Graphing Linear Functions

Graph a linear function: a step by step tutorial with examples and detailed solutions. Free graph paper is available.

Linear Functions

Any function of the form \[ f (x) = m x + b, \] is called a linear function. The domain of this function is the set of all real numbers. The range of \( f \) is the set of all real numbers. The graph of \( f \) is a line with slope \( m \) and \( y \) intercept \( b \).

Note: A function \( f (x) = b \), where \( b \) is a constant real number is called a constant function. Its graph is a horizontal line at \( y = b \) and its slope is undefined.

Example 1

Graph the linear function \( f \) given by \[ f (x) = 2 x + 4 \]

Solution to Example 1

You need only two points to graph a linear function. These points may be chosen as the x and y intercepts of the graph for example.
Determine the x intercept, set \( f(x) = 0 \) and solve for \( x \) \[ 2x + 4 = 0 \] to find \[ x = -2 \]
Determine the y intercept, set \( x = 0 \) to find \( f(0) \) \( f(0) = 4 \)
The graph of the above function is a line passing through the points (- 2 , 0) and (0 , 4) as shown below.

Matched Problem

Graph the linear function f given by \[ f (x) = x + 3 \]

Example 2

Graph the linear function f given by \[ f (x) = - (1 / 3) x - 1 / 2 \]

Solution to Example 2

Determine the x intercept, set \( f(x) = 0 \) and solve for x. \[ -(1 / 3) x - 1 / 2 = 0 \] to find \[ x = - 3 / 2 \]
Determine the y intercept, set \( x = 0 \) to find \[ f(0) = - \dfrac{1}{2} \].
The graph of the above function is a line passing through the points \( (-3 / 2 , 0) \) and \( (0 , -1 / 2) \) as shown below.

Matched Problem 2

Graph the linear function f given by \[ f (x) = - x / 5 + 1 / 3 \]