Question 1181247
<br>
There are probably quick and easy methods for finding the answer -- probably some process using transformation matrices.<br>
But I'm not familiar with those methods, so I will use more basic methods.<br>
The transformation is<br>
{{{y=af(b(x-c))+d}}}<br>
a is a vertical stretch or compression factor;
b is a horizontal stretch of compression factor;
c is a horizontal shift; and
d is a vertical shift<br>
We can easily find the horizontal and vertical stretch or compression factors:<br>
(1) The horizontal distance from B to C is 10; the horizontal distance from B' to C' is 5; the horizontal compression factor is b = 5/10 = 1/2.
(2) The vertical distance from A to B is -3; the vertical distance from A' to B' is 9; the vertical stretch factor is a = 9/-3 = -3.<br>
Parameters b and c affect the x coordinates; and we know b is 1/2.<br>
The transformation b(x-c) must change the x coordinate of each point into the x coordinate of its image.<br>
The x coordinate of A is -8; the x coordinate of A' is -6.  Knowing that b is 1/2, we have<br>
(1/2)(-8-c)=-6
-8-c=-12
c=4<br>
To check our method and our calculations, we can verify that b=1/2 and c=4 give the right x coordinates for the other two points:<br>
For point B, -6 should transform to -5:
(1/2)(-6-4)=(1/2)(-10)=-5<br>
And for point C, 4 should transform to 0:
(1/2)(4-4)=1/2(0)=0<br>
Parameters a and d affect the y coordinates; and we know a is -3.<br>
The transformation a(y)+d must change the y coordinate of each point into the y coordinate of its image.<br>
The y coordinate of A is -1; the y coordinate of A' is -3.  Knowing that a is -3, we have<br>
-3(-1)+d=-3
3+d=-3
d=-6<br>
And we should verify a=3 and d=-6 give the correct y coordinate for points B and C:<br>
-3(-4)-6=12-6=6<br>
Everything checks....<br>
ANSWER: y = -3((1/2)(x-4))-6<br>
Here are the transformations of the three points, one transformation at a time:<br><pre>

                                      A        B         C
  ------------------------------------------------------------
                                    (-8,-1)  (-6,-4)   (4,-4)
(1) shift left 4:                  (-12,-1) (-10,-4)   (0,-4)
(2) compress horizontally by 1/2:   (-6,-1)  (-5,-4)   (0,-4)
(3) stretch vertically by -3:       (-6,3)   (-5,12)   (0,12)
(4) shift down 6:                  (-6,-3)   (-5,6)    (0,6)<br></pre>