I need to rotate the line the given no. of degrees (a) about the x-intercept and (b) about the y-intercept need to write equation of each image. don't know how to start
y = 2x- 3;90degree
For y = -2x - 3, the slope is -2
So the x-intercept, let y=0, is
0 = -2x - 3
x = 3/2
After a rotation of 90º about this point, the slope of the new line is 1/2
and the new equation is
y = (1/2)x + b
but (3/2,0) lies on it
0 = (1/2)(3/2) + b
b = -3/4
so the new line is y = (1/2)x - 3/4
Follow the same method to rotate the original line about the y-intercept.
Let me know what you got
(It is easier, you should be able to write it without even doing any work)
correction:
0 = -2x - 3
x = -3/2
After a rotation of 90º about this point, the slope of the new line is 1/2
and the new equation is
y = (1/2)x + b
but (-3/2,0) lies on it
0 = (1/2)(-3/2) + b
b = 3/4
so the new line is y = (1/2)x + 3/4
To rotate a line about a specific point, you can use basic rotation formulas. Let's break down the process into two parts:
(a) Rotating about the x-intercept:
1. Find the coordinates of the x-intercept by setting y = 0 and solving for x.
For the given equation y = 2x - 3, we substitute y with 0:
0 = 2x - 3
2x = 3
x = 3/2
2. Substitute the x-intercept coordinates into the rotation formula.
The rotation formula for a point (x, y) about the origin (0, 0) by an angle θ is:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
Since we want to rotate about the x-intercept (3/2, 0), we need to translate the point to the origin first. The translated coordinates are (0, -3).
3. Plug the translated coordinates into the rotation formula.
x'' = x' * cos(θ) - y' * sin(θ)
y'' = x' * sin(θ) + y' * cos(θ)
Substituting (0, -3) into the formulas:
x'' = 0 * cos(90°) - (-3) * sin(90°)
y'' = 0 * sin(90°) + (-3) * cos(90°)
Simplifying these equations will give you the coordinates of the rotated line about the x-intercept.
(b) Rotating about the y-intercept:
1. Similar to step 1 above, find the coordinates of the y-intercept by setting x = 0 and solving for y.
For the given equation y = 2x - 3, we substitute x with 0:
y = 2(0) - 3
y = -3
2. Substitute the y-intercept coordinates into the rotation formula.
The rotation formula for a point (x, y) about the origin (0, 0) by an angle θ is:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
Since we want to rotate about the y-intercept (0, -3), we need to translate the point to the origin first. The translated coordinates are (-3, 0).
3. Plug the translated coordinates into the rotation formula.
x'' = x' * cos(θ) - y' * sin(θ)
y'' = x' * sin(θ) + y' * cos(θ)
Substituting (-3, 0) into the formulas:
x'' = (-3) * cos(90°) - 0 * sin(90°)
y'' = (-3) * sin(90°) + 0 * cos(90°)
Simplifying these equations will give you the coordinates of the rotated line about the y-intercept.
Once you have the coordinates of the rotated line, you can write the equation using the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept of the rotated line.