# Functions

# Graphing Relations

Graphing a finite relation just means graphing a bunch of ordered pairs at once. Don't freak out. You can still draw the dots one at a time. It would be amazing if you could draw them all in one fell swoop, but we're guessing you don't have that many hands.

### Sample Problem

Graph the relation {(1, 2), (3, 4)}.

This problem is telling you to graph the ordered pair (1, 2) and the ordered pair (3, 4) in the same picture. A treasure map leading to two treasures! You are one jolly Roger.

When there are multiple ordered pairs close to each other, we can write the ordered pair next to its corresponding point if we want to keep track of which pairs go with which points. Since it can get pretty crowded in these graphs, we don't recommend you writing with a crayon, as badly as you may want to label these guys in burnt sienna. A sewing needle dipped in black ink works best. Or, you can just use a pen.

### Sample Problem

Graph the relation {(1, 2), (3, 4)}.

This is asking us to graph the ordered pair (1, 2) and the ordered pair (3, 4) in the same picture. We already did this in the previous example, but now we'll also label the points. This way, we can glance at the graph and instantly know who is who. It will be like slapping a name tag on them at a high school reunion. "(3, 4), is that you? You haven't aged a bit!"

As with individual ordered pairs, we can also look at a graph and determine what relation is being graphed. To do this we make a set containing all the ordered pairs we see in the graph. For you tennis players out there: game, set... graph!

### Sample Problem

What relation is graphed here?

The points in the graph, starting with the highest and ending with the lowest, are

(-2, 1)

(-2, 0)

(-2, -1)

(-2, -2)

To get the relation being graphed, we put all those ordered pairs into a set. Therefore, the relation graphed is

{(-2, 1), (-2, 0), (-2, -1), (-2, -2)}.

We could also describe this relation by the equation *x* = -2, where *y* is an integer from -2 to 1. Take your pick, since either one is correct. Mostly, it depends on how much you love parentheses.

The above examples are simple enough, but what about when we have an infinite number of points? We're going to go through a lot of ink that way, aren't we?

When graphing an infinite relation, we can't graph *all* the ordered pairs in the relation. However, we can sometimes graph enough ordered pairs to get an idea of what the relation looks like. Just as you might read the first four pages of War and Peace so that you can get a gist of what it's about before delivering a 20-minute oral report on it in your World Literature class.

### Sample Problem

Graph the relation |*y*| = |*x*|.

Let's find some of the ordered pairs in this relation, starting with small whole numbers (a very good place to start). When *x* or *y* is 0, the other must also be 0, so the relation at least contains the ordered pair

(0, 0).

Whole numbers don't get much smaller than that. When *x* or *y* is 1 or -1, the other can be 1 or -1. This gives us some other ordered pairs in the relation:

(1, 1)

(1, -1)

(-1, 1)

(-1, -1)

Similarly, we find the ordered pairs

(2, 2)

(2, -2)

(-2, 2)

(-2, -2).

Let's graph what we have so far:

It looks sort of like the dots are making a giant X. Doesn't *x* get enough play in algebra? Couldn't this graph be a *b* or a *q* or a *w*? Share the wealth, function-hog.

Anyway, it would be great if this pattern keeps up, because then we would know exactly what this relation looks like, but we're not 100% convinced just yet. Let's find some more ordered pairs in between the ones we already have. The relation described by the equation |*y*| = |*x*| will also contain the points

(0.5, 0.5), (0.5, -0.5), (-0.5, 0.5), (-0.5, -0.5)

(1.5, 1.5), (1.5, -1.5), (-1.5, 1.5), (-1.5, -1.5)

When we graph these as well, we find

Yup, that definitely looks like a giant X. It appears someone is trying awfully hard to cross out this graph.

We have now drawn in enough points that we can be almost positive we have identified the pattern, and that we now know exactly what this drawing will look like. If we were to graph more points in between the ones we already have, we would fill in the X even more. If we graphed every point in the relation, or at least every point that can fit on a piece of paper or on your computer screen, we'd see this:

Looks like this spider lost a few of its legs.

One tool that can be helpful for graphing a relation is a **table of values**. This is not an organized list of moral virtues that you should strive to uphold. Demonstrating a healthy dose of compassion won't help you sail through your functions test, although it might allow you to feel more sympathy for the classmates who fail it.

A table of values is a way to write a bunch of ordered pairs without taking up too much room on the paper. No need to kill trees needlessly.

To get our table started, we set up one column for *x* values and one column for *y* values:

We put numbers into the columns depending on the relation we're working with. Each row of the table will be an ordered pair in the relation.

### Sample Problem

Use a table of values to graph the relation *y*^{2} = *x*.

First, we set up our table of values:

Next, we start plugging in numbers. For each non-zero value of *x*, we'll have two values of *y* (a negative and a positive square root). So there will be repeats of our *x*'s but no repeats of our *y*'s. Apparently, those are not yet in syndication.

Each row of the table is an ordered pair in the relation. So far, we have the ordered pairs

(0, 0), (1, 1), (1, -1), (4, 2), and (4, -2).

Let's graph these and see what we have.

What is this, a sideways trapezoid? Based on the points we've drawn so far, that's entirely possible, but if we were to continue filling in dots between the ones we currently have, you would start to see that our picture has some curves. In fact, it is *very* curvy. Move over, Marilyn Monroe.

The relation actually looks like this:

To figure out what numbers to put in the table of values, we usually start with *x* = 0 and see what values of *y* can be used. Then we move outward from there to see what can happen when *x* = 1 or *y* = 1. Then we move out further, to *x* = 2 and *y* = 2. Sometimes we need to pick values in between the integers to see how the shape is filling in.

The idea is to get a good mix of all sorts of pivotal or interesting numbers to be sure we aren't missing any piece of the pattern. See h w conf sing it can be w en you' e mis ing ce tain pie es?