Linear Functions

Significance

Linear functions are the easiest functions to study and linear equations are the easiest equations to solve. A key idea of differential calculus is to approximate more complicated functions by linear functions, calculate with the linear functions, and use the answers to study the more complicated functions.

Standard Forms

There are three standard forms for linear functions y = f(x):

f(x) = mx + b (The "slope-intercept" form),
y - y_o = m(x - x₀) or, equivalently, f(x) = y₀ + m(x - x₀) (The "point-slope" or "Taylor" form), and
Ax + By = C (The "general form") which defines y implicitly as a function of x as long as B 0.

Graphs

If f(x) is linear, the graph of y = f(x) is a straight line. The parameter m in the first two formulas is the slope of this line. In the general form, the slope is -A/B if B

0 and infinite if B = 0. In the slope-intercept form, the parameter b is the y-intercept. In the point-slope form, the point (x₀, y₀) is a point on the line y = f(x). Any point (x₀, y₀) on the line will lead to an equivalent equation.

Calculations

We will frequently have to perform the following calculations with linear equations:

Given the equation of a line, find the slope, x-intercept, and y-intercept of the line.
Given one of the following three pairs of data, find the equation of the line through those points:
- the slope of the line and the y-intercept,
- the slope of the line and a point (x₀,y₀) on the line.
- the coordinates of two points on the line.
Find the intersection of two lines from their equations.

Sample calculations are on the following page.