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# 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 $(\Delta x)$ and the vertical change $(\Delta y)$ 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 $(x_1,y_1)$ and $(x_2,y_2)$ is the ratio of the vertical change $(\Delta y)$ to the horizontal change $(\Delta x)$ between the points. The variable $m$ is most commonly used to represent slope.

$m=\dfrac{\Delta y}{\Delta x}$

The words rise and run are sometimes used to describe the slope of a line, especially when the line is given graphically. Rise corresponds to $\Delta y$ and run corresponds to $\Delta x.$

This gives the following definition for the slope of a line.

$m=\dfrac{\Delta y}{\Delta x}=\dfrac{\text{rise}}{\text{run}}$

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 (positive or negative) of each distance corresponds to the direction of the movement between points. Moving to the right, the run is positive; 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.

$m=\dfrac{y_2-y_1}{x_2-x_1}$

Here, $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line.
Rule

## Slope-Intercept Form

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

$y=mx+b$

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 $f(x)=2x+1,$ because it has a slope of $2$ and a $y$-intercept at $(0,1).$

Exercise

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

Solution
A line in slope-intercept form is given by the equation $y=mx+b,$ 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.
$m = \dfrac{y_2-y_1}{x_2-x_1}$
$m = \dfrac{{\color{#0000FF}{5}}-{\color{#009600}{1}}}{{\color{#0000FF}{5}}-{\color{#009600}{3}}}$
$m=\dfrac{4}{2}$
$m=2$
The equation, $y=mx+b,$ can now be rewritten using the knowledge that the slope, $m,$ is $2.$ $y=2x+b$ 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.
$y=2x+b$
${\color{#0000FF}{5}}=2\cdot {\color{#009600}{5}}+b$
$5=10+b$
$\text{-} 5=b$
$b=\text{-} 5$
Thus, the equation of the line passing through the two points is $y=2x-5.$
info Show solution Show solution
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, $y=mx+b,$

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

### 1

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 $(0,\text{-} 4).$

### 2

Replace $b$ with the $y$-intercept

The $y$-coordinate of the $y$-intercept can be substituted into $y=mx+b$ for $b.$ Here, substituting $b=\text{-} 4$ gives $y=mx-4.$

### 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 $m=\dfrac{\text{rise}}{\text{run}},$ 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 $y$-intercept and the arbitratily chosen $(2,2).$

From the lines drawn, it can be seen that the $\text{rise}=6$ and the $\text{run}=2.$ Therefore, the slope is $m=\dfrac{6}{2}=3.$

### 4

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. Here, substitute $m=3.$ $y=3x-4.$