Straight Lines

A straight line graph is made up of lots of and pairs known as co-ordinates or points. A point consists of an value and its corresponding value and is written in a bracket.

E.g. (3, 5) means that when , .

For example, consider the straight line .

The equation tells us the relationship between and .

You can choose any value of so there are an infinite number of pairs:

Let’s try some:

When , , , pair =

The above pairs (or co-ordinates) can be plotted on a graph and joined up to make a straight line.

Often we just want an idea of what the line will look like so that we can sketch it instead of plotting lots of pairs, and so we consider the general equation of a straight line.

The General Equation:

The general equation of a straight line is: . We can tell 2 things about the straight line from the equation:

The gradient of the linethe -intercept

(where it crosses the -axis)

Example:

Look at the straight line equation: . We know (just by looking) that this line will have a gradient of 3 and will cross the -axis at 2, so will look like this:

More about Gradients

A positive gradient is uphill:

A negative gradient is downhill:

A gradient of zero (0) is flat:

The gradient is found by dividing how much the line goes up () by how much it goes along (). We say “the change in over the change in ”.

A gradient of 1 means go along 1 and up 1.
A gradient of 2 means go along 1 and up 2 so it is steeper,
a gradient of 8 means go along 1 and up 8 so it’s getting very steep.
A gradient of means go along 1 and up so it is shallow…… etc

Examples:

(let the gradient = m)

m = 3m = 1m=m = 0m = m =

More about the axis-intercepts:

The -intercept can easily be seen from the equation but you need to do a little bit of rearranging to find the -intercept.

Remember: the line crosses the -axis when

the line crosses the -axis when

Example:

Look at the straight line equation: .

-intercept: (the number on it own)

-intercept:

When (we need to do some rearranging here)

Replace the with a 0 and rearrange to find

We know that the line crosses the -axis at 8 and the -axis at 4 (and the gradient is )

Sketching a straight line:

We can use all of the above information to sketch a straight line. When asked to sketch a straight line, there are 3 things that it is useful to know:

The gradient
The -intercept
The -intercept

Example:

Sketch the line

We can see that the gradient is 2 and it crosses the -axis at . We have 2 out of the 3 things straight away.

Itcrosses the -axis when = 0,

So replace with 0 in the equation:

We now have all the information we need to sketch the line:

Gradient = 2

-intercept =

Finding the Equation of a Straight Line

Sometimes you may be given 2 points on a line and asked to find the equation of the line. Remember that the general equation of a straight line is and so the 2 pieces of information you need to know to be able to write the equation are the gradient and the -intercept.

Gradient = =

-intercept: substitute one of the points in and solve.

Examples:

Find the equation of the straight line passing through the points (2, 12) and (3, 17).

If we call the 1st point 1 and the 2nd point 2 then we have: and

First find the gradient:

Gradient = = = =

We now know that the line has a gradient of 5 and so the equation looks like this:

Next find :

To complete the equation we need to work out . To do this substitute one of the points in (whichever you find easiest). In this example I’ll choose the 1st point (2, 12).

Substitute point in (replace with 2 and with 12):

We now have the equation of the straight line:

Find the equation of the straight line passing through the points and .

Gradient = = = =

So far we know that the equation looks like this:

To complete the equation we need to work out . We need to substitute one of the points in and I’ll choose the 1st point again (1, -4).

Substitute point in:

We now have the equation of the straight line:

N.b. if you are given the gradient and one point then the question is easier as you don’t need to do the 1st part (finding the gradient), just start from the substitution stage.

Example:

Find the equation of the straight line with gradient and passing through the point (1,1)

We are given the gradient so we know that the equation looks like this:

To complete the equation we need to work out . To do this we substitute the point in:

Substitute point in:

We now have the equation of the straight line: