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
Forms There are three standard forms for linear
functions y = f(x):
- f(x) = mx + b (The
- y -
yo = 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.
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
Sample calculations are on the following page.
William A. Bogley