Question

use your knowledge of reflections to complete parts (a) and (b) below
a. conjecture what the image of a point with coordinates (x,y) will be under the transformation
eflect in the line y = -x\. conjecture what the image of a point with coordinates (x,y) will be under the transformation
eflect in the x - axis.\
b. suppose a point p with coordinates (x,y) is reflected in the line y = -x and then its image is reflected in the x - axis to obtain p. what are the coordinates of p in terms of x and y?

a. the coordinates of the image point from the transformation
eflect in the line y = -x\ are
( (y,x) )
( (x,-y) )
( (-y,-x) )
( (-x,y) )

Explanation:

Step1: Recall reflection rules

When reflecting a point \((x,y)\) over the line \(y = x\), the \(x\) and \(y\) coordinates are swapped.

Step2: Apply the rule

For a point \((x,y)\) reflected over \(y=x\), the image is \((y,x)\).

Step3: Reflect over \(x -\)axis rule

When reflecting a point \((a,b)\) over the \(x -\)axis, the \(y -\)coordinate changes sign. So if we first reflect \((x,y)\) over \(y = x\) to get \((y,x)\) and then reflect \((y,x)\) over the \(x -\)axis, we change the sign of the second coordinate.

Step4: Calculate the final image

The image of \((y,x)\) after reflection over the \(x -\)axis is \((y,-x)\)

Answer:

a. \((y,x)\)
b. \((y, -x)\)