Question

for each representation of a dilation, list the scale factor and an algebraic rule for the dilation.
7.
8.
9.
pre - image\timage
p(-8, -4)\tp(-6, -3)
q(-8, 4)\tq(-6, 3)
r(8, 4)\tr(6, 3)
s(8, -4)\ts(6, -3)
scale factor:
rule:
scale factor:
rule:
scale factor:
rule:

Explanation:

Step1: Recall scale - factor formula

For a dilation of a point $(x,y)$ to $(x',y')$ with center at the origin, the scale factor $k$ is given by $k=\frac{x'}{x}=\frac{y'}{y}$ (when $x
eq0$ and $y
eq0$).

Step2: Solve for problem 7

Count the vertical or horizontal distances. Let's consider the vertical distance from the $x$ - axis to point $B$ and $B'$. The $y$ - coordinate of $B$ is 6 and of $B'$ is 3. So the scale factor $k = \frac{3}{6}=\frac{1}{2}$. The algebraic rule for a dilation with center at the origin and scale factor $k$ is $(x,y)\to(kx,ky)$. So the rule is $(x,y)\to(\frac{1}{2}x,\frac{1}{2}y)$.

Step3: Solve for problem 8

Take a point, say $P(-8,-4)$ and its image $P'(-6,-3)$. Calculate the scale factor $k=\frac{-6}{-8}=\frac{-3}{-4}=\frac{3}{4}$. The algebraic rule for the dilation is $(x,y)\to(\frac{3}{4}x,\frac{3}{4}y)$.

Step4: Solve for problem 9

Let's consider the horizontal distance of a point. For example, consider the left - most point $R(1,1)$ and its image $R'( \frac{5}{2}, \frac{5}{2})$. The scale factor $k=\frac{\frac{5}{2}}{1}=\frac{5}{2}$. The algebraic rule for the dilation is $(x,y)\to(\frac{5}{2}x,\frac{5}{2}y)$.

Answer:

1. Scale factor: $\frac{1}{2}$, Rule: $(x,y)\to(\frac{1}{2}x,\frac{1}{2}y)$
2. Scale factor: $\frac{3}{4}$, Rule: $(x,y)\to(\frac{3}{4}x,\frac{3}{4}y)$
3. Scale factor: $\frac{5}{2}$, Rule: $(x,y)\to(\frac{5}{2}x,\frac{5}{2}y)$