Question

complete the composition of functions step by step: translation 2 up and 3 left, followed by a reflection over the line y = x, and finished with a rotation of 90° (counter clockwise). (1 point)
start with point a (5,4)
use translation rule:
result of translation: a
use reflection rule:
result of reflection: a
use of rotation rule:
final resulting point: a

Explanation:

Step1: Apply translation rule

For a translation 2 up and 3 left, the rule is $(x,y)\to(x - 3,y + 2)$. Given point $A(5,4)$, we have $x=5$ and $y = 4$. So $A'=(5 - 3,4 + 2)=(2,6)$.

Step2: Apply reflection rule

The reflection rule over the line $y=x$ is $(x,y)\to(y,x)$. For point $A'(2,6)$, $A''=(6,2)$.

Step3: Apply rotation rule

The rotation rule for a 90 - degree counter - clockwise rotation about the origin is $(x,y)\to(-y,x)$. For point $A''(6,2)$, $A'''=(-2,6)$.

Answer:

$A'''(-2,6)$