Question

1. pre - image image p(-8, -4) p(-6, -3) q(-8, 4) q(-6, 3) r(8, 4) r(6, 3) s(8, -4) s(6, -3) scale factor: 1.5 rule:

Explanation:

Step1: Recall the rule for scale - factor in coordinate transformation

To find the scale - factor $k$ for a dilation in the coordinate plane, we can use the ratio of the coordinates of the image to the pre - image for a non - zero coordinate. For the $x$ - coordinates, if we take a point $P(-8,-4)$ and its image $P'(-6,-3)$, we use the formula $k=\frac{x_{image}}{x_{pre - image}}$ (similarly for $y$).

Step2: Calculate the scale - factor using $x$ - coordinates

For point $P$, $x_{pre - image}=-8$ and $x_{image}=-6$. Then $k=\frac{-6}{-8}=\frac{3}{4} = 0.75$. For point $R$, $x_{pre - image}=8$ and $x_{image}=6$, and $k=\frac{6}{8}=\frac{3}{4}=0.75$.

Step3: Calculate the scale - factor using $y$ - coordinates

For point $P$, $y_{pre - image}=-4$ and $y_{image}=-3$. Then $k = \frac{-3}{-4}=\frac{3}{4}=0.75$. For point $Q$, $y_{pre - image}=4$ and $y_{image}=3$, and $k=\frac{3}{4}=0.75$.

Step4: Write the rule for the dilation

The rule for a dilation with a scale - factor $k$ centered at the origin is $(x,y)\to(kx,ky)$. Since $k = 0.75=\frac{3}{4}$, the rule is $(x,y)\to(\frac{3}{4}x,\frac{3}{4}y)$.

Answer:

Scale factor: $0.75$
Rule: $(x,y)\to(\frac{3}{4}x,\frac{3}{4}y)$