Chapter 13.3A, Problem 8A

a. Explain why in a coordinate system a dilation with center at the origin and scale factor $r>0$ is given by $\left(x,y\right)\to \left(rx,ry\right)$.

b. The transformation $\left(x,y\right)\to \left(-2x,-2y\right)$ can be achieved by a dilation followed by an isometry. Find that dilation and the isometry.

c. Find the equations of the images of each of the following lines under the dilation in part (a) with the given scale factor r.

i. $y=2x,r=\frac{1}{2}$

ii. $y=2x,r=2$

iii. $y=2x+1,r=\frac{1}{3}$ 

iv. $y=-x-1,r=3$