Which phrase best describes the translation from the graph [tex]y=(x-5)^2+7[/tex] to the graph of [tex]y=(x+1)^2-2[/tex]?
A. 6 units left and 9 units down
B. 6 units right and 9 units down
C. 6 units left and 9 units up
D. 6 units right and 9 units up

The translation from the graph y = ( x − 5 ) 2 + 7 to y = ( x + 1 ) 2 − 2 involves moving 6 units left and 9 units down, which corresponds to option A.
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Answered by Anonymous | 2025-07-04

The vertex of the first parabola y = ( x − 5 ) 2 + 7 is ( 5 , 7 ) .
The vertex of the second parabola y = ( x + 1 ) 2 − 2 is ( − 1 , − 2 ) .
The horizontal translation is − 1 − 5 = − 6 , which means 6 units left.
The vertical translation is − 2 − 7 = − 9 , which means 9 units down. The translation is 6 units left and 9 units down ​ .

Explanation

Understanding the Problem We are given two parabolas, y = ( x − 5 ) 2 + 7 and y = ( x + 1 ) 2 − 2 . We want to find the translation that maps the first parabola to the second. The vertex form of a parabola is y = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola.

Finding the Vertex of the First Parabola The vertex of the first parabola, y = ( x − 5 ) 2 + 7 , is ( 5 , 7 ) .

Finding the Vertex of the Second Parabola The vertex of the second parabola, y = ( x + 1 ) 2 − 2 , is ( − 1 , − 2 ) .

Determining the Horizontal Translation To find the horizontal translation, we subtract the x-coordinate of the first vertex from the x-coordinate of the second vertex: − 1 − 5 = − 6 . This means the parabola is translated 6 units to the left.

Determining the Vertical Translation To find the vertical translation, we subtract the y-coordinate of the first vertex from the y-coordinate of the second vertex: − 2 − 7 = − 9 . This means the parabola is translated 9 units down.

Conclusion Therefore, the translation from the graph of y = ( x − 5 ) 2 + 7 to the graph of y = ( x + 1 ) 2 − 2 is 6 units left and 9 units down.


Examples
Understanding translations of graphs is crucial in various fields. For example, in physics, when analyzing projectile motion, understanding how the initial conditions affect the trajectory involves translating the parabolic path. Similarly, in engineering, adjusting parameters in control systems often involves shifting the system's response curve, which can be visualized as a translation of a graph. In computer graphics, translations are fundamental for moving objects around the screen. The translation of functions helps to model these real-world scenarios mathematically, allowing for predictions and adjustments to achieve desired outcomes.

Answered by GinnyAnswer | 2025-07-04