A translation moves an object from one place to another. Every point on the object moves the same distance in the same direction.

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

x' = x + h and y' = y + k are called transformation equations.

**Example 1 :**

Triangle OAB with vertices O(0, 0), A(2, 3) and B(-1, 2) is translated under the vector <3, 2>. Find the image vertices and illustrate the object and the image.

**Solution :**

To find vertices of image after translation, we use

x' = x+h and y' = y+k

Here (h, k) ==> (3, 2)

x' = x+3 and y' = y+2

Before translation O(0, 0) A(2, 3) B(-1, 2) |
After translation O'(0+3, 0+2) ==> O'(3, 2) A'(2+3, 3+2) ==> A'(5, 5) B'(-1+3, 2+2) ==> B'(2, 4) |

**Example 2 :**

Find the image equation when 2x-3y = 6 is translated under the vector <-1, 2>.

**Solution :**

Let us find any two point on the straight line 2x-3y = 6.

Converting 2x-3y = 6 into intercept form, we get

(2x/6)-(3y/6) = 6/6

x/3 - y/2 = 1

Point on x-intercept is (3, 0).

Point on y-intercept is (0, -2).

Here (h, k) ==> (-1, 2)

x' = x-1 and y' = y+2

Before translation A(3, 0) B(0, -2) |
After translation A'(3-1, 0+2) ==> A'(2, 2) B'(0-1, -2+2) ==> B'(-1, 0) |

**Example 3 :**

Find the image of y = 2x^{2} under a translation with vector <3, -2>.

**Solution :**

Let us find any three points on the parabola.

Vertex of the parabola is (0, 0)

If x = -1, then y = 2

If x = 1, then y = 2

So three point on the parabola are V(0, 0), A(-1, 2) and B(1, 2)

The image has to be translated along the vector <3, -2>.

(h, k) ==> (3,-2)

x' = x+3 and y' = y-2

Before translation V(0, 0) A(-1, 2) B(1, 2) |
After translation V'(0+3, 0-2) ==> V'(3, -2) A'(-1+3, 2-2) ==> A'(2, 0) B'(1+3, 2-2) ==> B'(4, 0) |

To find equation of new cure, let us find value of x and y from x' = x+3 and y' = y-2

x = x'-3 and y = y+2

Given curve :

y = 2x^{2}

Equation of curve after translation :

y+2 = 2(x-3)^{2}

y+2 = 2(x^{2}-6x+9)

y+2 = 2x^{2}-12x+18

y = 2x^{2}-12x+16

**Example 4 :**

Find the image of xy = 5 under a translation with vector <-4, 1>.

**Solution :**

xy = 5

Finding 4 point on the rectangular hyperbola,

y = 5/x

If x = 1, then y = 5 ==> A(1, 5)

If x = 5, then y = 1 ==> B(5, 1)

If x = -1, then y = -5 ==> C(-1, -5)

If x = -5, then y = -1 ==> D(-5, -1)

Here (h, k) ==> (-4, 1)

x' = x-4 and y' = y+1

Before translation A(1, 5) B(5, 1) C(-1, -5) D(-5, -1) |
After translation A'(1-4, 5+1) ==> A'(-3, 6) B'(5-4, 1+1) ==> B'(1, 2) C'(-1-4, -5+1) ==> B'(-5, -4) D'(-5-4, -1+1) ==> B'(-9, 0) |

Equation of curve before translation :

xy = 5

Equation of curve after translation :

x' = x-4 and y' = y+1

x = x'+4 and y = y'-1

By applying x as x'+4 and y as y-1, we get

(x+4)(y-1) = 5

xy-4x+4y-4-5 = 0

xy-4x+4y-9 = 0

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