a To rotate a line we need to know just two points on the line. To keep things simple, we can rotate the x-and y-intercept of the line. Since the slope-intercept form is y=2x-3, we know that the y-intercept is at (0,- 3). To find the x-intercept we set y equal to 0 and solve for x.
The x-intercept is at (1.5,0). To rotate a point by 90^(∘), 180^(∘), 270^(∘) and 360^(∘) about the origin, there are a number of rules that govern the new location of such a point.
90^(∘) Rotation:& (a,b)→ (- b,a)
180^(∘) Rotation:& (a,b)→ (- a,- b)
270^(∘) Rotation:& (a,b)→ (b,- a)
Let's apply these rules on the x- and y-intercept of the line.
|
|
Rotation Formula
|
x-intercept
|
Transformation
|
y-intercept
|
Transformation
|
| 90^(∘)
|
(a,b)→ (- b,a)
|
(1.5,0)
|
(0,1.5)
|
(0,- 3)
|
(3,0)
|
| 180^(∘)
|
(a,b)→ (- a,- b)
|
(1.5,0)
|
(- 1.5,0)
|
(0,- 3)
|
(0,3)
|
| 270^(∘)
|
(a,b)→ (b,- a)
|
(1.5,0)
|
(0,- 1.5)
|
(0,- 3)
|
(- 3,0)
|
Rotating something 360^(∘) about the origin will simply bring whatever is being rotated back to its original place. In other words, when we rotate the line 360^(∘), the x- and y-intercept will stay the same. Now we can graph each rotation.
From the graphs, we already know the y-intercept of each line that's been rotated. Additionally, we know that a 360^(∘) rotation maps the line onto itself so it's slope is m=2. So far we can write the different lines as.
90^(∘) Rotation:& y=mx+1.5
180^(∘) Rotation:& y=mx+3
270^(∘) Rotation:& y=mx-1.5
360^(∘) Rotation:& y= 2x-3
To complete the equations, we use the Slope Formula.
|
|
Points
|
y_2-y_1/x_2-x_1
|
m
|
| 90^(∘)
|
( 3,0) & ( 0,1.5)
|
0- 1.5/3- 0
|
- 1/2
|
| 180^(∘)
|
( 0,3) & ( - 1.5,0)
|
3- 0/0-( - 1.5)
|
2
|
| 270^(∘)
|
( 0,- 1.5) & ( - 3,0)
|
- 1.5- 0/0-( - 3)
|
- 1/2
|
Now we can write the equation of the line of each image.
90^(∘) Rotation:& y=- 1/2x+1.5 [0.8em]
180^(∘) Rotation:& y= 2x+3 [0.6em]
270^(∘) Rotation:& y=- 1/2x-1.5 [0.8em]
360^(∘) Rotation:& y= 2x-3
Since 2 and - 12 are negative reciprocals, we know that a rotation of 90^(∘) or 270^(∘) results in an image and preimage that are perpendicular. Additionally, when we do a 180^(∘) rotation, the preimage and image will be parallel. Finally, a 360^(∘) rotation results in an image and preimage that overlap.
