A(- 2,0), B(2,- 4), and C(4,2)
We want to draw the image of △ ABC after the indicated dilations centered at the origin with different scale factors.
A dilation with a scale factor - 2.
A dilation with a scale factor - 12.
A dilation with a scale factor - 3.
Let's consider each one separately.
Dilation With a Scale Factor - 2
To dilate △ ABC by a scale factor of - 2 centered at the origin, we will follow three steps.
Draw a ray from A through the origin and locate A' on AO such that A'O= - 2 AO.
Locate points B' and C' in the same way.
Draw △ A'B'C'.
Let's do it!
Dilation With a Scale Factor - 12
Now, we will draw the dilation of △ ABC centered at the origin with a scale factor of - 12. We will follow similar steps as we did before.
Dilation With a Scale Factor - 3
Last, we will dilate △ ABC by a scale factor of - 3. Again, we will follow the exact same steps.
Let's make a table showing the coordinates of each vertex after dilations.
Scale Factor
Coordinates
- 2
A'(4,0)
B'(- 4,8)
C'(- 8,- 4)
- 1/2
A''(1,0)
B''(- 1,2)
C''(- 2,- 1)
- 3
A'''(6,0)
B'''(- 6,12)
C'''(- 12,- 6)
b We want to make a conjecture about how the coordinates of the vertices of the figure change after a dilation centered at the origin with a negative scale factor. Here are the vertices of △ ABC.
A(- 2,0), B(2,- 4), and C(4,2)
Let's then take a look at the table from Part A.
Scale Factor
Coordinates
- 2
A'(4,0)
B'(- 4,8)
C'(- 8,- 4)
- 1/2
A''(1,0)
B''(- 1,2)
C''(- 2,- 1)
- 3
A'''(6,0)
B'''(- 6,12)
C'''(- 12,- 6)
If we compare the points with the vertices of △ ABC, we can see that the coordinates of each vertex were multiplied by the scale factor.
Scale Factor
Coordinates
- 2
A'( - 2(- 2), - 2(0))
B'( - 2(2), - 2(- 4))
C'( - 2(4), - 2(2))
- 1/2
A''( - 1/2(- 2), - 1/2(0))
B''( - 1/2(2), - 1/2(- 4))
C''( - 1/2(4), - 1/2(2))
- 3
A'''( - 3(- 2), - 3(0))
B'''( - 3(2), - 3(- 4))
C'''( - 3(4), - 3(2))
This is true for any dilation centered at the origin with a negative scale factor - k. This means we can find the coordinates of the image by multiplying each coordinate by - k.
(x,y) → ( - kx, - ky)
c Let's recall the function rule for a dilation centered at the origin with a scale factor k. See that here k is a positive value.
(x,y) → (kx,ky)
Now, if we take a negative value to be the scale factor, - k, the function rule looks the same. In this rule each coordinate is multiplied by the scale factor.
(x,y) → ( - kx, - ky)
Notice that it is the same rule as we found in Part B.
d In Parts B and C, we derived the function rule for a dilation centered at the origin with a negative scale factor - k.
(x,y) * k ⟶ (kx,ky) * - 1 ⟶ (- kx,- ky)
Here is what each operation means.
Multiplying the x- and y-coordinates by k means a dilation centered at the origin with a scale factor k.
Multiplying the x- and y-coordinates of the points by - 1 means a 180 ^(∘) rotation about the origin.
We can also see the composition of these two transformation in the applet below.
Therefore, the dilation of a figure centered at the origin with a scale factor of - k is the same as a dilation with center at the origin and scale factor k followed by a rotation 180^(∘) about the origin.
