Linear Functions and Models

Q What is a linear function?
A Briefly, a linear function is one whose graph is a straight line (hence the term "linear").

Q How do we recognize a linear function algebraically?
A As follows:
Linear Function

A linear function is one that can be written in the form

f(x) = mx + b Function formExample: f(x) = 3x - 1     m = 3, b = -1
y = mx + b Equation formExample: y = 3x - 1

where m and b are fixed numbers (the names m and b are traditional). The graph of a linear function is a straight line, as seen in the above example if we plot a few points of the equation y = 3x - 1:

Example Fill in the values for the linear function and press "Check." (The associated points will be plotted on the graph below the table at that time.) Hint: Press the Tab key to get from each cell to the next without having to point and click.
Points will be shown on the graph below when you enter values above and press "Check".
Role of m and b in the Linear Function f(x) = mx + b

Let us look carefully at the linear function

    f(x) = 0.5x + 2
This linear function has m = 0.5 and b = 2.

Role of b
Notice that that setting x = 0 gives y = 2, the value of b. On the graph (shown on the left below), the corresponding point (0, 2) is the point where the graph crosses the y-axis, and we say that b = 2 is the y-intercept of the graph.

Numerically, b is the value of y when x = 0
Graphically, b is the y-intercept of the graph

b is the y-intercept

m is the slope

Role of m
Notice from the graph (figure above on the right) that the value of y increases by m = 0.5 for every increase of 1 in x. This is caused by the term 0.5x in the formula. The result is that the graph of y = 0.5x + 2 "rises" by 0.5 units for every horizontal "run" of 1 unit, and we refer to 0.5 as the slope of the line.

Numerically, y changes by m units for every 1-unit increase of x.
Graphically (if m is positive) the graph rises by m units for every 1-unit move to the right; m is the slope of the line.

Here is a more general picture showing two "generic" lines; one with positive slope, and one with negative slope.

Graph of y = mx + b
Positive Slope Negative Slope

For positive m, the graph rises m units for every 1-unit move to the right. For negative m, the graph drops |m| units for every 1-unit move to the right.

The following questions are based on the linear equation The above answers tell us that, whenever x increases by 1 unit,

Further, whenever x increases by two units,

Now graph the above equation by clicking on two points on the graph as accurately as you can and then pressing [Hint: First click on the y-intercept, and then use the above information to find another point. The further apart the two points, the more accurate your graph will be.]

Click on two points to obtain a line.   

The table below shows three functions specified numerically. Only one of them is linear, the others are not:


More on the Slope

Let us take another look at the linear equation

Since the slope is m = 3, we now know that y increases by 3 for every 1-unit increase in x. Similarly, y increases by 3 × 2 = 6 for every 2-unit increase in x, and y increases by 3 × 3 = 9 for every 3-unit increase in x, and so on. Thus, in general, the change in y is 3 times the change in x.

Mathematicians traditionally use Δ (delta, the Greek equivalent of the Roman letter D) to stand for "difference," or "change in." For example, we write Δx to stand for "the change in x." Thus, in symbols, we have

Δy=x Change in y = 3 × Change in x

Here again is the graph of y = 3x - 1, showing two different choices for Δx and the associated Δy.

y = 3x - 1

Slope of a Line

Slope = m = \frac{Δy}{Δx} = \frac{Change in y}{Change in x}.   Graphically, the slope is the ratio \frac{Rise}{Run} shown in the following figure:

Computing the Slope from Coordinates

Press the buttons in the following figure to see Δx and Δy separately:


As you can see in the figure, the rise is Δy = y2 − y1, the change in the y-coordinate from the first point to the second, while the run is Δx = x2 &minus x1, the change in the x-coordinate. Thus,

    m = \frac{Δy}{Δx} = \frac{y_2 - y_1}{x_2 - x_1}.
1. The slope of the line through (x1, y1) = (1, 3) and (x2, y2) = (5, 11) is   m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{11 - 3}{5 - 1} = \frac{8}{4} = 2.
2. The slope of the line through (−1, 3) and (−1, 5) is   m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 3}{-1 - (-1)} = \frac{2}{0}, which is undefined, or infinite.

In each of the following, calculate the slope, if defined, of the line passing through the given pair of points. Fractions, decimals accurate to three decimal places, or valid technology formulas are permitted. Type if a particular slope is not defined.

Fill in the slopes of the following lines:

Getting Familiar with Slopes
It is useful to be able to recognize the slope of a line "at a glance." In particular, you should know how to recognize lines with slopes 0, 1, −1, and a few others.

Before trying the exercises for Section 1.3, you should go on to Part B of this tutorial by pressing the link on the side.

Last Updated: January, 2010
Copyright © 2010