Graphing Lines

Idea

Two points are enough to draw a line, so you never need to plot many values.

Find two points you are sure about, mark them, and draw a straight line through them. The only skill is choosing the two points that need the least arithmetic.

Rule

To sketch $ax + by + c = 0$ , take the two intercepts:

y = 0 \Rightarrow \text{the } x\text{-intercept}

x = 0 \Rightarrow \text{the } y\text{-intercept}

Plot both, join them, and extend past both ends - a line does not stop at the points you happened to plot.

How it is used

Choose the method that matches the form you are given.

Form	Fastest route
$ax + by + c = 0$	both intercepts
$y = mx + n$	start at $(0, n)$ , then step the slope: run right, rise up
Through the origin	the origin is one point; find any second
$x = c$ or $y = c$	draw the vertical or horizontal line directly

The stepping method is worth explaining.

Start at any point on the line. A slope of $\dfrac{3}{4}$ means go $4$ to the right and $3$ up. You land exactly on another point of the line. If the slope is negative, you go down instead of up.

Using whole numbers keeps you on grid corners, so you never have to guess a position.

Two checks catch most mistakes.

Plot a third point. If it does not sit on the same straight line, one of your three points is wrong.

Check the direction before anything else. A positive slope must rise to the right, and a negative slope must fall.

Worked example

Sketch the line $3x + 4y - 12 = 0$ using its intercepts. Set $y = 0$ to find where it crosses the $x$ -axis: $3x = 12$ , so $x = 4$ , giving $(4, 0)$ . Set $x = 0$ to find where it crosses the $y$ -axis: $4y = 12$ , so $y = 3$ , giving $(0, 3)$ . Plot $(4, 0)$ and $(0, 3)$ , join them, and extend both ways. As a check the slope should be negative, and indeed the line falls from $(0, 3)$ down to $(4, 0)$ :

m = -\frac{3}{4}

Explore

Put A and B on the two intercepts of a line you are drawing, and compare it with your own sketch. Then try the stepping method: leave A on the $y$ -intercept and move B by exactly the run and the rise. It stays on the same line.

A line through two points

Try this. Move the two points and compare the equation. Equal x values produce a vertical line; equal points cannot define one.

Point A · x-2Point A · y3Point B · x3Point B · y1

y = -0.4x + 2.2 · Slope = -0.4

Idea

Rule

How it is used

Worked example

Explore

Try it yourself.

Pause before the next idea.