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Linear Relationships

Writing Linear Functions

Concept

Linear Function

If a function has a constant rate of change, it is linear. Graphically, a linear function is a straight line.

Using that line, it's possible to determine the rate of change by finding the horizontal change and the vertical change between any two given points on the line. Any function whose graph is not a straight line cannot be linear.

Concept

Slope

The slope of a line passing through the points and is the ratio of the vertical change to the horizontal change between the points. The variable is most commonly used to represent the slope.

The phrase rise and run is sometimes used to describe the slope of a line, especially when the line is given graphically. Rise corresponds to and run corresponds to

With this information, the definition of the slope of a line can be written in terms of rise and run.

Because the rate of change of a linear function is constant, any two points on the line can be used to find the slope – either algebraically with the Slope Formula or graphically as shown below.
The sign of each distance corresponds to the direction of the movement between the points. Moving to the right, the run is positive. Conversely, moving to the left, the run is negative. Similarly, moving up yields a positive rise while moving down gives a negative rise.

Rule

Slope Formula

The slope of a line can be found algebraically using the following rule.

Here, and are two points on the line.

Concept

Slope-Intercept Form

One way to express linear function rules is called slope-intercept form.

Sometimes, is used. In either case, and describe the general characteristics of the line. indicates the slope, and indicates the -intercept. The linear function graphed below can be expressed as because it has a slope of and a -intercept at

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Exercise

Write the equation of the line that passes through the points and

Show Solution
Solution
A line in slope-intercept form is given by the equation where is the slope of the line and is the -intercept. Since two points on the line are known we can use the slope formula to determine the slope.
The equation, can now be rewritten using the knowledge that the slope, is The -intercept can now be found by replacing and in the equation with one of the points, for example and solve the resulting equation.
Thus, the equation of the line passing through the two points is

Method

Writing the Equation of a Line in Slope-Intercept Form from a Graph

To write the equation of the graph of a line in slope-intercept form,

the -intercept, , and the slope of the line, must be found. The following method can be used. As an example, consider the line shown.

1

Find the -intercept

The -intercept is the point where the graph intersects the -axis. From the diagram, it can be seen that the -intercept is

2

Replace with the -intercept

The -coordinate of the -intercept can be substituted into for Here, substituting gives

3

Find the slope

Next, the slope of the line must be determined. From a graph, the slope of a line can be expressed as where rise gives the vertical distance between two points, and run gives the horizontal distance. To find the slope, use any two points. Here, use the already marked -intercept and the arbitratily chosen

From the lines drawn, it can be seen that the and the Therefore, the slope is

4

Replace with the slope

The complete equation of the line can now be written by substituting the value of into the equation from Step 2. Here, substitute

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