Lesson 17. Linear function

The linear function is popular in economics. It is attractive because it is simple and easy to handle mathematically. It has many important applications.

Linear functions are those whose graph is a straight line.

A linear function has the following from

y = f(x) = a + bx

A linear function has one independent variable and one dependent variable. The independent variable is x and the dependent variable is y.

A is the constant term or the y intercept. It is the value of the dependent variable when x = 0.

b is the coefficient of the independent variable. It is also known as the slope and gives the rate of change of the dependent variable.

Point-Slope Formula.

We have seen that we can define the slope of a line given two points on the line, and use than information along with the y-intercept to graph the line.

If you don’t know the y-intercept, or the equation for the line you can use two points to define the equation of the line using the point-slope formula.

This is an important formula, as it will be used in other areas of college algebra and often in calculus to find the equation of a tangent line. We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one paint into the formula, we simplify it and write it in slope-intercept form.

Given one point and the slope, the point-slope formula will lead to the equation of a line:

y = mx.

EXAMPLE:

Write the equation of the line with slope m = –3 and passing through the point (4, 8). Write the final equation in slope-intercept form.

SOLUTION:

Using the point-slope formula, substitute –3 for m and the point (4, 8) for 

(x₁, y₁).

y – y₁ = m(x – x₁)

y – 8 = –3(x – 4)

y – 8 = –3x + 12

y = –3x + 20

Note that any point on the line can be used to find the equation. If done correctly, the same final equation will be obtained.

EXAMPLE:

Find the equation of the line passing through the points (3, 4) and  (0, –3). Write the final equation in slope-intercept form.

SOLUTION:

First, we calculate the slope using the slope formula and two points.

Next, we use the point-slope formula with the slope of ⁷/₃, and either point. Let’s pick the point (3, 4) for (x₁, y₁).

y – 4 = ⁷/₃(x – 3)

y – 4 = ⁷/₃ x – 7

y = ⁷/₃ x – 3

Distribute the ⁷/_3.

In slope-intercept form, the equation is written as

y = ⁷/₃ x – 3.

To prove that either point can be used, let us use the second point (0, –3) and see if we get the same equation.

y – (–3) = ⁷/₃(x – 0)

y + 3 = ⁷/₃ x

y = ⁷/₃ x – 3

We see that the same line will be obtained using either point. This makes 

sense because we used both points to calculate the slope.

Lesson 17

•  Test 1 
•  Test 2
•  Test 3