Point-Slope Form

A line is in point-slope form if it looks like

y – y₁ = m(x – x₁)

where y₁, x₁, and m are real numbers. Here (x₁, y₁) is a fixed point on the line, and m is the slope of the line. In fact, (x₁, y₁) is so fixed that it's never going to birth a litter. #petjokes

To graph an equation given in point-slope form, it's often easiest to rewrite the equation in slope-intercept form.

Sample Problem

Graph the equation y – 3 = 4(x – 0.5).

First we add 3 to each side:

y = 4(x – 0.5) + 3

Then simplify to get:

y = 4x + 1

From here, we can graph the equation using the y-intercept and the slope:

Point-slope form is most useful for finding the equation of a line when you're given either a graph or two points on the line. By the way, when you're given a graph, say "thank you" and don't ask for any more. You don't want to look a gift graph in the mouth.

Sample Problem

Find the equation of the line shown below.

First we need to pick a point (x₁, y₁). Let's take a point with nice, even integer coordinates. Yes, 14,838 and 372,410 are even numbers, but we can do better. Let (x₁, y₁) be the point (0, 1), so x₁ = 0 and y₁ = 1.

Now we need to find the slope, m, of the line. Pick another point on the line and look at the rise and run between the two points. Don't look at anything else if you can help it; this slope is a little self-conscious.

We can conclude that .

To write the equation for the line, we use the blueprint y – y₁ = m(x – x₁) and plug in the values x₁ = 0, y₁ = 1, and .

Rearrange that bad boy to get:

Here's a fun trick (and yeah, we're using "fun" very, very loosely): if we rearrange the point-slope equation y – y₁ = m(x – x₁), we find:

If we fix a point (x₁, y₁) on the line, then for any other point (x, y) on the line we can think of y – y₁ as the rise and x – x₁ as the run. We know how much you love your visual aids, and we would never dream of depriving you of them, so here you go:

Since m is the slope of the line, saying is really just saying , which we know is true. And just like that, we've got a handy new formula for finding the slope.