the question is to "rotate the line the given number of degrees a about the x-intercept and (b) about the y-intercept. Write the equation of each image"
and the problem is y=2x+2 ; 90 degrees
and i dont know what to do
To rotate a line about a point, you can use the following steps:
1. Find the coordinates of the point through which you want to rotate the line (a, b). In this case, we will find the coordinates of the x-intercept and the y-intercept.
2. Use the rotation formulas to calculate the new coordinates of each point on the line after rotation.
- To rotate a point (x, y) counterclockwise by a given angle θ about the origin (0, 0), the new coordinates (x', y') can be calculated using the following formulas:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
3. Substitute the new coordinates into the slope-intercept form of the line equation (y = mx + c) to find the equation of each image line.
Let's solve the problem step by step:
Step 1: Finding the coordinates of the x-intercept and y-intercept
To find the x-intercept, we set y = 0 in the equation of the line:
0 = 2x + 2
2x = -2
x = -1
So, the x-intercept is (-1, 0).
To find the y-intercept, we set x = 0 in the equation of the line:
y = 2(0) + 2
y = 2
So, the y-intercept is (0, 2).
Step 2: Calculating the new coordinates after rotation
Since we want to rotate the line 90 degrees, we can use the rotation formulas:
For rotating about the x-intercept:
x' = x * cos(90) - y * sin(90)
y' = x * sin(90) + y * cos(90)
For rotating about the y-intercept:
x' = x * cos(90) - y * sin(90)
y' = x * sin(90) + y * cos(90)
Step 3: Writing the equations of the image lines
Substituting the new coordinates into the slope-intercept form (y = mx + c), we can find the equations of the image lines.
For rotating about the x-intercept (-1, 0):
x' = 0 * cos(90) - 2 * sin(90) = 0
y' = 0 * sin(90) + 2 * cos(90) = 2
The new point after rotation is (0, 2).
Using the two points (-1, 0) and (0, 2), we can find the equation of the image line:
m = (y' - y) / (x' - x) = (2 - 0) / (0 - (-1)) = 2/1 = 2
Using the point-slope form (y - y1) = m(x - x1), where (x1, y1) = (-1, 0):
y - 0 = 2(x - (-1))
y = 2x + 2
So, the equation of the image line after rotating 90 degrees about the x-intercept is y = 2x + 2.
For rotating about the y-intercept (0, 2):
x' = 2 * cos(90) - 0 * sin(90) = 0
y' = 2 * sin(90) + 0 * cos(90) = 2
The new point after rotation is (0, 2).
Using the two points (0, 2) and (0, 2), we can find the equation of the image line:
m = (y' - y) / (x' - x) = (2 - 2) / (0 - 0) = 0/0 (undefined)
In this case, the slope is undefined, which means the line after rotation is vertical. Therefore, the equation of the image line after rotating 90 degrees about the y-intercept is x = 0.
In summary:
- After rotating 90 degrees about the x-intercept, the equation of the image line is y = 2x + 2.
- After rotating 90 degrees about the y-intercept, the equation of the image line is x = 0.
rotating the line 90° gives you a perpendicular line, thus with slope -1/2
So, find the desired intercept, and use that point and the new slope to determine the line (use the point-slope form)