Linear functions are those of the form, f(x) = ax + c, where, a and c are integers, and linear means a straight line.
Recall that, y = f(x)
Therefore, y = ax + c
Where, y is the dependent variable
a is the coefficient of x
x is the independent variable
c is the constant term.
Note also that, y = mx + c, equation of a line
Therefore, y = ax + c = mx + c
Where, m is the gradient of the line (ratio of the vertical rise over the horizontal run)
c is the point at which the line intercepts the y axis.
Having stated the form of linear functions, below is an example of how to draw linear functions.
Example
Draw the graph of the linear function, f(x) = 3x + 2, for the domain -2 ≤ x ≤ 2.
Method 1
Substitute the values given for x in the domain (-2, -1, 0, 1, 2), in the function, solving for the respective f(x)/ y values.
Given, f(x) = 3x + 2
Then, f(-2) = 3(-2) + 2 = -6 + 2 = -4
f(-1) = 3(-1) + 2 = -3 + 2 = -1
f(0) = 3(0) + 2 = 0 + 2 = 2
f(1) = 3(1) + 2 = 3 + 2 = 5
f(2) = 3(2) + 2 = 6 + 2 = 8
Therefore, the set of (x, y) values to be plotted and connected in forming the linear function are:
{(-2, -4), (-1, -1), (0, 2), (1, 5), (2, 8)}
Method 2
This method involves finding the x and y intercepts, that is the point at which the graph crosses the x and y axis. A linear function written in its correct form, that is, f(x) = mx + c, states the y intercept, c [in (x, y) form is (0, c)]. So, to find the x intercept, substitute 0 for y in the function and solve for x.
Given, f(x) = 3x + 2
Y intercept (point of intersection on the y axis) is, (0, 2)
X intercept (point of intersection on the x axis) is, y = f(x)
y = 3x + 2
0 = 3x +2 (substituting 0 for y)
3x = -2
x = -2/3 or -0.66
That is, x intercept is (-0.66, 0)
Using the x and y intercepts, (-0.66, 0) and (0, 2), the graph is plotted below.
