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Lines parallel to the axes

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Graphs

Lines parallel to the axes

Some lines have the simplest equations of all. x = 4 means every point on that line has an x-coordinate of 4 — whatever y does.

xy-4-224-4-224x = 4y = 2x = 4 is VERTICALevery point has x = 4,so it runs up and downy = 2 is HORIZONTALevery point has y = 2,so it runs left to right
It feels backwards: x = 4 is parallel to the y-axis.

Vertical lines: x = a

A line parallel to the y-AXIS is vertical. Every point on it has the SAME x-coordinate. If that value is 4, the line is called x = 4. It crosses the x-axis at 4. The points (4, 2), (4, −3) and (4, 100) are all on it — y can be anything at all.

Horizontal lines: y = b

A line parallel to the x-AXIS is horizontal. Every point on it has the SAME y-coordinate. If that value is 2, the line is y = 2. It crosses the y-axis at 2. The points (4, 2), (−2, 2) and (0, 2) are all on it — x can be anything.

The trap worth knowing

It feels backwards: the line x = 4 is PARALLEL to the y-axis, not the x-axis. Think it through rather than memorising. x = 4 fixes the x-value, so you can only move up and down — which is a vertical line, running alongside the y-axis. Similarly y = 2 fixes y, so you can only move left and right — a horizontal line.

Using them to describe a rectangle

A rectangle has vertices A(4, 2), B(4, −3), C(−2, −3) and D(−2, 2). A and B share x = 4, so that side is the line x = 4. C and D share x = −2, so that side is x = −2. A and D share y = 2, so that side is y = 2. B and C share y = −3, so that side is y = −3. Four equations describe the whole rectangle.

Worked examples

Naming a vertical line through 4

  1. Every point on it has x-coordinate 4.
  2. So the equation fixes x.

Answer: The line is x = 4

Naming a horizontal line through 2

  1. Every point on it has y-coordinate 2.
  2. So the equation fixes y.

Answer: The line is y = 2

Which axis is x = 4 parallel to?

  1. x is fixed at 4, so you can only move up and down.
  2. That is a vertical line.

Answer: Parallel to the y-axis

The side through A(4, 2) and B(4, −3)

  1. Both points have x = 4.
  2. Only the y-values differ.

Answer: The line x = 4

Is (7, 2) on the line y = 2?

  1. The line y = 2 needs the y-coordinate to be 2.
  2. The point (7, 2) has y = 2.

Answer: Yes — x can be anything

Where you see this in real life

  • Lines of latitude run parallel to the equator, all at a fixed distance north or south.
  • A street grid where every road runs either exactly north–south or exactly east–west.

Key words

Parallel
always the same distance apart, never meeting
x-axis
the horizontal axis
y-axis
the vertical axis
Vertex
a corner of a shape
Vertices
more than one vertex
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