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

Start from the parent function $f(x)=x^2$

In the first case, you are computing

$f(x-5)+7$

In the second case, you are computing

$f(x+1)-2 /tex] There are two translation going on: when you transform [tex] f(x) \to f(x+k)$, you translate the function horizontally, $k$ units left if $k>0$ and $k$ units right if $k$.

On the other hand, when you transform $f(x) \to f(x)+k$, you translate the function vertically, $k$ units up if $k>0$ and $k$ units down if $k$.

So, the first function is the "original" parabola $f(x)=x^2$, translated $5$ units right and $7$ units up. Likewise, the second function is the "original" parabola $f(x)=x^2$, translated $1$ units left and $2$ units down.

So, the transformation from $(x-5)^2+7$ to $(x+1)^2-2$ is: go $6$ units to the left and $2$ units down

Answer 2

Answer: Translation of shifted 6 unit left and 9 units down.

Step-by-step explanation: Original given function $y=(x-5)^2+7$.

Translated function $y=(x+1)^2-2$.

We can see $y=(x-5)^2+7$  is being transformed to $y=(x-5+6)^2+7-9.$

We can see $y=f(x+c)-d$ transformation rule is being applied there.

According to $y=f(x+c)-d$ transformation rule the function f(x) is being shifted c unit left and d units down.

Therefore, $y=(x-5+6)^2+7-9.$ is being shifted 6 unit left and 9 units down.