If P(x, y) is translated h units in the x-direction and k units in the y-direction to become P'(x', y'), then
x' = x + h and y' = y + k
Find the image equation of the following and give answer in the form of y = f(x).
Problem 1 :
3x + 2y = 8 under (-1, 3)
Solution :
By observing the translation factor, h = -1 and k = 3. Moving the line left of 1 unit and up 3 units.
x' = x + h and y' = y + k
x' = x - 1 and y' = y + 3
Given equation :
3x + 2y = 8
3(x - 1) + 2(y + 3) = 8
3x - 3 + 2y + 6 = 8
3x + 2y + 3 = 8
3x + 2y + 3 - 8 = 0
After the translation, the required equation will be,
3x + 2y - 5 = 0
Problem 2 :
2x - y = 6 under (-3, 0)
Solution :
By observing the translation factor, h = -3 and k = 0. Moving the line left of 3 units and no vertical move.
x' = x + h and y' = y + k
x' = x - 3 and y' = y + 0
Given equation :
2x - y = 6
2(x - 3) - y = 6
2x - 6 - y = 6
2x - y = 6 + 6
2x - y = 12
After the translation, the required equation will be,
2x - y = 12
Problem 3 :
y = x^{2} under (0, 3)
Solution :
By observing the translation factor, h = 0 and k = 3. There is no horizontal move and moving up 3 units.
x' = x + h and y' = y + k
x' = x + 0 and y' = y + 3
Given equation :
y = x^{2}
y + 3 = (x+0)^{2}
y + 3 = x^{2}
y = x^{2} + 3
After the translation, the required equation will be,
y = x^{2} + 3
Problem 4 :
xy = 5 under (-4, 1)
Solution :
By observing the translation factor, h = -4 and k = 1. There is no horizontal move and moving up 1 unit.
x' = x + h and y' = y + k
x' = x - 4 and y' = y + 1
Given equation :
xy = 5
(x - 4)(y + 1) = 5
xy + x - 4y - 4 = 5
xy + x - 4y = 4 + 5
xy + x - 4y = 9
After the translation, the required equation will be,
xy + x - 4y = 9
Problem 5 :
y = 2^{x} under (0, -3)
Solution :
By observing the translation factor, h = 0 and k = -3. There is no horizontal move and moving up units.
x' = x + h and y' = y + k
x' = x - 0 and y' = y - 3
Given equation :
y = 2^{x}
y - 3 = 2^{x}
y = 2^{x }+ 3
After the translation, the required equation will be,
y = 2^{x }+ 3
Problem 6 :
y = 3^{-x} under (2, 0)
Solution :
By observing the translation factor, h = 2 and k = 0. We move horizontally 2 units right and no vertical move.
x' = x + h and y' = y + k
x' = x + 2 and y' = y - 0
Given equation :
y = 3^{-x}
y = 3^{-(x + 2)}
y = 3^{-x - 2}
y = 3^{-x }3^{-2}
y = 3^{-x }/ 9
y = 1/9(3^{x})
After the translation, the required equation will be,
y = 1/9(3^{x})
Problem 7 :
What single transformation is equivalent to a translation of (2, 1) followed by translation of (3, 4) ?
Solution :
Let us consider the point involving here as (x, y). Moving this point with the translation of (2, 1)
So, it will become (x + 2, y + 1)
Moving this point with (3, 4). Then, we have to move 3 unit right and 4 units up.
(x + 2 + 3, y + 1 + 4)
(x + 5, y + 5)
So, the single transformation is translation of (5, 5).
Problem 8 :
Complete the missing translation factor :
Solution :
Initial position is (2, -1).
Translation factor is (3, 4), we have to move the point 3 units right and 4 units up.
New position will be,
= (2 + 3, -1 + 4)
= (5, 3)
Problem 9 :
Solution :
Initial position is (5, 2).
Translation factor is (-1, 4), we have to move the point 1 unit left and 4 units up.
New position will be,
= (5 - 1, 2 + 4)
= (4, 6)
Problem 10 :
Solution :
Initial position is (3, -2).
Translation factor is unknown, let the translation factor be (h, k)
New position is (3, 1)
(2 + h, -2 + k) --> (3, 1)
2 + h = 3 and -2 + k = 1
h = 1 and k = 3
So, moving the point 1 unit right and 3 units up.
Problem 11 :
Solution :
Initial position is unknown, let the initial position be (x, y).
New position is (-3, 2)
(x - 3, y + 1) --> (-3, 2)
x - 3 = -3 and y + 1 = 2
x = 0 and y = 1
So, there is no horizontal move and vertically moving 1 unit up.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM