## Straight Line Graphs Linear 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.

### Definitions

• 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: =  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 A horizontal line has a gradient of 0. m = 0 The gradient of a vertical line is undefined. m is undefined

### 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 = 1y = 2 x 1 + 1 = 3This gives the ordered pair (1, 3) Put x = 2y = 2 x 2 + 1 = 5This gives the ordered pair (2, 5) Check: When x = 3, y = 7The 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 = 03 x 0 + 2y = 62y = 6y = 3The y-intercept is 3. Put y = 03 x + 2 x 0 = 63x = 6x = 2The x-intercept is 2. 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. y = 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. 