9.02 Reflections
A reflection occurs when we flip an object or shape across a line, like in a mirror. The image of the reflection is congruent to the preimage, just flipped in position. 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 $xy$xy-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 $y$y-values of the coordinates remain the same and the $x$x values change sign.
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 $x$x-values have changed signs.In this diagram, the image is reflected across $y$y axis.
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 $x$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.
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.In this diagram, the image is reflected across $x$x axis.
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.
Exploration
Use this interactive below to further consolidate the ideas behind translations on the $xy$xy-plane.
Slide the right-most point to change the $y$y-intercept and/or the slope of the line of reflection. Slide points on the Object to change the shape of the preimage triangle.
What do you notice about the distance to the reflection line for corresponding points?
What happens when you reflect in the line $y=x$y=x? $y=-x$y=−x?
Practice questions
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.