Question

1. use the coordinate plane to complete parts (a) and (b).

name

1. point

eureka math

a. plot point a at (4, -3). reflect point a across the y - axis and label the reflection b. reflect
point b across the x - axis and label the reflection c.
b. what would happen if you reflected point a across the x - axis first to find point b and then
reflected point b across the y - axis to find point c?

Explanation:

Step1: Reflect point \(A(4, - 3)\) across the \(y\) - axis

When reflecting a point \((x,y)\) across the \(y\) - axis, the rule is \((x,y)\to(-x,y)\).
For point \(A(4,-3)\), substituting \(x = 4\) and \(y=-3\) into the rule, we get \(B(-4,-3)\).

Step2: Reflect point \(B(-4,-3)\) across the \(x\) - axis

When reflecting a point \((x,y)\) across the \(x\) - axis, the rule is \((x,y)\to(x,-y)\).
For point \(B(-4,-3)\), substituting \(x=-4\) and \(y = - 3\) into the rule, we get \(C(-4,3)\).

Step3: Analyze the double - reflection

If we first reflect point \(A(4,-3)\) across the \(x\) - axis (using the rule \((x,y)\to(x,-y)\)), we get \(A'(4,3)\). Then reflecting \(A'(4,3)\) across the \(y\) - axis (using the rule \((x,y)\to(-x,y)\)), we get \((-4,3)\) which is the same as point \(C\) from the previous part.

Answer:

a. Point \(B(-4,-3)\) (reflection of \(A(4,-3)\) across \(y\) - axis) and point \(C(-4,3)\) (reflection of \(B(-4,-3)\) across \(x\) - axis).
b. If we first reflect point \(A(4,-3)\) across the \(x\) - axis to get \((4,3)\) and then reflect across the \(y\) - axis, we also get the point \((-4,3)\) (same as point \(C\) obtained in part (a)). The order of reflecting across the \(x\) - axis and \(y\) - axis does not matter for the final result of a double - reflection (across two perpendicular axes) of a point.