Coordinates & Reflections

Coordinates are just a way of describing an exact spot on a grid using two numbers.

Reading and plotting a point

A coordinate is always written as a pair inside brackets, like (3, 2). The two numbers are separated by a comma, and the order matters.

• The first number is the x-coordinate — how far to go across (left or right).
• The second number is the y-coordinate — how far to go up or down.

The golden rule is: along the corridor, then up the stairs. You always walk along first, then go up. So to plot (3, 2), start at the origin (the point (0, 0) where the axes cross), go 3 across to the right, then 2 up.

Mixing up the two numbers is the single most common mistake. (3, 2) and (2, 3) are completely different points.

The four quadrants and negative numbers

When the grid extends below and to the left of the origin, the axes split it into four regions called quadrants. Points can now have negative coordinates:

• A negative x means go left instead of right.
• A negative y means go down instead of up.

So (-4, 3) means 4 to the left and 3 up. And (2, -5) means 2 to the right and 5 down. The number line rules still apply: -5 is lower than -2.

Reflecting a point in an axis

Reflecting means flipping a point to the mirror-image position on the other side of a line (the axis acts like a mirror). There are two cases, and the trick is knowing which coordinate changes:

• Reflect in the x-axis (the horizontal line): the point flips up/down, so keep the x the same and flip the sign of the y. Example: (4, 3) becomes (4, -3).
• Reflect in the y-axis (the vertical line): the point flips left/right, so keep the y the same and flip the sign of the x. Example: (4, 3) becomes (-4, 3).

A memory hook: the axis you mirror in tells you the coordinate that stays. Mirror in the x-axis → x stays. Mirror in the y-axis → y stays. The other one flips its sign.

Translations (sliding a point)

A translation slides a point without turning or flipping it. You just add or subtract:

• Right adds to x, left subtracts from x.
• Up adds to y, down subtracts from y.

Example: translate (1, 2) by 4 right and 3 up → x: 1 + 4 = 5, y: 2 + 3 = 5, giving (5, 5).

Finding the fourth corner

A favourite 11+ question gives you three corners of a rectangle or square and asks for the fourth. The trick: corners of a rectangle share coordinates in pairs. Two corners will share an x-value; two will share a y-value. The missing corner borrows the x from one point and the y from another.

Example: three corners are (1, 1), (5, 1) and (1, 4). The missing corner sits opposite (1, 1). It lines up under (1, 4) so its x is 5, and level with (5, 1) so its y is 4 — giving (5, 4). Sketching the three points first makes this obvious.

Common 11+ traps

⚠️ Order matters. (3, 5) is NOT the same as (5, 3). Along first, then up — every single time.

⚠️ Reflecting in the x-axis flips the y, not the x. Lots of children flip the wrong number. Say it out loud: "x-axis, so x stays."

⚠️ Watch the signs with negatives. Reflecting (-3, 4) in the x-axis gives (-3, -4) — the x keeps its minus sign, only the y flips.

⚠️ "Left" and "down" mean subtract. Sliding 6 left from (3, 5) gives x = 3 - 6 = -3, not 9.

Worked examples

Read through each example. Cover the steps and try it yourself first if you can.

1. 1

   Reflecting a point in the x-axis

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

   Steps

   1. The x-axis is the mirror, so the x-coordinate stays the same: x = -3.
   2. Flip the sign of the y-coordinate: 4 becomes -4.
   3. Put them back together: (-3, -4).

   Answer

   (-3, -4)

2. 2

   Finding the fourth corner of a rectangle

   Three corners of a rectangle are (1, 1), (5, 1) and (1, 4). Find the fourth corner.

   Steps

   1. Sketch the three points. The corner (1, 1) is missing its diagonal partner.
   2. The fourth corner lines up with (5, 1), so it shares that x-value: x = 5.
   3. It also lines up with (1, 4), so it shares that y-value: y = 4.

   Answer

   (5, 4)

3. 3

   Translating a point

   Translate the point (3, 5) by 6 left and 2 down.

   Steps

   1. Left means subtract from x: 3 - 6 = -3.
   2. Down means subtract from y: 5 - 2 = 3.
   3. Write the new point: (-3, 3).

   Answer

   (-3, 3)