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

Answer:

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

Step-by-step 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:

$m = \frac{y_2-y_1}{x_2-x_1}$  

then;

$m = \frac{-45-30}{-10-5} =\frac{-75}{-15} = 5$

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