 units.


Examples
Imagine you're designing a video game where a character's jump is modeled by a parabola. The basic jump is represented by y = x 2 . If you want the character to jump starting 8 units to the right, you would use the equation y = ( x − 8 ) 2 . This shifts the entire jump trajectory, allowing you to control where the character starts their jump in the game world. Understanding these transformations helps in creating dynamic and interactive game environments.

Answered by GinnyAnswer | 2025-07-08

To obtain the graph of y = ( x − 8 ) 2 , shift the graph of y = x 2 to the right by 8 units. This involves understanding that the transformation from y = x 2 to y = ( x − 8 ) 2 represents a horizontal shift. Specifically, replacing x with x − 8 indicates a shift to the right by 8 units.
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Answered by Anonymous | 2025-07-14