Gradients_and_Curves.jpgLinear functions have graphs which are straight lines.

The general equation of a linear function is ax + by + c = 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.

Sketching Straight Line Graphs
Special Types of Straight Line Graphs



  • Cartesian graph is a graph drawn on a number plane with two perpendicular axes.
  • Coordinates are the ordered pairs that locate points on a graph. The first number is the distance of the point along the x-axis (horizontal) and the second number is the distance of the point along the y-axis (vertical).
  • Each point is located by an ordered pair. e.g. (x, y), (3, 4)
  • The origin is the point where the two axes intersect. The origin has coordinates of (0, 0).
  • The intercepts of a graph are the points where the graph cuts the x-axis and y-axis.


The gradient of a straight line is a measure of its slope.

The gradient, m is defined as:





Types of gradient


A line sloping upwards from left to right has a positive gradient.


m is positive

A line sloping downwards from left to right has a negative gradient.


m is negative

Parallel lines have the same gradient.


m1 = m2

horizontal line has a gradient of 0.

m = 0

The gradient of a vertical line is undefined.

is undefined


Click Here for practice at calculating gradients.


Sketching Straight Line Graphs

There are several ways to sketch the graphs of linear functions.

Plot points. Choose two convenient values for x and complete the ordered pairs for the function. Join these two ordered pairs. Check that a third value for x also gives a point on the line.

e.g. y = 2x + 1


Put x = 1
y = 2 x 1 + 1 = 3
This gives the ordered pair (1, 3)

Put x = 2
y = 2 x 2 + 1 = 5
This gives the ordered pair (2, 5)

Check: When x = 3, y = 7
The point (3, 7) lies on the line.

It is sometimes useful to put these values in a table.


Intercept method.

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.

e.g. 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 x 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.

e.g. y = 2x + 1

By inspection of the above equation:

Gradient is 2

y-intercept is 1



Special Types of Straight-line Graphs

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


e.g. y = 2x

By inspection of the equation:

Gradient is 2.


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


e.g. y = 2

Note: no x-term.

Y10_Straight_Line_Graphs_11.gify = 2


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


e.g. x = 2

Note: no y-term.