**Problem 1 :**

Find the image equation when 3x+2y = 8 is translated under the vector <-1, 3>.

**Solution :**

Let us find any two points on the straight line 3x+2y = 8

Converting 3x+2y = 8 to intercept form, we get

(3x/8)+(2y/8) = 8/8

x/(8/3) + y/4 = 1

Point on x-intercept is A(8/3, 0).

Point on y-intercept is B(0, 4).

Here (h, k) ==> (-1, 3)

x' = x-1 and y' = y+3

Before translation A(8/3, 0) B(0, 4) |
After translation A'(8/3-1, 0+3) => A'(5/3, 3) B'(0-1, 4+3) ==> B'(-1, 7) |

**Problem 2 :**

Find the image equation when 2x-y = 6 is translated under the vector <-3, 0>.

**Solution :**

Let us find any two points on the straight line 2x-y = 6

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

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

x/3 - y/6 = 1

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

Point on y-intercept is B(0, -6).

Here (h, k) ==> (-3, 0)

x' = x-3 and y' = y+0

Before translation A(3, 0) B(0, -6) |
After translation A'(3-3, 0+0) => A'(0, 0) B'(0-3, -6+0) ==> B'(-3, -6) |

**Problem 3 :**

Find the image equation when y = x^{2} is translated under the vector <0, 3>.

**Solution :**

Let us find any three points on the parabola y = x^{2}

Finding vertex ==> V(0, 0)

If x = -2, then y = 4 ==> A(-2, 4)

If x = 2, then y = 4 ==> B(2, 4)

Here (h, k) ==> (0, 3)

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

**Problem 4 :**

Find the image equation when xy = -8 is translated under the vector <3, -2>.

**Solution :**

Let us find any three points on the rectangular parabola xy = -8

Finding 4 point on the rectangular hyperbola,

y = -8/x

If x = -2, then y = 4 ==> A(-2, 4)

If x = -1, then y = 8 ==> B(-1, 8)

If x = 1, then y = -8 ==> C(1, -8)

If x = 2, then y = -4 ==> D(2, -4)

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

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

x coordinate is +3, so we have to move old x coordinate 3 units right.

y coordinate is -2, so we have to move old y coordinate 2 units down.

Before translation A(-2, 4) B(-1, 8) C(1, -8) D(2, -4) |
After translation A'(-2+3, 4-2) ==> A'(1, 2) B'(-1+3, 8-2) ==> B'(2, 6) C'(1+3, -8-2) ==> C'(4, -10) D'(2+3, -4-2) ==> D'(5, -6) |

**Problem 5 :**

Find the image equation when y = 2^{x} is translated under the vector <0, -3>.

**Solution :**

Finding 3 points on the exponential function.

y = 2^{x}

If x = -1, then y = 1/2 ==> A(-1, 1/2)

If x = 0, then y = 1 ==> B(0, 1)

If x = 1, then y = 2 ==> C(1, 2)

Here (h, k) ==> (0, -3)

x' = x+0 and y' = y-3

No need of horizontal movements, but move the y-coordinate 3 units down.

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

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