# Functions

### Topics

We already have a place set up and waiting for us to graph all over it, so let's take advantage. We'll start by graphing the ordered pair (2, 3). In order to do so, we start at the origin, go 2 to the right, 3 up, then draw a dot where we landed. It's almost as if we're following a treasure map. First we travel left or right along the *x*-axis to find where "*x*" marks the spot, then we travel up or down along the *y*-axis when we want to know "why" the treasure wasn't in the chest where it was supposed to be, and we are trying to track down the wise guy who gave us this bogus map in the first place.

The first number in an ordered pair tells us how far to go left or right on the *x*-axis (horizontal number line), and the second number in the ordered pair tells us how far to go up or down on the *y*-axis (vertical number line). Since *x* comes before *y* in the alphabet, *x* goes with the first number and *y* goes with the second number.

These numbers are called **coordinates**. They "coordinate" with one another to arrive at a certain spot on the graph. The first number in an ordered pair is the *x*-coordinate and the second number is the *y*-coordinate. The dot we draw to represent the ordered pair is called a **point**. You can look at a point, but don't point at it. That's rude.

When we graph a point by traveling along the *x*-axis and then the *y*-axis, it is almost as if we are traveling along two sides of an imaginary rectangle. It should come as no surprise then that we are using something called the **Rectangular Coordinate System**, also known as the **Cartesian Coordinate System**. You may see this referred to as "Cartesian Coordinate System" more often, which is unfortunate, as there is no shape called a cartesle. However, we can just pretend that there is one, and that it looks exactly like a rectangle.

### Sample Problem

Graph the ordered pair (5, -2).

The *x*-coordinate is 5 and the *y*-coordinate is -2, which means we start at the origin, count 5 to the right on the *x*-axis and then count 2 down on the *y*-axis. We have a negative *y*-coordinate this time, so our yo-yo will be headed downward.

Technically, a point is what we get when we graph an ordered pair. In practice, the phrase "ordered pair'' and the word "point'' are used interchangeably. You can try working this into everyday conversation. "Hm... you have a good ordered pair there, " or "Could you ordered pair me in the direction of the Post Office?"

Okay, so maybe it doesn't work as well in English.

We may talk about the "point'' (3, 4), which has coordinates 3 and 4. We may be asked to graph a point, instead of an ordered pair. You can't go wrong as long as you remember that they are one and the same.

In addition to using coordinates to graph a point, we can also go backwards; that is, we can look at a point on a graph and figure out its coordinates. It's like starting with a treasure and then looking for the treasure map. We're not sure who in their right mind would do things in that order, but here goes. To set our minds at ease, we'll assume for the time being that this process is of more value when dealing with functions than when dealing with gold doubloons.

### Sample Problem

What are the coordinates of the point graphed below?

To get to this point from the origin we have to go 1 right (along the *x*-axis) and 2 up (along the *y*-axis). Therefore, the coordinates of the point are (1, 2). At least it isn't a long journey from the origin and there are no layovers. It would be a pain if we had to stop at (1, 1) for a couple hours while waiting for a connecting coordinate.

So far, all the points we've been graphing have had integer coordinates. These points are easy to graph, but in more advanced problems we'll also need to graph points with non-integer coordinates. On the downside, things will get a little trickier. On the upside, now that we don't have to stick to a grid, we will be able to graph some more interesting pictures.

As with the number line, we can draw points with non-integer coordinates in approximately the right place, and then label the points so other people know exactly where they are. Hopefully nobody will get out a ruler just to prove that your dot is off by a half-millimeter. If they do, they have too much time on their hands.