Question

use the function rules to find the coordinates of $delta abc$ and predict the quadrant it will be in after the transformations. $delta abc$, with coordinates $a(-8,4), b(-5,8)$, and $c(-3,3)$, is transformed by $(x,y)\to(x + 4,y - 6)$ to produce $delta abc$. $delta abc$ is transformed by the rule $(x,y)\to(x + 5,y - 3)$. create the rule for the composition transformation and predict what quadrant the image will be in. (4 points)

Explanation:

Step1: Find the composition transformation rule

The first transformation is $(x,y)\to(x + 4,y - 6)$ and the second is $(x,y)\to(x + 5,y - 3)$. For the composition, we apply the first - then the second. If we start with $(x,y)$, after the first transformation we get $(x_1,y_1)=(x + 4,y - 6)$. Then applying the second transformation to $(x_1,y_1)$ gives $(x_2,y_2)=(x_1+5,y_1 - 3)=(x + 4+5,y - 6-3)=(x + 9,y - 9)$.

Step2: Apply the composition rule to point A

For point $A(-8,4)$, substituting $x=-8$ and $y = 4$ into the composition rule $(x + 9,y - 9)$ gives $A''=(-8 + 9,4 - 9)=(1,-5)$.

Step3: Apply the composition rule to point B

For point $B(-5,8)$, substituting $x=-5$ and $y = 8$ into the composition rule $(x + 9,y - 9)$ gives $B''=(-5 + 9,8 - 9)=(4,-1)$.

Step4: Apply the composition rule to point C

For point $C(-3,3)$, substituting $x=-3$ and $y = 3$ into the composition rule $(x + 9,y - 9)$ gives $C''=(-3 + 9,3 - 9)=(6,-6)$.

Step5: Determine the quadrant

Since the $x$-coordinates of $A''$, $B''$, and $C''$ are positive and the $y$-coordinates are negative, $\triangle A''B''C''$ is in the fourth - quadrant.

Answer:

The composition transformation rule is $(x,y)\to(x + 9,y - 9)$ and $\triangle A''B''C''$ is in the fourth quadrant.