BestMaths Online

Straight Line Graphs

Linear functions have graphs which are straight lines.

The general equation of a linear function is ax + by + c = 0.         e.g. 3x + 2y + 5 = 0

The exponents of both the x term and the y term are 1.

Another often used form of linear functions is y = mx + c.            e.g. y = 2x − 1

Plotting Points

For a linear function, a table could be set up of several x values and then the coordinates produced plotted on a graph.

Example

For the function y = 2x + 1, plot points to sketch its graph.

x 2x + 1 y Coordinates -2 2 × -2 + 1 -3 (-2, -3) -1 2 × -1 + 1 -1 (-1, -1) 0 2 × 0 + 1 1 (0, 1) 1 2 × 1 + 1 3 (1, 3) 2 2 × 2 + 1 5 (2, 5) The points can be joined to produce a straight line.

The example above can also be done on a spreadsheet:

This can be done quickly using the fill down and make chart features of your spreadsheet program. Notice the slightly different way our spreadsheet program labels the axes and has a different scale for each axis.

Sketching Straight Line Graphs

There are quicker ways to sketch the graphs of linear functions than plotting points.

Intercept method.

Only two points need to be found to draw the graph. These can then be joined to produce the line.

 

Find the two intercepts by putting x = 0 into the equation and finding the corresponding y-value.

Then put y = 0 into the equation and find the corresponding x-value.

This gives the values of x and y where the graph cuts the axes.

Example 3x + 2y = 6 Put x = 0 3 x 0 + 2y = 6 2y = 6 y = 3 The y-intercept is 3. Put y = 0 3 x + 2 × 0 = 6 3x = 6 x = 2 The x-intercept is 2.

 

Gradient / Intercept method.

If the equation is written in the form y = mx + c, where y is the subject of the equation, then:

m, the coefficient of x, is the gradient of the line,

and c, the constant term, is the y-intercept.

Example y = 2x + 1 inspection of the above equation: Gradient, m, is 2 y-intercept, c, is 1

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Special Types of Straight-line Graphs

 

Lines passing through the origin. If the equation has no constant term, it passes through (0, 0).

 

Example y = 2x By inspection of the equation: Gradient is 2. There is no constant term so  c = 0 and this type of line passes through the origin.

 

Lines parallel to the x-axis. These lines have equations of the type y = c, where c is a constant.

 

Example y = 2 Note: no x-term. These lines are horizontal.

y = 2

 

 

Lines parallel to the y-axis. These lines have equations of the type x = c, where c is a constant.

 

Example x = 2 Note: no y-term. These lines are vertical.