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Lesson 14 ~40 min Linear Relationships C · Path +90 XP

Reflections in the Axes

A reflection sends every point to the opposite side of a mirror line, the same distance away. When the mirror is an axis, that description collapses into a single sign change, and which coordinate changes is the only thing to keep straight.

Today's hook: Reflect a point in the x-axis and it flips vertically. Reflect it in the y-axis and it flips horizontally. Both rules change exactly one sign, and swapping them is the single commonest error in this topic.
0/5QUESTS
Think First
warm-up

Plot $(3, 4)$. Now fold the page along the x-axis. Where does the point land? Fold along the y-axis instead. Where now? Which coordinate changed in each case?

Record your answer in your workbook.
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The Big Idea
+5 XP to read

Reflecting in the x-axis flips a point vertically, so the y-coordinate changes sign. Reflecting in the y-axis flips it horizontally, so the x-coordinate changes sign. The mirror is the axis whose coordinate stays.

$$\text{x-axis}: (x, y) \to (x, -y) \qquad \text{y-axis}: (x, y) \to (-x, y)$$

The rule to remember is which coordinate survives. Reflecting in the x-axis, points slide up and down but not sideways, so $x$ is untouched. Reflecting in the y-axis, points slide sideways only, so $y$ is untouched.

A A′ mirror equal distances, opposite sides
$(x, -y) \quad \text{or} \quad (-x, y)$
Name the mirror, then the survivor
Mirror is the x-axis, so $x$ survives and $y$ flips. Saying it that way makes the rule hard to get backwards.
Points on the mirror stay
A point already on the axis is its own image, since its flipping coordinate is zero.
Orientation reverses
A triangle labelled clockwise comes back labelled anticlockwise. That is what tells a reflection from a rotation.
2
What You'll Master
objectives

Know

  • that reflection in the x-axis sends $P(x, y)$ to $P'(x, -y)$
  • that reflection in the y-axis sends $P(x, y)$ to $P'(-x, y)$
  • that reflecting in the x-axis and then the y-axis gives $(-x, -y)$, which is a half turn about the origin

Understand

  • why the coordinate belonging to the mirror axis is the one left unchanged
  • why a point lying on the mirror line is an invariant point
  • why a reflection reverses orientation while a translation does not

Can Do

  • find the image of a point or a shape reflected in either axis
  • identify which reflection maps a given point onto a given image
  • describe the single transformation equivalent to two successive reflections in the axes
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Words You Need
vocabulary
ReflectionA transformation that maps each point to the opposite side of a fixed line, the same perpendicular distance away.
Mirror lineThe fixed line of a reflection. Also called the axis of reflection.
Invariant pointA point left unmoved by a transformation. Every point on the mirror line is invariant.
OrientationThe sense in which the vertices of a shape are labelled, clockwise or anticlockwise.
Composite transformationThe single transformation equivalent to doing two or more transformations in turn.
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What a Reflection Does
+5 XP to read

A reflection is defined by a mirror line. Each point is sent to the point on the opposite side of the line, at the same perpendicular distance from it.

Two consequences follow immediately. Lengths and angles are unchanged, so the image is congruent. But orientation reverses: a triangle whose vertices read $A$, $B$, $C$ clockwise has an image whose vertices read $A'$, $B'$, $C'$ anticlockwise.

That reversal is the practical signature of a reflection. Translations and rotations preserve orientation; reflections do not, which is why a reflected letter looks wrong in a way that a rotated one does not.

Points on the mirror line do not move at all, since their distance from the line is zero. A reflection therefore always has invariant points, unlike a translation.

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Reflection in the x-axis
+5 XP to read

The x-axis is horizontal, so reflecting in it moves points straight up or straight down. Nothing moves sideways.

A point at height $y$ above the axis goes to height $y$ below it, which is the point at $-y$. The x-coordinate is untouched. So

$P(x, y) \to P'(x, -y)$

The point $(3, 4)$ maps to $(3, -4)$, and the point $(-2, -5)$ maps to $(-2, 5)$, since the negative of $-5$ is $5$.

The invariant points are those with $y = 0$, which is exactly the x-axis itself, as expected.

6
Reflection in the y-axis
+5 XP to read

The y-axis is vertical, so reflecting in it moves points sideways only. Nothing moves up or down.

