Question

visit www.bigideasmathvideos.com to watch the flipped video instruction for the \try this\ problem(s) below. try this video for example 5 – performing a composition 5) graph \\(\overline{rs}\\) with endpoints \\(r(-8, 5)\\) and \\(s(-6, 8)\\) and its image after the composition. translation: \\((x, y) \to (x - 1, y + 4)\\) translation: \\((x, y) \to (x + 4, y - 6)\\)

Explanation:

Step1: Translate point R first

Apply $(x,y)\to(x-1,y+4)$ to $R(-8,5)$:
$x' = -8 - 1 = -9$, $y' = 5 + 4 = 9$
Result: $R'(-9,9)$

Step2: Translate R' second

Apply $(x,y)\to(x+4,y-6)$ to $R'(-9,9)$:
$x'' = -9 + 4 = -5$, $y'' = 9 - 6 = 3$
Result: $R''(-5,3)$

Step3: Translate point S first

Apply $(x,y)\to(x-1,y+4)$ to $S(-6,8)$:
$x' = -6 - 1 = -7$, $y' = 8 + 4 = 12$
Result: $S'(-7,12)$

Step4: Translate S' second

Apply $(x,y)\to(x+4,y-6)$ to $S'(-7,12)$:
$x'' = -7 + 4 = -3$, $y'' = 12 - 6 = 6$
Result: $S''(-3,6)$

Step5: Combine translations (simplify)

Combine the two translations:
$(x,y)\to(x-1+4,y+4-6)=(x+3,y-2)$
Verify with R: $(-8+3,5-2)=(-5,3)$, matches $R''$
Verify with S: $(-6+3,8-2)=(-3,6)$, matches $S''$

Answer:

1. Original segment $\overline{RS}$ has endpoints $R(-8,5)$ and $S(-6,8)$.
2. The image segment $\overline{R''S''}$ after the composition has endpoints $R''(-5,3)$ and $S''(-3,6)$. To graph:

• Plot $R(-8,5)$ and $S(-6,8)$, connect them for the original segment.
• Plot $R''(-5,3)$ and $S''(-3,6)$, connect them for the image segment.