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

Identify the vertex of the first parabola: ( 5 , 7 ) .
Identify the vertex of the second parabola: ( − 1 , − 2 ) .
Calculate the horizontal translation: − 1 − 5 = − 6 (6 units left).
Calculate the vertical translation: − 2 − 7 = − 9 (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 describe the translation from the first parabola to the second.

Vertex Form of a Parabola The vertex form of a parabola is given by y = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola.

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

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

Horizontal Translation To find the horizontal translation, we calculate the difference in the x-coordinates of the vertices: − 1 − 5 = − 6 . This means the parabola is translated 6 units to the left.

Vertical Translation To find the vertical translation, we calculate the difference in the y-coordinates of the vertices: − 2 − 7 = − 9 . This means the parabola is translated 9 units down.

Final Answer 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 useful in many fields, such as physics and engineering. For example, if you are modeling the trajectory of a projectile, a translation of the graph could represent a change in the initial conditions, such as the starting position or velocity. Similarly, in signal processing, translations can represent time delays or frequency shifts.

Answered by GinnyAnswer | 2025-07-03

The translation from the graph of y = ( x − 5 ) 2 + 7 to y = ( x + 1 ) 2 − 2 is 6 units left and 9 units down. The appropriate choice is option A. This is derived from calculating the differences in the vertices of the two parabolas.
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Answered by Anonymous | 2025-07-04