Question

line l passes through the points 1,6 and -2,-9. write an equation of the image of l after a dilation with a scale factor of 5 centered at the origin

Answer 1

Equation of image I is, y = 5x + 5

Explanation:

An Equation of line passing through the two points
`(x_1, y_1)` and
`(x_2, y_2)` is given by;

`y-y_1 = m(x-x_1)` where m is the slope of the line.

Given: Line I passes through the points (1, 6) and (-2, -9)

To find an equation of the image of I after a dilation of scale factor 5 centered at origin.

Dilation: A transformation in which a image grows larger. It may be with respect to a point or with respect to the axis of a graph.

Since, dilation requires a center point and a scale factor k.

The rule of dilation with a scale factor k =5 centered at origin is given by:

`(x, y) \rightarrow (5x , 5y)`

Now, to dilate the points of I are;

`(1, 6) \rightarrow (5 \cdot 1 , 5 \cdot 6)` = (5 , 30)

`(-2, -9) \rightarrow (5 \cdot -2 , 5 \cdot -9)` = (-10 , -45)

The points of image I are; (5, 30) and (-10 , -30)

First calculate the slope:

Slope(m) of the Image I is given by:

`m = (y_2-y_1)/(x_2-x_1)`

then;

`m = (-45-30)/(-10-5) =(-75)/(-15) = 5`

Then, the equation of image I is;

`y-30 = 5(x-5)`

Using distributive property;
`a \cdot (b+c) = a\cdot b + a\cdot c`

y -30 =5x -25

Add 30 to both sides we get;

y -30+30 = 5x -25 +30

Simplify:

y = 5x + 5

The equation of the image I after a dilation with scale factor of 5 centered at the origin is, y = 5x + 5