Reflections on the Cartesian Plane
Reflections
As we saw in the previous lesson on Flips, A reflection occurs when we flip an object or shape across a line. Like a mirror, the object is exactly the same size, just flipped in position. So what was on the left may now appear on the right. Every point on the object or shape has a corresponding point on the image, and they will both have the same distance from the reflection line. We can reflect points, lines, polygons on the Cartesian plane by flipping them across an axis line or another line in the plane.
If we reflect horizontally across the $y$y axis, then the y values of the coordinates remain the same and the $x$x values change sign.
In this diagram, the image is reflected across $y$y axis.
Note how the point $\left(-2,1\right)$(−2,1) becomes $\left(2,1\right)$(2,1). The $y$y values have not changed and the x values have changed signs.
Similarly the point $\left(-6,3\right)$(−6,3) becomes $\left(6,3\right)$(6,3). The $y$y values have not changed and the x values have changed signs.
If we reflect vertically across the $x$x axis, then the $x$x values of the coordinates remain the same and the $y$y values change sign.
In this diagram, the image is reflected across $x$x axis.
Note how the point $\left(4,3\right)$(4,3) becomes $\left(4,-3\right)$(4,−3). The $x$x values have not changed and the $y$y values have changed signs.
Similarly the point $\left(0,5\right)$(0,5) becomes $\left(0,-5\right)$(0,−5). The $x$x values have not changed and the $y$y values have changed signs.
Have a quick play with this interactive to further consolidate the ideas behind translations on the Cartesian plane.
Let's have a look at these worked examples.
Question 1
Consider the point $A\left(7,3\right)$A(7,3).
Plot point $A$A on the number plane.
Loading Graph...Now plot point $A'$A′, a reflection of point $A$A across the $x$x-axis.
Loading Graph...
Question 2
Consider the point $A\left(-7,-3\right)$A(−7,−3).
Plot point $A$A on the number plane.
Loading Graph...Now plot point $A'$A′, a reflection of point $A$A across the $y$y-axis.
Loading Graph...
Question 3
Consider the line segment $AB$AB, where the endpoints are $A$A$\left(-4,-2\right)$(−4,−2) and $B$B$\left(6,7\right)$(6,7).
Plot the line segment $AB$AB on the number plane.
Loading Graph...Now plot the reflection of the line segment $AB$AB across the $x$x-axis.
Loading Graph...
Question 4
Consider the graph of the triangle and the line $x=-3$x=−3.
The three points of the triangle, $A$A $\left(-1,7\right)$(−1,7), $B$B $\left(3,-2\right)$(3,−2) and $C$C $\left(0,-6\right)$(0,−6) are reflected across the line $x=-3$x=−3 to produce the points $A'$A′, $B'$B′ and $C'$C′.
What are the coordinates of the new points?
$A'$A′$\left(-4,7\right)$(−4,7),$B'$B′$\left(0,-2\right)$(0,−2), $C'$C′$\left(-3,-6\right)$(−3,−6)
A$A'$A′$\left(-5,7\right)$(−5,7), $B'$B′$\left(-9,-2\right)$(−9,−2), $C'$C′$\left(-6,-6\right)$(−6,−6)
B$A'$A′$\left(-5,-4\right)$(−5,−4), $B'$B′$\left(-9,0\right)$(−9,0), $C'$C′$\left(-6,-3\right)$(−6,−3)
CPlot the new triangle formed by reflecting the given triangle across the line $x=-3$x=−3.
Loading Graph...A yellow-shaded triangle has its vertices highlighted with blue dots at the coordinates $\left(-5,7\right)$(−5,7), $\left(-9,-2\right)$(−9,−2), and $\left(-6,-6\right)$(−6,−6) on the Cartesian coordinate plane. The axes, labeled "x" and "y," extend from -10 to 10, with major tick marks at intervals of 5 and minor tick marks at intervals of 1. The major tick marks are labeled with numbers to indicate their value on both axes. A vertical line at $x=-3$x=−3 is also plotted, serving as a mirror line for the triangle.