College Preparatory Mathematics

Section 2.4 Point-Slope Form

The slope-intercept form has the advantage of being simple to remember and use, however, it has one major disadvantage: we must know the y-intercept in order to use it! Generally, we do not know the y-intercept, we only know one or more points (that are not the y-intercept). In these cases, we can’t use the slope intercept equation, so we will use a different more flexible formula. If we let the slope of an equation be \(m\text{,}\) and a specific point on the line be \((x_1, y_1), \) and any other point on the line be \((x, y) \text{.}\) We can use the slope formula to make a second equation.

If we know the slope, \(m\) of an equation and any point on the line \((x_1, y_1) \) we can easily plug these values into the equation above which will be called the point slope formula.

Point - Slope Formula: \(y - y_1 = m(x - x_1)\)

Often, we will prefer final answers be written in slope intercept form. If the directions ask for the answer in slope-intercept form we will simply distribute the slope, then solve for \(y\text{.}\)

An important thing to observe about the point slope formula is that the operation between the \(x\)’s and \(y\)’s is subtraction. This means when you simplify the signs you will have the opposite of the numbers in the point. We need to be very careful with signs as we use the point-slope formula.

In order to find the equation of a line we will always need to know the slope. If we don’t know the slope to begin with, we will have to do some work to find it first before we can get an equation.

World View Note: The city of Konigsberg (now Kaliningrad, Russia) had a river that flowed through the city breaking it into several parts. There were 7 bridges that connected the parts of the city. In 1735 Leonhard Euler considered the question of whether it was possible to cross each bridge exactly once and only once. It turned out that this problem was impossible, but the work laid the foundation of what would become graph theory