Math 6 · 6.NS.C.6B

Points that differ only in sign: reflections across the axes

Two mirrors standing along the axes turn one point into four. Knowing the first point’s pair of numbers, what are the pairs of its mirror images?

You need

Opposites and zero pairsMath 6 Plotting in all four quadrantsMath 6

Where you are 4 in The coordinate plane

Geometry11

Coordinate geometry9

The coordinate plane4

Ordered pairs and the first quadrant Plotting in all four quadrants

Points that differ only in sign: reflections across the axes

Two number patterns side by side and graphing their pairs

Why this exists

A kaleidoscope works with mirrors. Stand two flat mirrors at right angles, drop a bead between them, and you see four beads: the bead, its image in each mirror, and in the corner an image of an image. A pattern designer working on a grid wants the same trick on paper: draw one shape, and have its mirror copies appear in the other three corners.

Plotting in all four quadrants gives every point of the plane a pair of numbers. This lesson shows that copies made by mirrors along the axes need no measuring at all. A point and its mirror image have the same two numbers, except that one or both have changed sign. Changing the sign of the second number flips a point across the horizontal axis, changing the first flips it across the vertical axis, and changing both turns it half a turn round the origin.

Definition

To change the sign of a number is to replace it with its opposite, the number the same distance from 0 on the other side of 0. Changing the sign of 3 gives -3, changing the sign of -3 gives 3, and changing the sign of 0 leaves 0, because 0 is its own opposite.

The mirror image of a point across a line is the point straight across the line from it, on the other side and the same distance away. If the paper were folded along the line, the point and its mirror image would land on top of each other. A point on the line is its own mirror image. Moving a point to its mirror image is reflecting it across the line, which is called the line of reflection or mirror line. The new point is the image, often named with a prime mark: the image of P is P', read “P prime”.

Two points differ only in sign when their pairs have the same sizes of numbers in the same places, with one or both signs changed. There are three ways this can happen, and each has a name. The arrow \to is read “goes to”.

sign changed	example	name
the second number’s	(2, 4) \to (2, -4)	reflection across the horizontal axis (x-axis)
the first number’s	(2, 4) \to (-2, 4)	reflection across the vertical axis (y-axis)
both	(2, 4) \to (-2, -4)	reflection through the origin, a half turn

Intuition

Figure 1P(2, 4) and its three images. Q(2, -4) is across the horizontal axis, R(-2, 4) across the vertical axis, and S(-2, -4) is half a turn round the origin.

Fold the grid along the horizontal axis. The top half lands on the bottom half, and P, which is 4 units above the axis, lands 4 units below it, on the same vertical grid line. Its first number is still 2, and its second has gone from 4 to -4: that is Q(2, -4).

Fold along the vertical axis instead. P is 2 units right of the axis and lands 2 units left of it, on the same horizontal grid line: R(-2, 4).

Fold both ways and P ends up in the far corner, at S(-2, -4). The dashed line from P to S runs through the origin, and the origin is exactly halfway along it. Pin the paper at the origin and turn it half a turn: P lands on S.

A way to remember which number changes: a horizontal mirror flips up and down, and up and down is what the second number measures. A vertical mirror flips left and right, which is the first number.

Results

Across the horizontal axis. Keeping the first number and changing the sign of the second gives the mirror image across the horizontal axis.

Across the vertical axis. Changing the sign of the first number and keeping the second gives the mirror image across the vertical axis.

Through the origin. Changing the sign of both numbers gives the point half a turn round the origin. It is the same as reflecting across one axis and then across the other, in either order.

Points that stay put. A point on the horizontal axis is its own image across the horizontal axis, a point on the vertical axis is its own image across the vertical axis, and the origin is its own image under all three.

Undoing. Doing the same reflection twice returns the starting point, because the opposite of the opposite of a number is the number itself.

Quadrants. A point in a quadrant always lands in a different quadrant:

start	across x-axis	across y-axis	through origin
I	IV	II	III
II	III	I	IV
III	II	IV	I
IV	I	III	II

Why it is true

Across the horizontal axis. The mirror line is horizontal, so the way from a point to its image runs straight up or down, at right angles to the mirror. The point and its image are on the same vertical grid line, and every point on one vertical grid line has the same first number. So the first number does not change.