A point at horizontal position $x$ to the right goes to the same distance to the left, which is $-x$. The y-coordinate is untouched. So

$P(x, y) \to P'(-x, y)$

The point $(3, 4)$ maps to $(-3, 4)$, and the point $(-2, -5)$ maps to $(2, -5)$.

Comparing the two rules, the coordinate that survives is always the one named by the mirror. Mirror in the x-axis, $x$ survives; mirror in the y-axis, $y$ survives. Stating it in that form is the reliable way to avoid swapping them.

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Reflecting a Whole Shape
+5 XP to read

Apply the rule to each vertex and join the images in the same order.

Reflecting triangle $A(1, 2)$, $B(5, 3)$, $C(2, 6)$ in the y-axis gives $A'(-1, 2)$, $B'(-5, 3)$, $C'(-2, 6)$.

Any vertex sitting on the mirror line stays where it is. If a vertex were at $(0, 4)$, its image under reflection in the y-axis would be $(0, 4)$ again, and the original and image shapes would touch at that point.

Check the orientation as a final test. If $ABC$ ran clockwise and your $A'B'C'$ also runs clockwise, something has gone wrong, because a reflection must reverse it.

8
Doing Two Reflections
+5 XP to read

Reflect in the x-axis and then in the y-axis:

$(x, y) \to (x, -y) \to (-x, -y)$

The result is $(-x, -y)$, which negates both coordinates. That is a rotation of $180°$ about the origin, as the next lesson confirms.

Doing them in the other order gives $(x, y) \to (-x, y) \to (-x, -y)$, the same result. So for these two particular reflections the order does not matter, because the two mirrors are perpendicular.

Two reflections in the same line return every point to where it started, since each point crosses the mirror and crosses back. Reflecting twice in one line is doing nothing at all, which is worth knowing when a question chains transformations together.

Note also that orientation reverses twice and so ends up unchanged, which fits: the composite is a rotation, and rotations preserve orientation.

Watch Me Solve It · 3 examples

Watch Me Solve It · One point, two mirrors
+15 XP per step
Q1
PROBLEM
Find the image of $A(3, -5)$ after reflection in the x-axis, and after reflection in the y-axis.
  1. 1
    Reflect in the x-axis
    $(x, y) \to (x, -y)$
    $A(3, -5) \to (3, 5)$
    The mirror is the x-axis, so $x$ survives and the y-coordinate changes sign. The negative of $-5$ is $5$.
  2. 2
    Reflect in the y-axis
    $(x, y) \to (-x, y)$
    $A(3, -5) \to (-3, -5)$
    The mirror is the y-axis, so $y$ survives and the x-coordinate changes sign.
  3. 3
    Check against a sketch
    The original is in the fourth quadrant. The first image is in the first quadrant, directly above it; the second is in the third quadrant, directly left of it. Both are consistent with the direction each mirror flips in.
Answer$(3, 5)$ in the x-axis; $(-3, -5)$ in the y-axis
Watch Me Solve It · Reflecting a triangle with a vertex on the mirror
+15 XP per step
Q2
PROBLEM
The triangle $PQR$ has $P(0, 4)$, $Q(5, 1)$ and $R(3, 7)$. Reflect it in the y-axis and state which vertex is invariant.
  1. 1
    Apply the y-axis rule to each vertex
    $P(0, 4) \to P'(0, 4)$
    $Q(5, 1) \to Q'(-5, 1)$
    $R(3, 7) \to R'(-3, 7)$
    Negate the x-coordinate each time. The negative of $0$ is $0$.
  2. 2
    Identify the invariant point
    $P$ lies on the y-axis, which is the mirror line, so it does not move. It is an invariant point, and $P'$ coincides with $P$.
  3. 3
    Check the orientation reversed
    Reading $P$, $Q$, $R$ in order traces one rotational sense, and $P'$, $Q'$, $R'$ traces the opposite one, as a reflection requires.
Answer$P'(0, 4)$, $Q'(-5, 1)$, $R'(-3, 7)$, with $P$ invariant
Watch Me Solve It · Two reflections in a row
+15 XP per step
Q3
PROBLEM
The point $B(-2, 6)$ is reflected in the x-axis to give $B'$, and $B'$ is then reflected in the y-axis to give $B''$. Find $B''$ and describe the single transformation equivalent to the pair.
  1. 1
    First reflection
    $B(-2, 6) \to B'(-2, -6)$
    Mirror is the x-axis, so the y-coordinate changes sign.
  2. 2
    Second reflection
    $B'(-2, -6) \to B''(2, -6)$
    Mirror is the y-axis, so the x-coordinate changes sign.
  3. 3
    Describe the composite in general
    $(x, y) \to (x, -y) \to (-x, -y)$
    Both coordinates end up negated, so the composite sends $(x, y)$ to $(-x, -y)$.
  4. 4
    Name it
    Negating both coordinates is a rotation of $180°$ about the origin. Checking on the numbers: $(-2, 6)$ becomes $(2, -6)$, which is the point diametrically opposite through the origin.
Answer$B''(2, -6)$; the pair is equivalent to a $180°$ rotation about the origin

