Exercise 6 Page 557

We will first write an equation for the coordinates of the point $P(x,y)$ reflected across the $y\text{-}$axis.

Reflection Across the $y\text{-}$axis

Let's find the coordinates of the point $P(x,y)$ reflected across the $y\text{-}$axis. We will draw a diagram to find the location of the reflected point $P^{\prime }.$ To do so, we will move along the line through $P$ that is perpendicular to the line of reflection. Then, we stop when the distances of $P$ and $P^{\prime }$ to the line of reflection are the same. We see that the point $P^{\prime }(\text{-}x,y)$ is the image of $P(x,y).$ Therefore, we can write the equation for the coordinates of the point $P(x,y)$ reflected across the $y\text{-}$axis as follows.

Reflection Across the $x\text{-}$axis

We will now find the coordinates of a point $P(x,y)$ reflected across the $x\text{-}$axis. Let's draw a diagram to find the location of the reflected point. To do so, we will move along the line through $P$ that is perpendicular to the line of reflection and then stop when the distances of $P$ and $P^{\prime }$ to the line of reflection are the same. We see that the point $P^{\prime }(x,\text{-}y)$ is the image of $P(x,y).$ Therefore, we can write the equation for the coordinates of the point $P(x,y)$ reflected across the $x\text{-}$axis as follows.