High School Geometry Unlocked (2016)

Lesson 1.3. Reflection

Reflection is the term for a flipped version of an image. The way we think about reflection in real life, with mirrors, is very much the same way that reflection works in geometry. The reflected image is the same shape and size as the pre-image, but it’s backwards.

Examples of Reflections

As with translations, reflections are usually worked in the coordinate plane. Common exercises include drawing a reflection, identifying a reflection line, or writing an algebraic expression for a reflection.

Here is how you may see point reflection on the ACT.

In the standard (x, y) coordinate plane, Z (−2, −4) is reflected over the x-axis. What are the coordinates of the image of Z?

A.(4, −2)

B.(4, 2)

C.(2, −4)

D.(−2, 4)

E.(−4, 2)

One way to create a reflection is to use a folded piece of paper. Draw a figure on one side of the paper; then flip the folded paper over, and trace the image on the other side of the fold. The two figures will be congruent, but flipped. The fold in the paper serves as the line of reflection—the line across which the image is flipped.

In an image-editing
program, you can use
the “flip horizontal” or
“flip vertical” feature to
perform a reflection.

If you draw straight line segments between each pair of reflected points, as shown above, those segments will be parallel to each other. Additionally, the line of reflection forms a perpendicular bisector (a line that is perpendicular to another line segment, and intersects at its midpoint) with each of these segments. Each point in the pre-image is the same distance from the reflection line as its mirror image point.

In the image below, point A′ is the result of reflecting point A across the y-axis. That is, the reflection line is the y-axis itself.

When reflecting a point across an axis, move straight across—that is, perpendicular to—the axis. Make sure the image and pre-image point are the same distance from the axis.

Draw the standard (x, y) coordinate plane and plot point Z at an x-coordinate of −2 and a y-coordinate of −4. It will look like the image on the left, below.

A reflection of a point over an axis means to use that axis as a sort of mirror line of reflection. If the line of reflection is the x-axis, the image of Z will be above the x-axis. It will still be 2 units from the y-axis and 4 units from the x-axis, like the original point, as shown on the right, below.

The x-coordinate of the image point is still − 2, and the y-coordinate is now 4, so (D) is the credited response.

If triangle ABC is reflected across the y-axis, what are the coordinates of the reflected image’s vertices?

Earlier in this chapter, we learned that to translate a polygon, you should just perform the translation on each of its vertices. The same is true for reflection. So, for triangle ABC, we’re going to perform the reflection on each of its vertices. One way to do this is to use the grid and visually count to where the new vertices will be.

Starting with point A, count over to the y-axis, which is 2 units from point A. So, count over 2 more units, and place point A’ at coordinate (−2, 5).

With point B, count over to the y-axis, which is 4 units from point B. So, count over 4 more units, and place point B’ at coordinate (−4, 8).

With point C, count over to the y-axis, which is 7 units from point C. So, count over 7 more units, and place point C’ at coordinate (−7, 3).

Note that the image and pre-image are congruent, but flipped.

Once you’re comfortable with how reflection works, you can memorize this rule for reflection across the y-axis:

When reflecting across the y-axis, the effect is that the y-coordinate(s) will be the same as in the pre-image, but the x-coordinates have opposite signs.

To apply this rule to triangle ABC above, change the sign of each x-coordinate, while leaving the y-coordinates the same.

If triangle ABC is reflected across the x-axis, what are the coordinates of the reflected image’s vertices?

This time, the line of reflection is the x-axis. Let’s count off on the grid. Starting with point A, count down to the x-axis, which is 5 units from point A. So, count down 5 more units, and place point A′ at coordinate (2, −5).

With point B, count down to the x-axis, which is 8 units from point B. So, count down 8 more units, and place point B′ at coordinate (4, −8).

With point C, count down to the x-axis, which is 3 units from point C. So, count down 3 more units, and place point C′ at coordinate (7, −3).

You can also memorize this rule for reflection across the x-axis:

When reflecting across the x-axis, the effect is that the x-coordinate(s) are the same as in the pre-image, but the y-coordinates have opposite signs.

To apply this rule to triangle ABC above, change the sign of each y-coordinate, while leaving the x-coordinates the same.

Reflection Across the Axis

When reflecting across the y-axis, the effect is that the y-coordinate(s) will be the same as in the pre-image, but the x-coordinates have opposite signs.

When reflecting across the x-axis, the effect is that the x-coordinate(s) are the same as in the pre-image, but the y-coordinates have opposite signs.

If triangle ABC is reflected across the line y = x, what are the coordinates of the reflected image’s vertices?

In this example, the line of reflection is the line y = x. That line is the set of points for which the x- and y-coordinates are equal—for example, (1, 1), (5, 5), (−5, −5), and so on.

To reflect a coordinate across the line y = x, use this simple rule:

When reflecting a point across the line y = x, the effect is that the x-coordinate and y-coordinate are switched.

To apply this rule to triangle ABC above, simply switch the x- and y-coordinates of each point.

REFLECTING FUNCTIONS

The graph above shows the function f(x) = x + 3. What is the result of reflecting this function about the y-axis?

To reflect a function that forms a line, you can choose a few coordinate pairs from the line, and apply the reflection rules to those points. For example, what is f(2) on this function? f(2) = 2 + 3, so f(2) = 5. That gives us the coordinate pair (2, 5). Great!

To reflect the coordinate (2, 5) across the y-axis, count over, or just change the sign of the x-coordinate. The reflected point is (−2, 5).

We’ll need at least one more coordinate to make a line. How about f(−2)? f(−2) = −2 + 3 = 1. That gives us the coordinate pair (−2, 1). To reflect that coordinate across the y-axis, count over, or just change the sign of the x-coordinate. The reflected point is (2, 1).

Try a few more points for practice! Then, draw a line between your reflected points.

The reflected line looks like this:

The graph above shows the function f(x) = x + 3. What is the result of reflecting this function about the x-axis?

As in the previous example, we’ll just choose a few points from the graph, and apply the reflection rules to those points. Let’s try f(3), which is f(3) = 3 + 3, so f(3) = 6. That gives us the coordinate pair (3, 6).

To reflect the coordinate (3, 6) across the x-axis, count downward, or just change the sign of the y-coordinate. The reflected point is (3, −6).

We’ll need at least one more coordinate to make a line. Try f(−4). f(−4) = −4 + 3 = −1. That gives us the coordinate pair (−4, −1). To reflect that coordinate across the x-axis, count upward, or just change the sign of the y-coordinate. The reflected point is (4, 1).

Try a few more points for practice. Then, draw a line between your reflected points.

The reflected line looks like this:

To recap, if you’re reflecting across the x-axis, you’re changing the sign of each of the y-coordinates of the figure. If you’re reflecting across the y-axis, you’re changing the sign of each of the x-coordinates of your figure.

The rules for reflecting across the x-axis and y-axis can be expressed as follows:

Reflection Rules for Functions

Compared to the graph of y = f(x),

y = −f(x) reflects f(x) across x-axis

y = f(−x) reflects f(x) across y-axis

To see how function
reflection is tested on
the ACT, access your
Student Tools online.