Brain Trainer · 5 problems

D
Brain Trainer · Which sign flips
5 problems

Five items. Name the surviving coordinate before you write anything down.

  1. 1 Reflect $(7, 2)$ in the x-axis.

    Mirror is the x-axis, so $x$ survives.$(7, -2)$

  2. 2 Reflect $(7, 2)$ in the y-axis.

    Mirror is the y-axis, so $y$ survives.$(-7, 2)$

  3. 3 Reflect $(-4, -9)$ in the x-axis.

    Negate the y-coordinate, and the negative of $-9$ is $9$.$(-4, 9)$

  4. 4 $(6, 3)$ maps to $(-6, 3)$. Which reflection was it?

    The y-coordinate survived, so the mirror is the y-axis.In the y-axis

  5. 5 Which points are invariant under reflection in the x-axis?

    Those whose flipping coordinate is already zero.All points on the x-axis
Complete in your workbook.

Multiple Choice · 5 questions

MC1
The x-axis rule
+10 XP

Reflected in the x-axis, the point $(-5, 8)$ maps to:

MC2
Naming the reflection
+10 XP

A reflection maps $(4, -3)$ to $(-4, -3)$. The mirror line is:

MC3
What reverses
+10 XP

Which property distinguishes a reflection from a translation?

MC4
Two in a row
+10 XP

A point is reflected in the x-axis and then in the y-axis. The combined effect on $(x, y)$ is:

MC5
Invariant vertices
+10 XP

A triangle has a vertex at $(0, -7)$. Under reflection in the y-axis, this vertex:

Short Answer · 3 questions

Q6
Reflect a shape and check
+15 XP
Q6
SHORT ANSWER
The triangle $ABC$ has $A(2, 1)$, $B(6, 3)$ and $C(4, 8)$. Find the image after reflection in the x-axis, verify that one side length is unchanged, and state what has happened to the orientation.
Write your working in your book.
Q7
Identify the transformation
+15 XP
Q7
SHORT ANSWER
A transformation maps $P(3, 5)$ to $P'(3, -5)$ and $Q(-2, 4)$ to $Q'(-2, -4)$. Identify the transformation, state the coordinate rule, and give the coordinates of any invariant points among the points $(0, 0)$, $(7, 0)$ and $(0, 7)$.
Write your working in your book.
Q8
Chain three transformations
+15 XP
Q8
SHORT ANSWER
The point $A(-4, 3)$ is reflected in the y-axis to give $A'$, then reflected in the x-axis to give $A''$, then translated $2$ right and $5$ down to give $A'''$. Find all three images, and state the single transformation equivalent to the first two.
Write your working in your book.
S
Stretch Challenge · Reflecting a line, and the mirrors that leave it alone
+25 XP
S
CHALLENGE
Find the equation of the image of the line $y = 3x - 6$ after reflection in the x-axis, and after reflection in the y-axis. Then determine which lines are unchanged as a whole by reflection in the y-axis, even though their individual points move.
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Quick Review
recap

The mirror names the survivor

Reflecting in the x-axis gives $(x, -y)$; reflecting in the y-axis gives $(-x, y)$. The coordinate matching the mirror's name is the one that stays.

Points on the mirror do not move

A reflection always has invariant points, namely the whole mirror line. A translation, by contrast, moves every point without exception.

Orientation reverses

A clockwise-labelled triangle comes back anticlockwise. This is the property that distinguishes a reflection from a translation or a rotation.

Two axis reflections make a half turn

$(x, y) \to (x, -y) \to (-x, -y)$, in either order, which is a rotation of $180°$ about the origin. Reflecting twice in the same line changes nothing at all.

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