To understand the translation from the graph of [tex]\( y = (x-5)^2 + 7 \)[/tex] to the graph of [tex]\( y = (x+1)^2 - 2 \)[/tex], we need to analyze how the transformations affect the graph.
1. Identify Horizontal Translation:
- The original graph is [tex]\( y = (x-5)^2 + 7 \)[/tex].
- Notice the [tex]\( x \)[/tex]-term shift: it changes from [tex]\( (x-5) \)[/tex] to [tex]\( (x+1) \)[/tex].
To understand the movement:
[tex]\[
x - 5 \rightarrow x + 1
\][/tex]
This means that [tex]\( (x-5) \)[/tex] needs to change to [tex]\( (x+1) \)[/tex].
We see that [tex]\( x-5 \)[/tex] relates to [tex]\( x+1 \)[/tex] if you move 6 units right:
[tex]\[
(x-5) + 6 = x+1
\][/tex]
So, the graph shifts 6 units right.
2. Identify Vertical Translation:
- The original constant term is +7.
- The new constant term is -2.
To understand the movement:
[tex]\[
y = (x-5)^2 + 7 \rightarrow y = (x+1)^2 - 2
\][/tex]
To get from 7 to -2, you need to move down:
[tex]\[
7 - 9 = -2
\][/tex]
So, the graph shifts 9 units down.
Thus, the best phrase to describe the complete translation from the graph [tex]\( y = (x-5)^2 + 7 \)[/tex] to [tex]\( y = (x+1)^2 - 2 \)[/tex] is:
6 units right and 9 units down.
