what is the image of point A ( 7, -2) after each transformation

Question

Reflection over the Y-axis 

And reflection over the line Y= -1

Answer 1

To reflect a point about the y-axis, make the x-coordinate negative:

From (7,-2), we have (-7,-2).

 

Now let's reflect it about the line y = -1.  How far is your point away from this line?  We can use absolute values to find out:

 

|-2 - (-1)| =

|-2 + 1| =

| -1 | =

1.

 

So, we're one unit away from y = -1.  If we're one unit ABOVE y=-1, we need to move the point to be one unit BELOW it instead.  And if we're one unit BELOW y = -1, we need to move the point to be one unit ABOVE it. 

 

Since the point has a y-coordinate of -2, its BELOW the line y = -1, by 1 unit.  Now we change the y-coordinate so its instead ABOVE y = -1, by 1 unit:

 

(y-coordinate of reflection line) + (number of units above that line) =

-1 + 1 =

0.

 

Our new y-coordinate is 0, so the point is now at

(7, 0).

 

A more mechanical, but less intuitive approach is as follows: 

Let L = the y-coordinate of the line, and

P = the y-coordinate of the point.

The new y-coordinate is 2L - P.

In this case, L = -1, and P = -2.  So we have

2L - P =

2(-1) - (-2) =

-2 -(-2) =

0.

 

Answer 2

Reflection over the y-axis, (x,y)→(-x, y)

Therefore,

A(7, -2)→A'(-7, -2)

Reflection over the line y=-1

 

|-2-(-1)| will give you the distance from A to the line of reflection, y= -1

which is 1.

Add twice the distance to the y co-ordinate of point A to obtain A'.

A(7,-2) →A'(7, -2 +2(1) )

A'(7,0)