3.02 Reflections on the coordinate plane
Reflections
A reflection is what 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 be the same distance from the reflection line.
Have a quick play with this interactive. Here you can change the shape of the object and the position of the mirror line.
Practice questions
Question 1
Which image below represents the reflection for the following shape about the vertical mirror line?
A trapezoid is shaded within a grid of evenly spaced horizontal and vertical lines and is positioned to the right of a vertical mirror line. The longer vertical side of the trapezoid, measuring 3 units, is two square units to the right of the vertical mirror line, and the shorter vertical side, measuring 2 units, is one square unit to the right of the mirror line. The bottom horizontal side, connecting the upper endpoints of the two vertical sides, measures 1 unit. The top side of the trapezoid is slanted upward from left to right, connecting the two vertical sides, and its closest point is one square unit away from the vertical mirror line. A
A trapezoid is shaded within a grid of evenly spaced horizontal and vertical lines and is positioned to the right of a vertical mirror line. The shorter vertical side of the trapezoid, measuring 2 units, is two square units to the right of the vertical mirror line, and the longer vertical side, measuring 3 units, is one square unit to the right of the mirror line. The top horizontal side, connecting the upper endpoints of two vertical vertical sides, measures 1 unit. The bottom side of the trapezoid is slanted downward from right to left, connecting the two vertical sides, and its closest point is one square unit away from the vertical mirror line. B
A trapezoid is shaded within a grid of evenly spaced horizontal and vertical lines and is positioned to the right of a vertical mirror line. The longer vertical side of the trapezoid, measuring 3 units, is two square units to the right of the vertical mirror line, and the shorter vertical side, measuring 2 units, is one square unit to the right of the mirror line. The top horizontal side, connecting the upper endpoints of the two vertical sides, measures 1 unit. The bottom side of the trapezoid is slanted downward from left to right, connecting the two vertical sides, and its closest point is one square unit away from the vertical mirror line.
C
A trapezoid is shaded within a grid of evenly spaced horizontal and vertical lines and is positioned to the right of a vertical mirror line. The shorter vertical side of the trapezoid, measuring 1 unit, is one and a half square units to the right of the vertical mirror line, and the longer vertical side, measuring 1 and a half units, is one square unit to the right of the mirror line. The top horizontal side, connecting the upper endpoints of two vertical vertical sides, measures half a unit. The bottom side of the trapezoid is slanted downward from right to left, connecting the two vertical sides, and its closest point is one square unit away from the vertical mirror line. D
Question 2
Which three of the following diagrams show a reflection across the given line?
A
B
C
D
E
Reflections across an axis
As we saw above, a reflection occurs when we flip an object or shape across a line like a mirror. We can reflect points, lines, or polygons on a graph by flipping them across an axis or another line in the plane.
Reflecting over the $y$y-axis
Note how the point $\left(-2,1\right)$(−2,1) becomes $\left(2,1\right)$(2,1). The $y$y -value has stayed the same while the $x$x -value has 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 -value have stayed the same and the $x$x -value has changed signs.
Reflecting over the $x$x-axis
Note how the point $\left(4,3\right)$(4,3) becomes $\left(4,-3\right)$(4,−3). The $x$x -value has stayed the same and the $y$y -value has changed signs.
Similarly, the point $\left(0,5\right)$(0,5) becomes $\left(0,-5\right)$(0,−5). The $x$x -value have stayed the same and the $y$y -values has changed signs.
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.
If we reflect vertically across the $x$x -axis, the $x$x -values of the coordinates will remain the same and the $y$y -values will change sign.
Practice questions
Question 3
Plot the following.
Plot the point $A$A$\left(2,-2\right)$(2,−2).
Loading Graph...Now plot point $A'$A′, which is a reflection of point $A$A across the $x$x-axis.
Loading Graph...
Question 4
Plot the following.
Plot the line segment $AB$AB, where the endpoints are $A$A$\left(-6,-1\right)$(−6,−1) and $B$B$\left(10,8\right)$(10,8).
Loading Graph...Now plot the reflection of the line segment about the $y$y-axis.
Loading Graph...