# Functions

# Point-Slope Form

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

*y* – *y*_{1} = *m*(*x* – *x*_{1})

where *y*_{1}, *x*_{1}, and *m* are real numbers. Here (*x*_{1}, *y*_{1}) is a fixed point on the line, and *m* is the slope of the line. (*x*_{1}, *y*_{1}) 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* = 4*x* + 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 *– *y*_{1} = *m*(*x *– *x*_{1})

where *m* is the slope of the line and (*x*_{1}, *y*_{1}) is a point on the line.

### Sample Problem

Find the equation of the line shown below.

First we need to pick a point (*x*_{1}, *y*_{1}). 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 (*x*_{1}, *y*_{1}) be the point (0, 1), so *x*_{1} = 0 and *y*_{1} = 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 – *y*_{1} = m(x-*x*_{1})

and plug in the values *x*_{1} = 0, *y*_{1} = 1, . We get the equation

which can be rearranged to give

*.*

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

*y* – *y*_{1} = m(*x*-*x*_{1}),

we find

.

If we fix a point (*x*_{1}, *y*_{1}) on the line, then for any other point (*x*, *y*) on the line we can think of *y* – *y*_{1} as the rise and *x* – *x*_{1} 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.