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This topic is part of the TCS FREE high school mathematics 'How-to Library'. It introduces the straight line function, y = mx + c, discusses 'm' and 'c', and shows you how to draw graphs of some different forms of this function. (See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

Theory:

Any function: y = f(x) that graphs as a straight line has an equation of this form:

y = mx + c

NOTE: This does not include vertical lines, because they are not functions of 'x': y = f(x). A function of 'x' cannot have more than one point in the same vertical line.

'x' and 'y' are the coordinates of the points that satisfy the function and so lie on the straight line graph.

'm' is the gradient of the straight line graph, and

'c' is the 'y intercept' of the straight line graph.

About 'gradient'...

'Gradient' is a number that represents the steepness of a straight line. A horizontal line has gradient zero. A 45º line has gradient 1, and a vertical line has infinite gradient.

This diagram shows some different lines and their gradients (the 'm' values):

The sign of the gradient is important. Positive gradients (like those in the diagram above) mean that the line is sloping uphill as you go left to right. If a line slopes downhill going left to right, then it has a negative gradient, as shown below:

Gradient is an exact quantity. This is how it is defined mathematically...

1. Start with any two points on a straight line, (x₁,y₁) and (x,y):

2. Starting at (x₁,y₁) on the left, then moving to the right along the line to (x,y), we measure the change in 'x', and the change in 'y'.

The change in 'x'  =  x - x₁ while the change in 'y' =   y - y₁

Gradient 'm' is defined as follows:

The Greek capital letter 'Delta' : D is often used in mathematics to mean a change in some quantity. So the gradient definition can be written like this:

To measure the change in 'x' and 'y' between the two points in the graph above, we draw a right triangle beneath the graph. The length of the vertical leg is: (y-y₁), and the horizontal leg has length (x-x₁)...

 So the gradient of this line is: 

About 'y' intercept...

The 'y' intercept is the 'y' coordinate of where the straight line graph cuts the 'y' axis:

The symbol 'c' is used to represent the 'y' intercept. So in the example above, c = 2.

Note: at the 'y' intercept, the 'x' coordinate is zero. 

So the equation of the example straight line graph: y = mx + c is:

y = 0.5x + 2

Method:

Maths Helper Plus can find the equation of a straight line given two points. It will calculate the gradient and y intercept showing the working steps. It also displays a labelled diagram of the situation.  

Step 2  Display the parameters box

Press the F5 key to display the parameters box:

The coordinates of two (x,y) points are entered into the four edit boxes 'A', 'D', 'B' and 'X', as follows:

x₁ = A,  y₁ = D,  x = B, and y = X.

To find the equation of the straight line through two points, as well as calculate the gradient 'm' and y intercept '(0,c)', enter the coordinates of two points into the edit boxes.

To enter the coordinates, click on the 'A' edit box with the mouse, then type the x₁ coordinate. Now click on the 'D' edit box and type the y₁ coordinate. Continue until the four coordinates are entered, then click the 'Update' button.

Step 3  Adjust the scale of the labelled diagram

If the points lie off the graphing area of the screen, the scale needs to be reduced. In this case, briefly press the F10 key enough times until the two plotted points are seen.

You can make the diagram bigger by holding down 'Ctrl' while you press F10.