# Functions

### Topics

A linear equation in **slope-intercept form** looks like

*y* = *mx* + *b*

where *m* and *b* are real numbers and *m* is the slope of the line. Get to know slope-intercept form, because it's going to be one of your new best friends. You two are going to have sleepovers, do each other's hair and play Fantasy Date. Then slope-intercept form is going to go behind your back and kiss Davy Phillips, and it's going to cause a whole big thing. We're getting ahead of ourselves.

When *x* = 0, we find

*y* = *m*(0) + *b* = *b*

so *b* is the *y*-intercept (the value of *y* when *x* = 0).

To graph a line in slope-intercept form, we graph the *y*-intercept and then use the slope to plot another point or two. Or seven, if you have the time.

### Sample Problem

Graph the line *y* = 2*x* + 5.

When *x* = 0, *y* = 5, so the *y*-intercept is at 5.

The slope is 2, so as *x* moves right by 1, *y* moves up by 2 to give us the point (1, 7):

Then we connect the dots:

To check our work, we can plug in some other value of *x* and make sure the point we get is actually on the line. If the point was on the line but is no longer, it probably has AT&T.

When *x* = 3,

*y* = 2(3) + 5 = 11,

so we should have the point (3, 11) on the graph. Thankfully, we do.

To read off the equation of a line in slope-intercept form, we need to figure out the *y*-intercept, the slope of the line, and the *z*-intercept. Psych! There is no *z*-intercept. We're just checking to see if you're paying attention.

### Sample Problem

Find the equation of the line graphed below.

We can see from the picture that the *y*-intercept is 2, so the equation for the line is

*y* = *mx* + 2.

Now we need to figure out *m*, the slope of the line. We can see from the picture that as *x* moves right by 1, *y* increases by 3.

The slope of the line is *m* = 3, so the final equation for the line is

*y* = 3*x* + 2.

We can also find the equation of a line in slope-intercept form if we're given two points but are not given the *y*-intercept. First we find the slope, then we find the *y*-intercept by sneaky methods. Put on your black mask and grab your throwing stars. We're going ninja.

### Sample Problem

Find the equation of the line in slope-intercept form.

First, we find the slope. Slope is , which we can figure out from the graph:

The slope is . Slope is *m* in the equation

*y* = *mx* + *b*

so we know the equation for this line is

.

All we need to do is figure out *b*. Our "Big Book of What Variables Equal" is in the shop, so we'll have to do this manually. Since we know the points (1, 4) and (10, 7) are on this line, each of them must satisfy the equation. We can use one point to find *b*, and the other to check that we're right.

To find *b*, we know the point with *x* = 1 and *y* = 4 must be on the line, so let's plug those values into the equation:

.

Now we solve for *b* to find

.

The equation for the line should be

*.*

To make sure we're right, let's check that the other point (the one with coordinates *x* = 10 and *y* = 7) satisfies this equation. If not, maybe we can buy it a steak.

The left-hand side of the equation is 7

and the right-hand side is ,

which is also 7. Both the points (1, 4) and (10, 7) satisfy this equation, so we found the right equation for the line. Oh, good. We didn't want to have to shell out for that steak.