Question

triangle qrs has coordinates q(-8,4), r(-5,8) and s(-3,3). triangle qrs is transformed by (x,y)→(x + 4,y - 6)to produce △qrs. then △qrs is transformed by the rule (x,y)→(x + 5,y - 3) to produce △qrs. a. the final image, △qrs, will be in which quadrant? b. create the rule for the composition transformation. example (x+2,y - 4)

Explanation:

Step1: First transformation for point Q

For point Q(-8,4), applying the rule $(x,y)\to(x + 4,y - 6)$:
$x_1=-8 + 4=-4$
$y_1=4-6=-2$
So $Q'(-4,-2)$

Step2: Second transformation for Q'

Applying the rule $(x,y)\to(x + 5,y - 3)$ to $Q'(-4,-2)$:
$x_2=-4+5 = 1$
$y_2=-2-3=-5$
So $Q''(1,-5)$

For point R(-5,8), first - transformation:
$x_3=-5 + 4=-1$
$y_3=8-6 = 2$
So $R'(-1,2)$
Second - transformation:
$x_4=-1+5 = 4$
$y_4=2-3=-1$
So $R''(4,-1)$

For point S(-3,3), first - transformation:
$x_5=-3 + 4=1$
$y_5=3-6=-3$
So $S'(1,-3)$
Second - transformation:
$x_6=1+5 = 6$
$y_6=-3-3=-6$
So $S''(6,-6)$

Since all the x - coordinates of $Q''$, $R''$ and $S''$ are positive and all the y - coordinates are negative, the final image $\triangle Q''R''S''$ is in the fourth quadrant.

Step3: Find the composition rule

If we start with $(x,y)$ and first apply $(x,y)\to(x + 4,y - 6)$ and then $(x,y)\to(x + 5,y - 3)$
For the x - coordinate: First we add 4 and then 5, so the total change in x is $4 + 5=9$.
For the y - coordinate: First we subtract 6 and then 3, so the total change in y is $-6-3=-9$.
The composition rule is $(x,y)\to(x + 9,y-9)$

Answer:

a. Fourth quadrant
b. $(x + 9,y-9)$