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Introduction to :

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

yy1 = m(xx1)

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. (x1, y1) is so fixed, as a matter of fact, that it is never going to birth a litter.

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

Sample Problem

Graph the equation

y – 3 = 4(x – 0.5).

Add 3 to each side to find

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 are given either a graph or two points on the line. By the way, when you are 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.

Point-slope form is

y y1 = m(x x1)

where m is the slope of the line and (x1, y1) is a point on the line.

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, . We get the equation

which can be rearranged to give

.

Point-slope form makes sense, because if we rearrange the point-slope equation

yy1 = 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 yy1 as the rise and xx1 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.

Example 1

Find the equation of the line shown below.


Exercise 1

Rewrite the equation to be in slope-intercept form: y + 1 = 2(x – 0.5).

Exercise 2

Rewrite the equation to be in slope-intercept form: y-2 = 7(x + 3).

Exercise 3

Rewrite the equation to be in slope-intercept form: y + 3 = 0.5(x + 3).

Exercise 4

Find the point-slope equation for the following line, using the left-most point (the point with the smaller x-coordinate) as (x1, y1).

Exercise 5

Find the point-slope equation for the following line, using the left-most point (the point with the smaller x-coordinate) as (x1, y1).

Exercise 6

Find the point-slope equation for the following line, using the left-most point (the point with the smaller x-coordinate) as (x1, y1). 

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