# 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 - 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.

#### 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.  Exercises Field Guide HUB CQ Directory CQ Resources