Geometry and Measurement · Position and Transformation · Lesson 2
Plotting Points and Reflecting Shapes on the Coordinate Plane
A coordinate plane lets you give every point on a page an exact address, just like a house number and street. In this lesson you will plot points and shapes, reflect them in the axes and in lines such as x = 3, y = x and y = −x, and discover the simple coordinate rules that let you find an image without drawing.
By the end you will be able to
- ✔Plot and read points in all four quadrants of the coordinate plane.
- ✔Draw and name lines such as x = 0, y = 0, y = x, y = −x, x = k and y = k.
- ✔Draw the image of a point or shape under reflection in a given line.
- ✔Use the rules (a, b) → (a, −b), (−a, b), (b, a) and (−b, −a) for reflections in the axes and the lines y = x and y = −x.
- ✔Describe the properties of the image: congruent, reversed, same distance from the mirror line.
Key words
- Coordinates
- A pair of numbers (x, y) that gives the position of a point: x units across, y units up or down from the origin.
- Origin
- The point (0, 0) where the x-axis and y-axis cross.
- Reflection
- A transformation that flips a shape over a mirror line, so that each point and its image are the same distance from the line on opposite sides.
- Object and image
- The object is the original shape; the image is the shape after the transformation. We name image points with a dash: A → A′.
- Invariant point
- A point that does not move under a transformation. Points on the mirror line are invariant under reflection.
Curriculum reference
Content standard: B7.3.3.1 Perform a single transformation (i.e. reflection and translation) on a 2D shape using graph paper (including technology) and describe the properties of the image under the transformation (i.e. congruence, similarity, etc.)
Indicator: Plot points and shapes (i.e. plane figures) on a coordinate plane and draw their images under reflection in given lines.
Starter
The map of the school compound
Kwame draws a map of his school compound on squared paper. The gate is at (0, 0), the classroom block at (3, 2) and the football field at (−4, −3). 'To find the tree, go three steps east and two steps north,' he tells Ama. The same idea guides ships, mobile phone maps and the screen of your favourite game. When Kwame then folds the paper along the road to copy the buildings to the other side, he is doing a reflection.
1. The coordinate plane and plotting points
The coordinate plane is made from two number lines that cross at right angles: the horizontal x-axis and the vertical y-axis. They meet at the origin (0, 0). Every point has an address (x, y) called its coordinates. The first number tells you how far to move across (right if positive, left if negative); the second tells you how far to move up or down (up if positive, down if negative).
▶ Follow the arrows: along first, then up or down. Notice the sign in each quadrant.
| Quadrant | Position | Signs of (x, y) | Example |
|---|---|---|---|
| 1 | top right | (+, +) | (3, 2) |
| 2 | top left | (−, +) | (−3, 2) |
| 3 | bottom left | (−, −) | (−3, −2) |
| 4 | bottom right | (+, −) | (3, −2) |
To plot a shape, plot all its corners, then join them in the given order with a ruler. The curriculum gives the points A(3, 1), B(3, 3), C(4, 3), D(4, 2), E(5, 2), F(5, 3), H(6, 3) and I(6, 1). Joining them in order and back to A makes a castle-wall shape, as you will see in section 4.
2. Named lines on the plane
Reflection needs a mirror line, so we need to name lines. A line parallel to the y-axis has the same x-coordinate at every point, so its equation is x = k. A line parallel to the x-axis has the same y-coordinate everywhere, so its equation is y = k.
| Line | Equation | What it looks like |
|---|---|---|
| The y-axis | x = 0 | Vertical line through the origin |
| The x-axis | y = 0 | Horizontal line through the origin |
| x = 3 | x = 3 | Vertical line 3 units right of the y-axis |
| y = −1 | y = −1 | Horizontal line 1 unit below the x-axis |
| y = x | y = x | Diagonal through (0, 0), (1, 1), (2, 2) … |
| y = −x | y = −x | Diagonal through (0, 0), (1, −1), (2, −2) … |
On a graph, label each line clearly, for example Line 1: x = 0, Line 2: y = 0, Line 3: y = x, Line 4: y = −x and Line 5: x = −1. Teachers and examiners mark neatness and labelling.
3. Reflecting in the axes and in y = x and y = −x
A reflection flips a shape over a mirror line. To reflect a point: measure its perpendicular distance to the mirror line, then go the same distance on the other side. The point and its image are joined by a line that is perpendicular to the mirror line, and the mirror line cuts that joining line in half.
▶ Reflect triangle ABC in the y-axis. Notice that A(1, 1) goes to A′(−1, 1): the x-coordinate changes sign.
▶ Reflect in the x-axis. Notice (2, 1) goes to (2, −1): the y-coordinate changes sign.
