Question

1. graph $overline{vw}$ with endpoints $v(-6, -4)$ and $w(-3, 1)$ and its image after the composition.

translation: $(x,y)\to(x + 3,y + 1)$
translation: $(x,y)\to(x - 6,y - 4)$

1. in example 6, you move 2 tiles right and 3 tiles up. then you move 1 tile left and 1 tile down. rewrite the composition as a single translation.

Explanation:

Step1: Apply first translation to point V

For point $V(-6,-4)$, using the translation $(x,y)\to(x + 3,y + 1)$, we get $V_1=(-6+3,-4 + 1)=(-3,-3)$.

Step2: Apply second translation to $V_1$

Using the translation $(x,y)\to(x-6,y - 4)$ on $V_1(-3,-3)$, we get $V_2=(-3-6,-3-4)=(-9,-7)$.

Step3: Apply first translation to point W

For point $W(-3,1)$, using the translation $(x,y)\to(x + 3,y + 1)$, we get $W_1=(-3+3,1 + 1)=(0,2)$.

Step4: Apply second translation to $W_1$

Using the translation $(x,y)\to(x-6,y - 4)$ on $W_1(0,2)$, we get $W_2=(0-6,2-4)=(-6,-2)$.

Step5: Find single translation for problem 7

Moving 2 tiles right and 3 tiles up then 1 tile left and 1 tile down. The net change in x - direction is $2-1=1$ (right) and net change in y - direction is $3-1 = 2$ (up). So the single translation is $(x,y)\to(x + 1,y+2)$.

Answer:

For problem 6: The endpoints of the image are $V_2(-9,-7)$ and $W_2(-6,-2)$.
For problem 7: The single - translation is $(x,y)\to(x + 1,y+2)$.