Reflections in the coordinate plane

Warm-up

Three quick questions from earlier lessons.

Discover

A point is 3 units above the $x$ -axis. Its mirror image across the $x$ -axis is…

3 units below the axis, straight down
3 units to the left

Key idea

The line of reflection is the perpendicular bisector of each segment joining a point to its image.

Across the

x

-axis:

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

. Across the

y

-axis:

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

. Across

y = x

:

(x, y) \to (y, x)

.

Worked example

Reflect $B(-2, 5)$ in three lines.

Step 1Triangle ABC above the x-axis.
Step 2From each vertex, go straight across the axis, the same distance on the other side.
Step 3Join the images: (x, y) → (x, −y).

x

-axis:

(-2, -5)

Negate y.

y

-axis:

(2, 5)

Negate x.

y = x

:

(5, -2)

Swap.

Your turn, guided

Your turn: the x-coordinate of $(4, -3)$ reflected across the $y$ -axis.

-4

Negate x.

Practice ★

Reflect $A(3, 2)$ across the $x$ -axis. Tap its image.
Reflect $A(5, 6)$ across the $x$ -axis. Tap its image.
Reflect $A(-4, 1)$ across the $x$ -axis. Tap its image.

Watch out!

Practice ★★

Reflect $A(5, -1)$ across the line $y = x$ . Tap its image.
Reflect $A(-5, -3)$ across the line $y = x$ . Tap its image.

A point is 5 units from a line of reflection. How far is it from its image?

Reflect triangle ABC across the $y$ -axis. Tap the images A′, B′ and C′.

Exit ticket

Three questions to check you have got it.

Reflect $A(3, 2)$ across the $x$ -axis. Tap its image.
Reflect $A(5, 6)$ across the $x$ -axis. Tap its image.
Reflect $A(-4, 1)$ across the $x$ -axis. Tap its image.

Summary card

Reflections

$x$ -axis: $(x, -y)$ ; $y$ -axis: $(-x, y)$ ; $y = x$ : $(y, x)$ .

Watch out: across the $x$ -axis, it is $y$ that changes sign.

About this lesson

Reflections in the coordinate plane

Warm-up

Discover

Key idea

Worked example

Your turn, guided

Practice ★

Watch out!

Practice ★★

Exit ticket

Summary card

End of the lesson

Answers: Reflections in the coordinate plane