Question

defining reflections a reflection is a rigid motion that uses a line like a mirror to reflect a figure. this line is called the line of reflection. example: triangle abc is reflected across the y - axis to form triangle a’b’c’. the y - axis is the line of reflection. what do you notice about the coordinates of the image and pre - image for each reflection? reflect across the x - axis pre - image coordinates a (3,5) b (4,2) c (1,1) image coordinates x - axis reflection rule: (x,y) → ____

Explanation:

Step1: Recall x - axis reflection rule

The rule for reflecting a point \((x,y)\) across the \(x\) - axis is that the \(x\) - coordinate remains the same and the \(y\) - coordinate changes its sign. So the transformation rule is \((x,y)\to(x, - y)\).

Step2: Find image of point A

For point \(A(3,5)\), applying the \(x\) - axis reflection rule \((x,y)\to(x, - y)\), we substitute \(x = 3\) and \(y = 5\). So the image of \(A\) is \((3,-5)\).

Step3: Find image of point B

For point \(B(4,2)\), applying the rule \((x,y)\to(x, - y)\), we substitute \(x = 4\) and \(y = 2\). So the image of \(B\) is \((4,-2)\).

Step4: Find image of point C

For point \(C(1,1)\), applying the rule \((x,y)\to(x, - y)\), we substitute \(x = 1\) and \(y = 1\). So the image of \(C\) is \((1,-1)\).

Answer:

The \(x\) - axis reflection rule is \((x,y)\to\boldsymbol{(x, - y)}\). The image coordinates are: \(A(3, - 5)\), \(B(4, - 2)\), \(C(1, - 1)\). When reflecting across the \(x\) - axis, the \(x\) - coordinate of each point remains unchanged and the \(y\) - coordinate is negated.