Comparing many examples gives you these rules. If (a, b) is the object point, then:
| Mirror line | Image of (a, b) | Example: (3, 5) |
|---|---|---|
| x-axis (y = 0) | (a, −b) | (3, −5) |
| y-axis (x = 0) | (−a, b) | (−3, 5) |
| y = x | (b, a) | (5, 3) |
| y = −x | (−b, −a) | (−5, −3) |
▶ Reflection in y = x swaps the coordinates: (1, 0) → (0, 1), (3, 1) → (1, 3), (1, 4) → (4, 1).
▶ Reflection in y = −x: (1, 1) → (−1, −1). Swap the coordinates and change both signs.
Reflection in a point is also possible: every point is sent through a centre point to the same distance on the opposite side, which for the origin gives (a, b) → (−a, −b). In this lesson we mainly use reflection in a line. Always label the object ABC and the image A′B′C′ so that your teacher can see which corner went where.
4. Reflecting in x = k and y = k, and describing the image
Mirror lines do not have to pass through the origin. For a line x = k, the image of (a, b) is (2k − a, b): the height stays the same, and the distance to the line stays the same on the other side. For a line y = k, the image of (a, b) is (a, 2k − b).
▶ Mirror line x = 3. A(1, 1) is 2 units to the left of the line, so A′ is 2 units to the right at (5, 1).
▶ Mirror line y = −1. A(1, 1) is 2 units above the line, so A′ is 2 units below it at (1, −3).
▶ Plot the points, join them, then reflect each point in the y-axis. Compare the two shapes.
Properties of the image under a reflection: (1) the image is congruent to the object (same lengths, same angles, same area); (2) the image is a mirror image: the order of the letters A, B, C goes clockwise on one shape and anticlockwise on the other; (3) each object point and its image point are the same distance from the mirror line; (4) points on the mirror line do not move.
Worked examples
Try each one yourself first, then open the solution and compare every step.
Example 1
Four reflections of one point
Show the solution
- Step 1(a) x-axis: (a, b) → (a, −b), so A → (3, −5).
- Step 2(b) y-axis: (a, b) → (−a, b), so A → (−3, 5).
- Step 3(c) y = x: (a, b) → (b, a), so A → (5, 3).
- Step 4(d) y = −x: (a, b) → (−b, −a), so A → (−5, −3).
Answer: (3, −5), (−3, 5), (5, 3), (−5, −3)
Example 2
Reflecting a triangle in the x-axis
▶ Each point goes to the other side of the x-axis at the same distance.
Show the solution
- Step 1Rule: (a, b) → (a, −b).
- Step 2A(2, 1) → A′(2, −1).
- Step 3B(5, 1) → B′(5, −1).
- Step 4C(2, 4) → C′(2, −4).
- Step 5Plot the points and join. Check: A′B′ = 3 = AB; the image is congruent ✓
Answer: A′(2, −1), B′(5, −1), C′(2, −4)
Example 3
Reflection in the line x = 3
Show the solution
- Step 1For the line x = k the image of (a, b) is (2k − a, b). Here k = 3, so 2k = 6.
- Step 2P(1, 2) → (6 − 1, 2) = (5, 2).
- Step 3Q(4, −1) → (6 − 4, −1) = (2, −1).
- Step 4R(3, 5) is on the mirror line, so it does not move: R′ = (3, 5).
- Step 5Check P: P is 2 left of the line and P′ is 2 right ✓
Answer: P′(5, 2), Q′(2, −1), R′(3, 5)
Example 4
Reflection in the line y = −1
▶ The mirror line y = −1 lies below the object, so the image appears further down.
Show the solution
- Step 1For y = k the image of (a, b) is (a, 2k − b). Here 2k = −2.
- Step 2M(2, 3) → (2, −2 − 3) = (2, −5).
- Step 3N(−3, −4) → (−3, −2 − (−4)) = (−3, 2).
- Step 4Check M: M is 4 above the line (3 − (−1) = 4) and M′ is 4 below (−1 − (−5) = 4) ✓.
Answer: M′(2, −5), N′(−3, 2)
Example 5
Finding the mirror line
▶ The mirror line y = 2 is exactly halfway between each point and its image.
Show the solution
- Step 1The x-coordinates are equal, so the movement is vertical and the mirror line is horizontal: y = k.
- Step 2k is the midpoint of the y-coordinates: (5 + (−1)) ÷ 2 = 2.
- Step 3The line is y = 2.
- Step 4Check B: the midpoint of 0 and 4 is 2 ✓.
Answer: y = 2
Summary
- Coordinates (x, y): across first, then up or down. The axes meet at the origin (0, 0).
- Vertical lines are x = k; horizontal lines are y = k; the diagonals through the origin are y = x and y = −x.
- Rules: x-axis (a, b) → (a, −b); y-axis → (−a, b); y = x → (b, a); y = −x → (−b, −a).
- In x = k: (a, b) → (2k − a, b). In y = k: (a, b) → (a, 2k − b).
- The image under reflection is congruent to the object but reversed; points on the mirror line are invariant.
Practice: 15 questions
5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.
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