That vertical grid line works as a number line of its own, with 0 where it crosses the horizontal axis, and the second number says where on it a point sits. The image is the same distance from the axis as the point, on the other side. On a number line, the number the same distance from 0 on the other side is the opposite (Placing and comparing integers on a number line). So the image’s second number is the opposite of the point’s: its sign has changed. If the second number is 0, the point is on the mirror, and the opposite of 0 is 0, so it stays where it is.

Across the vertical axis. The same argument with the roles swapped: the image is on the same horizontal grid line, so the second number stays, and along that line the first number changes to its opposite.

Through the origin. Reflecting across the horizontal axis changes the second sign, and reflecting that image across the vertical axis changes the first. After both, both signs have changed; doing them in the other order changes the same two signs, so it ends at the same point. To see the half turn, compare trips from the origin. (2, 4) is 2 right and 4 up, and (-2, -4) is 2 left and 4 down: the same trip with every direction reversed. A half turn round the origin turns right into left and up into down, so it carries the first trip onto the second.

Quadrants. Use the sign table from Plotting in all four quadrants. Quadrant I is positive, positive. Changing the second sign gives positive, negative, which is Quadrant IV; changing the first gives negative, positive, Quadrant II; changing both gives negative, negative, Quadrant III. The other rows work the same way. Every reflection changes at least one sign, and a point in a quadrant has no zero coordinate to stay put, so it always moves to a different quadrant.

Worked examples

Example 1

The bead in the kaleidoscope

Zia’s kaleidoscope has two mirrors at right angles, standing along the axes of a grid. A bead sits at (4, -2). Find the three images she sees, say which mirror makes each, and name the quadrant of the bead and of each image.

Solution

The bead has a positive first number and a negative second number, so it is in Quadrant IV.

The mirror along the horizontal axis keeps the first number and changes the sign of the second: (4, 2), in Quadrant I.
The mirror along the vertical axis changes the sign of the first number: (-4, -2), in Quadrant III.
The image in the corner is the first image seen in the second mirror, so both signs change: (-4, 2), in Quadrant II, half a turn round the origin from the bead.

Each of the four beads is 4 units from the vertical axis and 2 units from the horizontal one, which is why the pattern looks the same in all four corners.

Example 2

Naming the reflection

How is each pair of points related? (a) (-6, 1) and (6, 1) (b) (2, -9) and (2, 9) (c) (-1, -8) and (1, 8) (d) (5, 3) and (3, 5) (e) (0, -6) and (0, 6)

Solution

Compare the first numbers, then the second numbers.

(a) The first numbers are opposites and the second numbers match, so the points are reflections of each other across the vertical axis.
(b) The first numbers match and the second numbers are opposites: across the horizontal axis.
(c) Both numbers have changed sign: through the origin.
(d) No sign changed; the two numbers swapped places. These points do not differ only in sign, so neither is the other’s image across an axis or through the origin.
(e) The second numbers are opposites. The first numbers are both 0, and 0 is its own opposite, so the first number has both stayed the same and changed sign. The points are images of each other across the horizontal axis, and also through the origin. Both lie on the vertical axis, where those two moves agree.

Example 3

Lights on a pinball table

Aria designs the lights for a pinball table drawn on a grid, with the origin at the center and the vertical axis running down the middle. The lights must be symmetric left to right: every light on the left half needs a partner on the right that is its mirror image across the vertical axis. The left-half lights are A(-3, 6), B(-5, -2) and C(-1.5, 2.5), and one more light, D(0, -5), sits on the middle line. Where do the partners go?

Solution

A partner across the vertical axis keeps the second number and changes the sign of the first.

A gives A'(3, 6).
B gives B'(5, -2).
C gives C'(1.5, 2.5). The halves carry over, so C' sits in the middle of a grid square on the right just as C does on the left.
D has first number 0, which stays 0. D is on the mirror line and is its own partner, so no extra light is needed.

Figure 2The pinball lights of Example 3. Each light and its partner sit on one horizontal grid line, the same distance either side of the vertical axis.

Each light and its partner sit on one horizontal grid line, as the dashed lines show.

