Reflection
We are going to determine how to reflect over the line
y = - x -1
First, if we are going to undertand how to reflect it over y = -x
What happens when y = - x?
If we want to reflect the point (-4, 0) over y = -x, graphically we know that its reflection is (0, 4).
It would be tha same for any point (x, 0), its reflection would be (0, -x).
Similarly, if we want to reflect the point (0, -5) over y = -x. Its reflection would be (5,0).
It would be the same for any point (0, y), its reflection would be (-y, 0)
We can generalize it: if we have a point (x, y) its reflection over y = -x would be (-y, -x)
(x, y) → (-y, -x)
Now that we have undertood when it is reflected over y = -x. We can find how to do it over y = -x -1
Reflection over y = - x -1
Doing the same case as y = -x
If we see the figure we will observe that
(x, y) reflected point would be (- y - 1, - x -1)
(x, y) → (-y - 1, -x - 1)
Reflecting the figure
We can observe that the figure has 5 points
(-4, -1), (-4, -4), (-2, -9), (-7, -2), (-8, -8)
We are going to reflect them using the last formula
(x, y) → (-y - 1, -x - 1)
(-4, -1) → ( -(-1) - 1, - (-4) -1)
= ( 1 -1, 4 - 1) = (0, 3)
(-4, -4) → (- (-4) -1, - (-4) -1)
= (4 -1, 4 - 1) = (3, 3)
(-2, -9) → (- (-9) -1, - (-2) -1)
= (9 - 1, 2 - 1) = (8, 1)
(-7, -2) → (- (-2) -1, - (-7) -1)
= (2 -1, 7 -1) = (1, 6)
(-8, -8) → (- (-8) -1, - (-8) -1)
= (8 - 1, 8 -1) = (7, 7)
Then the new points are:
(0, 3), (3, 3), (8, 1), (1, 6), (7, 7)