Definition of Linear Function

A function f is a linear function if f(x) = ax + b , for real numbers a and b.

When a linear function is written in the form Ax + By = C, it is said to be in standard form.

The graph of a linear function is a straight line. To graph a linear function, find at least two of its ordered pairs, plot them, and draw a line through them.

Example. Graph 5x – 2y = 10

Solution.

Let x = 0. 5(0) – 2y = 10	Letting x = 0 will give us the y-intercept
- 2y = 10	Solve for y.
y = -5
(0,-5)
Let y = 0. 5x – 2(0) = 10	Letting y = 0 will give us the x-intercept.
5x = 10
x = 2
(2,0)
Let x = 4. 5(4) – 2y = 10	Third point is a check
20 – 2y = 10 - 2y = -10 y = 5 (4,5)

These points would be plotted and a line drawn through them to complete the graph.

Slope of a Line

The slope is a numerical measure of the steepness of a line. Slope compares the vertical change (the rise) to the horizontal change (the run) when moving from one point to another along the line. To calculate the slope, we use a ratio comparing the change in y, D y, to the change in x, D x. The slope m of the line through the points and is

Example. Find the slope of the line through the points (-3, -1) and (-2,4).

Solution. = (-3, -1) and = (-2, 4).

The slope of a vertical line is undefined.

The slope of a horizontal line is 0.

The Difference Quotient

If h is any nonzero number, then the quotient is called the difference quotient. The difference quotient is used in calculus to find the steepness of a curve at a point.

Example. For the function f(x) = x - 4x + 7, find .

Solution. f(x + h) is found by replacing x with x + h in the formula for f(x).

f(x + h) = (x + h) - 4(x + h) + 7 = x + 2xh + h - 4x – 4h + 7

We will substitute this result into the Difference Quotient formula.

=	Difference Quotient; substitution of f(x + h) and f(x)
	Simplify
=
=
= 2x + h – 4