Example 4

Completing a kite

Cleo is drawing a tile for a kaleidoscope pattern: a four-sided shape ABCD that is symmetric about the horizontal axis, meaning that folding along the axis lays the shape exactly on itself. Three corners are A(-3, 0), B(1, 4) and C(6, 0). Find D.

Solution

Folding along the axis must carry every corner onto a corner. A and C have second number 0, so they are on the mirror and are their own images. B is above the axis, so its image is a different corner, and the only one left is D. Changing the sign of the second number of B gives D(1, -4).

Figure 3Cleo’s tile in Example 4. Folding along the horizontal axis lays B on D, while A and C stay where they are.

Check: B and D are on the same vertical grid line, through 1, one 4 units above the axis and the other 4 units below it. The tile has the shape of a kite.

Example 5

Which quadrant, without the point

A point is somewhere in Quadrant II, but its numbers are not known. In which quadrant is its image across the horizontal axis, across the vertical axis, and through the origin?

Solution

A point in Quadrant II has a negative first number and a positive second number. The exact numbers do not matter, only their signs.

Across the horizontal axis the second number changes from positive to negative. Negative, negative is Quadrant III.
Across the vertical axis the first number changes from negative to positive. Positive, positive is Quadrant I.
Through the origin both change, giving positive, negative: Quadrant IV.

The three images fill the other three quadrants, one each.

Example 6

Working backwards

Jem reflects a point across the vertical axis, then reflects the new point across the horizontal axis, and ends at (-3.5, 6). Where did the point start, and what single move takes it straight from the start to the end?

Solution

Undo the moves in reverse order. A reflection undoes itself, because changing a sign twice gives the number back. The last move was across the horizontal axis, so just before it the second number had the other sign: the point was at (-3.5, -6). The first move was across the vertical axis, so before that the first number had the other sign: the point started at (3.5, -6).

Compare the start (3.5, -6) with the end (-3.5, 6): both signs have changed. So the single move is a reflection through the origin, a half turn. Two reflections across the two axes make one half turn.

Check, going forwards: across the vertical axis (3.5, -6) \to (-3.5, -6), then across the horizontal axis (-3.5, -6) \to (-3.5, 6).

Boundaries

A sign change reflects only across the axes. Changing a sign moves a number to the same distance on the other side of 0, so the mirror has to pass through 0. A reflection across another line, such as the vertical grid line through 3, does not just change a sign.
Swapping the two numbers, as with (5, 3) and (3, 5), is not a sign change. It is a reflection across a slanted line through the origin, which belongs to later work.
A reflection through the origin is a half turn, not a reflection across any one line. A mirror turns a shape back to front, like writing seen in a mirror; a half turn spins it round without flipping it.
Each axis must keep the same steps on both sides of 0, as every number line does, or the image will not be the same distance away. The two axes may still have different scales.
The quadrant table holds only for points in a quadrant. A point on an axis is in no quadrant, and it is its own image across that axis.

Where this goes next

Horizontal and vertical distance measures how far apart two points on one grid line are, such as a point and its mirror image across an axis. Later, reflections are treated as moves of whole figures, with mirror lines that need not be the axes, including the slanted line that swaps the two numbers. Example 6, where two reflections across crossing mirrors make a half turn, is the first case of Two reflections in a row.

Exercises

Reflecting a point

Reflect (6, 2) across the horizontal axis.
Reflect (-3, -7) across the vertical axis.
Reflect (-8, 5) through the origin.
Reflect (1.5, -4) across the vertical axis.
Reflect (0, 7) across the horizontal axis, across the vertical axis, and through the origin.
Reflect \displaystyle \left(-\frac{3}{4}, 2\right) across the horizontal axis.

Naming the reflection

Say how each pair of points is related, or say that neither is the other’s image across an axis or through the origin.

(7, -1) and (-7, -1)
(-4, -9) and (4, 9)
(2.5, 8) and (2.5, -8)
(8, -2) and (-2, 8)
(-5, 0) and (5, 0)

Where the image lands

A point in Quadrant IV is reflected across the vertical axis. In which quadrant is its image?
A point in Quadrant III is reflected through the origin. In which quadrant is its image?
A point in Quadrant I is reflected and lands in Quadrant IV. Which reflection was it?
Without plotting, name the quadrant of the image of (-12, 30) across the horizontal axis.

