Question

Which phrase best describes the translation from the graph y = (x – 5)2 + 7 to the graph of y = (x + 1)2 – 2?

Answer 1

To best describe the translation that moves the graph of

$y=(x-5)^2+7$

to the graph of

$y =(x+1)^2-2$ , we have to consider first their relationship to the graph of $y=x^2$.

The graph of $y=x^2$ has a vertex at the origin, i.e (0,0).

The graph of  $y=(x-5)^2+7$  represents the graph of   $y=x^2$ translated by moving it 7 units in the positive y direction and 5 units in the positive x direction. The vertex of  $y=(x-5)^2+7$ is (5,7).

Similarly, the graph of $y =(x+1)^2-2$ represents the graph of $y=x^2$  shifted by 2 units in the negative y direction and  1 unit in the negative x direction. The vertex of this graph is at (-1,-2).

To describe the full transformation that of  one graph to another, we just have to describe the what this transformation does to the vertex of the first graph to get to the second one. The transformation moves the vertex of   $y=(x-5)^2+7$  9 units in the negative y direction and 6 units in the negative x direction.

The phrase which best describes this translation is " a move of 6 units to the left and a move of 9 units down.$\left[\begin{array}{c}-6\\-9\\\end{array}\right]$