If is a line whose equation is y = 2x - 1, find the equation of the image of under each of the following trans?
a. (x, y) → (x, y - 2)
b. (x, y) → (x +3, y)
c. (x, y) → (x - 3, y + 2)
a. To find the equation of the image of the line y = 2x - 1 under the transformation (x, y) → (x, y - 2), we need to substitute y with y - 2 in the equation.
Substituting y with y - 2 in the equation y = 2x - 1, we get:
y - 2 = 2x - 1
Simplifying the equation gives us the equation of the image line:
y = 2x + 1
So, the equation of the image line under the transformation (x, y) → (x, y - 2) is y = 2x + 1.
b. To find the equation of the image of the line y = 2x - 1 under the transformation (x, y) → (x + 3, y), we need to substitute x with x + 3 in the equation.
Substituting x with x + 3 in the equation y = 2x - 1, we get:
y = 2(x + 3) - 1
Simplifying the equation gives us the equation of the image line:
y = 2x + 5
So, the equation of the image line under the transformation (x, y) → (x + 3, y) is y = 2x + 5.
c. To find the equation of the image of the line y = 2x - 1 under the transformation (x, y) → (x - 3, y + 2), we need to substitute x with x - 3 and y with y + 2 in the equation.
Substituting x with x - 3 and y with y + 2 in the equation y = 2x - 1, we get:
y + 2 = 2(x - 3) - 1
Simplifying the equation gives us the equation of the image line:
y = 2x - 5
So, the equation of the image line under the transformation (x, y) → (x - 3, y + 2) is y = 2x - 5.
To find the equation of the image of a line under a given transformation, we will apply the transformation to each point on the original line and then find the equation of the transformed line using the new set of points.
a. Transformation: (x, y) → (x, y - 2)
To apply the transformation to each point on the line y = 2x - 1, we substitute the x and y values into the transformation equations.
Let's take an arbitrary point on the line, (x, y) = (a, 2a - 1).
Applying the transformation, (a, 2a - 1) → (a, (2a - 1) - 2) = (a, 2a - 3).
Now, we have a new set of points on the transformed line: (x, y) = (a, 2a - 3).
To find the equation of the transformed line, we express y in terms of x using these new points.
y = 2a - 3
Therefore, the equation of the image under the transformation (x, y) → (x, y - 2) is y = 2x - 3.
b. Transformation: (x, y) → (x + 3, y)
Following the same process as above, we substitute the x and y values into the transformation equations.
Let's take an arbitrary point on the line, (x, y) = (a, 2a - 1).
Applying the transformation, (a, 2a - 1) → (a + 3, 2a - 1).
Now, we have a new set of points on the transformed line: (x, y) = (a + 3, 2a - 1).
Expressing y in terms of x using these new points gives us:
y = 2(x + 3) - 1
Simplifying, we get: y = 2x + 5.
Therefore, the equation of the image under the transformation (x, y) → (x + 3, y) is y = 2x + 5.
c. Transformation: (x, y) → (x - 3, y + 2)
Again, we substitute the x and y values into the transformation equations.
Let's take an arbitrary point on the line, (x, y) = (a, 2a - 1).
Applying the transformation, (a, 2a - 1) → (a - 3, 2a - 1 + 2) = (a - 3, 2a + 1).
Now, we have a new set of points on the transformed line: (x, y) = (a - 3, 2a + 1).
Expressing y in terms of x using these new points gives us:
y = 2(a - 3) + 1
Simplifying, we get: y = 2a - 5.
Therefore, the equation of the image under the transformation (x, y) → (x - 3, y + 2) is y = 2x - 5.