Question 1181247: The graph of y= f(x) with points A( -8,-1), B ( -6,-4) , and C (4,-4) is transformed to give the image points A'( -6,-3), B' ( -5,6) and C'( 0,6) . Plot the points and determine the equation of the image function in the form
y=af[b(x-c)) +d.
I appreciate any help.
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
There are probably quick and easy methods for finding the answer -- probably some process using transformation matrices.
But I'm not familiar with those methods, so I will use more basic methods.
The transformation is
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
We can easily find the horizontal and vertical stretch or compression factors:
(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.
Parameters b and c affect the x coordinates; and we know b is 1/2.
The transformation b(x-c) must change the x coordinate of each point into the x coordinate of its image.
The x coordinate of A is -8; the x coordinate of A' is -6. Knowing that b is 1/2, we have
(1/2)(-8-c)=-6
-8-c=-12
c=4
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:
For point B, -6 should transform to -5:
(1/2)(-6-4)=(1/2)(-10)=-5
And for point C, 4 should transform to 0:
(1/2)(4-4)=1/2(0)=0
Parameters a and d affect the y coordinates; and we know a is -3.
The transformation a(y)+d must change the y coordinate of each point into the y coordinate of its image.
The y coordinate of A is -1; the y coordinate of A' is -3. Knowing that a is -3, we have
-3(-1)+d=-3
3+d=-3
d=-6
And we should verify a=3 and d=-6 give the correct y coordinate for points B and C:
-3(-4)-6=12-6=6
Everything checks....
ANSWER: y = -3((1/2)(x-4))-6
Here are the transformations of the three points, one transformation at a time:
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)
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