Pinball lights and kaleidoscope tiles

Beau’s pinball table is symmetric about the vertical axis. The left-half lights are at (-1, 7), (-6, -4) and (-2.5, 0.5). Where are their partners?

Figure 4Half of the tile in Exercise 17. The other half is its mirror image across the vertical axis.

The figure shows half of a tile that is symmetric about the vertical axis, with corners A(0, 5), B(-4, 3), C(-3, -4) and D(0, -2). Find the corners of the other half.
Aria’s kaleidoscope tile is a rectangle symmetric about both axes, and one corner is at (-7, 3). Find the other three corners.
A triangle is symmetric about the horizontal axis, and two of its corners are (4, 6) and (4, -6). Where can the third corner be?

Reasoning

Explain why reflecting a point across the horizontal axis never changes its first number.
Jem reflects a point across the horizontal axis, then reflects the image across the horizontal axis again. Where does it end up? Explain.
Cleo reflects (-9, 4) across the vertical axis and then across the horizontal axis. Zia reflects the same point across the horizontal axis and then across the vertical axis. Find where each ends, and explain why they agree.
Explain why a point on the vertical axis has the same image across the horizontal axis as through the origin.
A point’s image through the origin is the origin itself. What was the point? Explain.

Answers

(6, -2). The first number stays and the second changes sign.
(3, -7). The first number changes sign and the second stays.
(8, -5). Both numbers change sign.
(-1.5, -4). Only the first number changes sign.
Across the horizontal axis, (0, -7). Across the vertical axis, (0, 7): the point is on that axis, so it is its own image. Through the origin, (0, -7), the same as across the horizontal axis, because the sign of 0 changes nothing.
\displaystyle \left(-\frac{3}{4}, -2\right). The first number stays as it is; the second changes sign.
Across the vertical axis: the first numbers are opposites and the second numbers match.
Through the origin: both numbers have changed sign.
Across the horizontal axis: the first numbers match and the second numbers are opposites.
Not a reflection across an axis or through the origin. No sign changed; the numbers swapped places.
Across the vertical axis, and also through the origin. The first numbers are opposites, and the second numbers are both 0, its own opposite. Both points are on the horizontal axis.
Quadrant III. Positive, negative becomes negative, negative.
Quadrant I. Negative, negative becomes positive, positive.
Across the horizontal axis. Quadrant I is positive, positive and Quadrant IV is positive, negative, so only the second sign changed.
Quadrant III. The image is (-12, -30), with both numbers negative.
(1, 7), (6, -4) and (2.5, 0.5). The first number changes sign; the second stays.
(4, 3) and (3, -4), the images of B and C. A and D are on the vertical axis, so they are their own images.
(7, 3) across the vertical axis, (-7, -3) across the horizontal axis, and (7, -3) through the origin.
Anywhere on the horizontal axis except (4, 0). Folding along the axis carries the triangle onto itself, so the third corner’s image is a corner. It cannot be (4, 6), since then the third corner would be the image of (4, 6), which is (4, -6), already taken; the same goes for (4, -6). So the third corner is its own image, which puts it on the horizontal axis. At (4, 0) it would sit on the line through the other two, and there would be no triangle.
The mirror is horizontal, so a point and its image are straight above and below each other, on one vertical grid line. Every point on a vertical grid line has the same first number.
Back where it started. The first reflection changes the sign of the second number and the second reflection changes it back, since the opposite of the opposite of a number is the number itself. The first number never changes.
Both end at (9, -4). Cleo goes to (9, 4) and then to (9, -4); Zia goes to (-9, -4) and then to (9, -4). Each reflection changes one sign, and either way each sign is changed once.
A point on the vertical axis has first number 0. Across the horizontal axis only the second sign changes; through the origin the first sign changes too, but changing the sign of 0 leaves 0. So both moves give the same point.
The origin, (0, 0). Through the origin each number is replaced by its opposite, and the only number whose opposite is 0 is 0: any other number is some distance from 0, and so is its opposite.