3.05 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 shaded four-sided polygon is situated on a square grid. The whole figure is drawn on three squares on the sixth column of the grid. Two squares on the sixth column are shaded. On the square above these two, a right triangle is shaded with the hypotenuse being the line when the bottom left and top right vertex of the square is connected. The shaded right triangle at the top of the figure is located at the second row from the top and the shaded square at the bottom of the figure is located at the third row from the bottom. A vertical mirror line is drawn in between the line between the fourth and fifth column. A
A shaded four-sided polygon is situated on a square grid. The whole figure is drawn on three squares on the sixth column of the grid. Two squares on the sixth column are shaded. On the square below these two, a right triangle is shaded with the hypotenuse being the line when the top right and bottom left vertex of the square is connected. The shaded square at the top of the figure is located at the second row from the top and the shaded right triangle of the figure is located at the third row from the bottom. A vertical mirror line is drawn in between the line between the fourth and fifth column. B
A shaded four-sided polygon is situated on a square grid. The whole figure is drawn on three squares on the sixth column of the grid. Two squares on the sixth column are shaded. On the square below these two, a right triangle is shaded with the hypotenuse being the line when the top left and bottom right vertex of the square is connected. The shaded square at the top of the figure is located at the second row from the top and the shaded right triangle of the figure is located at the third row from the bottom. A vertical mirror line is drawn in between the line between the fourth and fifth column. C
A shaded four-sided polygon is situated on a square grid. The whole figure is drawn on two squares on the sixth column of the grid. The left half of one squares on the sixth column is shaded. On the square below this, a right triangle is shaded with the hypotenuse being the line when the point from the middle of the left side of the square is connected to the bottom right vertex of the shaded part of the square above. The shaded square at the top of the figure is located at the third row from the top and the shaded right triangle of the figure is located at the third row from the bottom. A vertical mirror line is drawn in between the line between the fourth and fifth column. 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...