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     Example: f(x) = 3x  1 m = 3, b = 1 
y = mx + b     Example: 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.)

Role of m and b in the Linear Function f(x) = mx + b
Let us look carefully at the linear function
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 yaxis, and we say that b = 2 is the yintercept of the graph.
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.
Here is a more general picture showing two "generic" lines; one with positive slope, and one with negative slope.
Graph of y = mx + b 
 




For positive m, the graph rises m units for every 1unit move to the right. For negative m, the graph drops m units for every 1unit 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 yintercept, and then use the above information to find another point. The further apart the two points, the more accurate your graph will be.]
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 1unit increase in x. Similarly, y increases by 3 × 2 = 6 for every 2unit increase in x, and y increases by 3 × 3 = 9 for every 3unit 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

or
Slope  =  3  =  Δy
Δx 
Here again is the graph of
y = 3x  1, showing two different choices for Δ
x and the associated Δ
y.
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 = y_{2} − y_{1}, the change in the ycoordinate from the first point to the second, while the run is Δx = x_{2} &minus x_{1}, the change in the xcoordinate. Thus,
m = \frac{Δy}{Δx} = \frac{y_2  y_1}{x_2  x_1}.

Examples
1. The slope of the line through (x_{1}, y_{1}) = (1, 3) and (x_{2}, y_{2}) = (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

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
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