SlopePoint form: $y  y_1 = m (x  x_1)$
What is point slope form?
The pointslope form (or slope point form) is a way that straightline equations are written. Other examples of ways straightline equations are written includes standard form equations, and slope intercept form equations.
What is given when you’re faced with a question dealing with an equation in pointslope form is that you’ll get a point (an xcoordinate and a ycoordinate), and a slope called m.
The point slope form formula looks like this:
y – y1 = m(x – x1)
The x1 and y1 are the specific x and y from the point that you are given to work with. The regular x and y with no numbered subscripts are just the regular x and y that are always in equations for lines. Don’t be too worried about these unknowns as we’ll learn how to work with them! Just know that for now, the point slope form helps show us the equation of a line.
How to write point slope form
What exactly does pointslope form look like when we put it into writing? What better way to explore that than through an example question?
Question:
Given this point and the slope, write the equation in point slope form and sketch the graph:
(3, 2), m = 2/3
Solution
Let’s go back to the formula that we had for the pointslope form, which was:
y – y1 = m(x – x1)
Like we previously mentioned, the regular y and x (with no numerical subscript beside it) will be kept as variables. So you won’t have to solve for them here. Then, let’s just plug in the info we’re given.
y – y1 = m(x – x1)
y  2 = (2/3)(x3)
y  2 = (2x/3)  (6/3)
y = 2/3x  2 + 2
y = 2/3x
What we did was try to isolate the y onto one side, while keeping the x on the other. This equation can now help you sketch a graph! You may also notice that this equation is currently in the slopeintercept form now. As a refresher, the slopeintercept form is
y = mx + b
It’s obvious in this equation that there is no yintercept since the “b” from our now slopeintercept form equation doesn’t exist. This means that we’ll have our first point at the origin (0, 0). The slope tells us to go up 2 and 3 to the right. This is the slope formula coming into play. That puts a point at (3, 2), which matches the point that we were originally given in the question. This validates that we’re on the right track since that point should be on the graph.
Then just connect the two points and you’ll have a graph that is always increasing to the right, and decreasing to the left.
As you can see, when we are given a set of coordinates and the slope, we can put them into a point slope form equation to help us turn it into a slopeintercept form equation through isolating the y to one side of the equation. This in turn helps us with graphing out the equation. Again, at first it may seem daunting to have two sets of unknowns, but once you realize you won’t actually have to solve one set of them, it makes it easier.
You can find an interactive chart at the bottom of this page that can let you see the relationship between point coordinates and the slope of a line, and the point slope equation. Play around with it and watch how the two changes one another!
SlopePoint form: $y  y_1 = m (x  x_1)$
Lessons

a)
Overview: Slopes of lines

b)
Slope intercept form VS. General form VS. Slopepoint form


3.
Given the info of slope and two other points on the graph, find the missing variable.

4.
Given a point and the slope, write the equation in slopepoint form and sketch the graph.