Linear Functions and Models

In the last tutorial we studied the meaning of the slope, m, and intercept, b, of a linear function f(x) = mx + b, but we did not spend much time talking about how to obtain a linear function.

Q So, how do I get the equation of a linear function?
A That depends on the information you have about the linear function. If, for example, you know the slope and intercept, then you can simply write down the function without the need for any computation.

Example The equation of the line with slope 5 and y-intercept −3 is

Q What if I am not given the slope and intercept?
A The most systematic way of obtaining the equation of a line—and this is a method that always works—is to use the point-slope formula, for which we need to know two things about the line:

(This is all the information we need to know: Knowing the slope tells us the direction of the line, and knowing a point fixes its position in space.)
Point-Slope Formula

The equation of the line through the point (x_1, y_1) with slope m is given by

    y = mx + b
    b = y_1 - mx_1

When to Apply the Point-Slope Formula
Apply the point-slope formula to find the equation of a line whenever you are given information about a point and the slope of the line. The formula does not apply if the slope is undefined (in the case of a vertical line; see below).


The line through (2, 3) with slope 4 has

    m = 4
    b = y_1 - mx_1
    = 3 − (4)(2) = −5
So the equation of the line is
    y = 4x - 5.
Some for you:
Vertical Lines

The point-slope formula does not apply to vertical lines, as these lines have undefined slope. For example, the vertical line through (2, −3) has equation

    x = 2.

Q Where does that formula for the y-intercept come from?
A Once we know the slope m of a line and also the coordinates (x_1, y_1) of a point, then we can calculate its intercept b as follows: The equation of the line is

where b is as yet unknown. To determine b we use the fact that the line must pass through the point (x_1, y_1), and so (x_1, y_1) satisfies the equation y = mx + b. In other words, Solving for b gives

Finding the Equation of a Line When a Point and Slope are Not Given Directly

Often, we need to find a linear function when we are not given the "point and slope" information directly. For instance, we may be given two points and asked to find the line through them. The way to approach problems like this is to use the given information to first find a point and the slope, and then to proceed as above.

You can enter all answers using either fractions or decimals accurate to at least 4 places. For instance, -7x/3 - 2/3 or (7/3)x - 2/3 or 2.3333x - 0.6667

1. The line through has:

2. The line through has:

3. The line through has: [Hint: Parallel lines have the same slope.]

4. The line through has:

Use proper technology notation. For instance, enter "mx − bm" as   m*x-b*m

A certain quantity Q is a linear function of x, and decreases by units per increase in x. Further, when x is the value of Q is Thus, Q is specified by:

You can now try the non-application exercises in Section 1.3 of , or finish the material in Section 1.3 by going on to Part C of this tutorial (press the link on the side).

Last Updated: January, 2010
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