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Linear Relationships

Writing Linear Functions

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Linear Function

A linear function is a function with a constant rate of change. A linear function can be represented by a linear equation in two variables. Graphically, a linear function is a nonvertical line.

Using that line, the rate of change can be determined by finding the horizontal change and the vertical change between any two points on the line. Any function whose graph is not a straight line cannot be linear.



The slope of a line passing through the points (x1,y1) and (x2,y2) is the ratio of the vertical change to the horizontal change between the points. The variable m is most commonly used to represent the slope.

The phrase rise over 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

The definition of the slope of a line can be written in terms of its rise over its 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. That can be done 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. If the direction moves to the right, the run is positive. Conversely, if the direction move to the left, the run is negative. Similarly, moving up yields a positive rise while moving down gives a negative rise.


Slope Formula

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

Here, (x1,y1) and (x2,y2) are two points on the line.


Slope-Intercept Form

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


Sometimes, f(x)=mx+b is used. In either case, m and b describe the general characteristics of the line. m indicates the slope, and b indicates the y-intercept. The linear function graphed below can be expressed as
because it has a slope of 2 and a y-intercept at (0,1).


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


Write the equation of the line that passes through the points (3,1) and (5,5).

Show Solution expand_more
A line in slope-intercept form is given by the equation
where m is the slope of the line and b is the y-intercept. Since two points on the line are known we can use the slope formula to determine the slope.
The equation, y=mx+b, can now be rewritten using the knowledge that the slope, m, is 2.
The y-intercept can now be found by replacing x and y in the equation with one of the points, for example (5,5), and solve the resulting equation.
Thus, the equation of the line passing through the two points is


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

To write the equation in slope-intercept form of the graph of a line, the y-intercept b and the slope m of the line must be found.
The following method can be used. As an example, consider the line shown.
A line on a coordinate plane
Find the y-Intercept

The y-intercept is the point where the graph intersects the y-axis. From the diagram, it can be seen that the y-intercept is (0,-4).

The y-intercept of the line is identified on the graph
Replace b With the y-Intercept
The y-coordinate of the y-intercept can be substituted into the slope-intercept form equation for b. In the considered example, substituting -4 for b results in the following equation.
Find the Slope
Next, the slope of the line must be determined. By the definition, the slope of a line is expressed as the quotient of the rise and run of the line.
Here, rise is the vertical distance between two points, and run is the horizontal distance. To find the slope, use any two points. On the diagram, the previously-found y-intercept and the point (2,2) were chosen.
Rise and run between the two chosen points are determined
As seen above, the rise equals 6 and the run equals 2. By substituting these values into the formula, the slope can be calculated.
Replace m With the Slope
The complete equation of the line can now be written by substituting the value of m into the equation from Step 2. In this case, substitute m=3.
The equation is now complete.
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