1.3 Linear Functions and Models
Part A: Basics: Slope and Intercept

Q What is a linear function?
A 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 form Example: f(x) = 3x - 1     m = 3, b = -1 y = mx + b Equation form Example:y = 3x - 1 where m and b are fixed numbers (the names m and b are traditional). Example Here is a partial table of values of the linear function f(x) = 3x - 1. Fill in the missing values and press "Check." x -4 -3 -2 -1 0 1 2 3 4 y -13 -10 -7 -1 2 5     Plotting a few of these points gives the following graph.

The Role of b in the equation y = mx + b
Let us look more closely at the above linear function, y = 3x - 1, and its graph, shown above. This linear equation has m = 3 and b = -1.

Notice that that setting x = 0 gives y = -1, the value of b.

On the graph, the corresponding point (0, -1) is the point where the graph crosses the y-axis, and we say that b = -1 is the y-intercept of the graph

The Role of m in the equation y = mx + b
Notice from the table that the value of y increases by m = 3 for every increase of 1 in x. This is caused by the term 3x in the formula: for every increase of 1 in x we get an increase of 31 = 3 in y.

On the graph, the value of y increases by exactly 3 for every increase of 1 in x, the graph is a straight line rising by 3 for every 1 we go to the right. We say that we have a rise of 3 units for each run of 1 unit. Similarly, we have a rise of 6 for a run of 2, a rise of 9 for a run of 3, and so on. Thus we see that m = 3 is a measure of the steepness of the line; we call m the slope of the line.

Here is the graph of y = 0.5x + 2, so that b = 2 (y-intercept) and m = 0.5 (slope).

Notice that the graph cuts the y-axis at b = 2, and goes up 0.5 units for every one unit to the right. Here is a more general picture showing two "generic" lines; one with positive slope, and one wqith negative slope.

Graph of y = mx + b
Positive Slope	Negative Slope

Let y = -1.5x + 4.

Q

m =

b =

Q The above answers tell us that, whenever x increases by 1 unit,

Select one y decreases by 4 units y increases by 4 units y decreases by 1.5 units y increases by 1.5 units None of the above

Q Similarly, whenever x increases by 2 units,

Select one y decreases by 3 units y increases by 3 units y decreases by 2.5 units y increases by 2.5 units None of the above

Q Select the correct graph of y = -1.5x + 4. (Gridlines are spaced one unit apart.)

   

Here are two functions specified numerically. One of them is linear, the other is not.

x	1	3	5	7	9
f(x)	-4	4	20	45	76
g(x)	-4.5	-1.5	1.5	4.5	7.5

Q The function

f g

is linear.

Q

The linear function above has equation y =

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

Let us take another look at the linear equation

y = 3x - 1

Now we know that y increases by 3 for every 1-unit increase in x.
Similarly, y increases by 32 = 6 for every 3-unit increase in x.
. . . .
In general, y increases by 3x units for every x-unit change in x.

Using symbols,

	y	=	3x		Change in y = 3 Change in x
or	y x	=	3 = slope

Q How do these changes show up on the graph?
A Here again is the graph of y = 3x - 1 , showing two different choices for x and the associated y.

y = 3x - 1

To summarize:

Slope of a Line The slope of a line is given by the ratio m = y = 3x = Change in y Change in x = Rise Run = Slope Definition of the Slope For positive m, the graph rises m units for every 1-unit move to the right, and rises y = mx units for every x units moved to the right. For negative m, the graph drops |m| units for every 1-unit move to the right, and drops |m|x units for every x units moved to the right. Graph of y = mx + b Positive Slope Negative Slope

Fill in the slopes of the following lines.

m =	m =	m =

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

Computing the Slope of a Line

Q Two points, say (x₁, y₁) and (x₂, y₂), determine a line in the xy-plane. How do we find its slope?
A Look at the following figure.

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

To summarize:

Computing the Slope of a Line Computing the Slope of a Line We can compute the slope m of the line through the points (x1, y1) and (x2, y2) using m = y x = y2 - y1 x2 - x1 Examples 1. The slope of the line through (x1, y1) = (1, 3) and (x2, y2) = (5, 11) is m = y2 - y1 x2 - x1 = 11 - 3 5 - 1 = 8 4 = 2 2. The slope of the line through (-1, 3) and (-1, 5) is m = y2 - y1 x2 - x1 = 5 - 3 -1 - (-1) = 2 0 , which is undefined, or infinite. 3. The slope of the line through (-1, 0) and (2, -2) is m =

Before trying the exercises, you should go on to the next tutorial: Part B: Finding the Equation of a Line.