Slope Intercept Form

The Slope Intercept Form for an equation of a line is $y=mx+b,\:$where $m=the\:slope\:$and $\left(0,b\right)=\:the\:y-intercept.$

Let's review what the slope is:

Slope

The slope m of a line through points $\left(x_1y_1\right)\:$and $\left(x_2,y_2\right)\:$is defined as follows.

$m=\frac{rise}{run}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1},\:\:where\:\:\Delta x\ne0$

That is, the slope of a line is the change in y divided by the corresponding change in x, where the change in x is not 0.

Visual Representation of Slope

The slope is a numerical measure of the steepness and orientation of a straight line. Slope is independent of the choice of points on a line. That is, the slope is the same no matter which pair of distinct points we use to find it.

1. A line with positive slope rises from left to right. The corresponding linear function is increasing on its entire domain.
2. A line with decreasing slope falls from left to right. The corresponding linear function is decreasing on its entire domain.
3. A line with a slope of zero is flat from left to right. The corresponding linear function is constant on its entire domain.
4. A line with undefined slope is vertical. There is no corresponding linear function, as this relation would not pass the vertical line test.

Slope as an Average Rate of Change

Since the slope is the quotient of the vertical change is y divided by the change in x, slope gives the average change in y for a unit change in x.

$Average\:rate\:of\:change\:on\:the\:inclusive\:interval\:from\:a\:to\:b=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

This is another way to write the slope definition using function notation.

As well as using it for linear functions, sometimes this formula is used to find the average rate of change of a nonlinear function, given two points on the graph of the function. 

Finding the Slope of a Line Given Two Points

Find the slope of a line through the points $\left(-7,2\right)\:$and $\left(5,3\right).$

Solution

Using the definition of slope,  

$Slope\:=\:m\:=\:\frac{\Delta y}{\Delta x}=\frac{3-2}{5-\left(-7\right)}=\frac{1}{5+7}=\frac{1}{12}$

Slope-Intercept Form

The slope-intercept form for the equation of a line is:

$f\left(x\right)=mx+b,\:$or $y=mx+b,\:$where $m=the\:slope\:$and $\left(0,b\right)=\:the\:y-intercept.$

When an equation for a line is given in this form, we use the following method for graphing the line:

1. Graph the y-intercept, $\left(0,b\right).$
2. Use the fact that $m=the\:slope\:$to find one or more other points on the line.
3. Connect the points and graph the line.

Example

Here is an example:

Graph $y=3x-6.$

Solution