A systematic way of graphing functions
Mathematical equations or functions often represent lines or curves in a plane. It's important to be able to see an image of those lines or curves. To do that, we'll need a systematic way to represent their equations on a graph. We'll be sticking with 2-dimensional graphs, but three or more dimensions are also possible.
We owe the graphing system we've used for more than 300 years to French mathematician and philosopher Rene Descartes [Day-kart'] (1596-1650). After him, we call it the Cartesian coordinate system.
Descartes gave us a few other pieces of our math toolkit, including our lettering system. For the most part, we use the letters:
- $i, j, k, l, m, n$ to represent integers,
- $x, y, z$ to represent real numbers,
- $z$ to represent complex numbers, and
- $f, g, h$ for function names, as in $f(x) = x^2 + 1.$
Descartes is also famous for his philosophical work. One of his best known statements, on what it means to exist, is "I think, therefore I am."
Definitions
The Cartesian coordinate system consists of two axes (plural of axis) drawn at right angles to one-another. The vertical axis is called the ordinate, and is usually labeled as the y-axis. The horizontal axis is called the abscissa and is usually labeled as the x-axis. We don't usually use "ordinate" and "abscissa" too much these days, but you should know that you might hear them.
The two axes divide the Cartesian plane into four quadrants, which we label, beginning with the upper right and moving around counterclockwise, as Quadrants I, II, III and IV, using Roman numerals. If each axis is like a 1-dimensional number line, where x is positive on the right, negative on the left, and y is positive above the x-axis and negative below it, we have these results for each quadrant:
| Quadrant | x & y |
|---|---|
| I | x positive, y positive |
| II | x |
| III | x |
| IV | x positive, y |
The center of the Cartesian grid, the intersection point of the axes, is called the origin of the graph.
Any point in the Cartesian plane can be located by specifying its two coordinates, x and y.
These are usually given in an ordered pair, (x, y), where the x-coordinate is always written first. We do this as a long-standing convention to avoid confusion
In the graph above, four points are plotted. The point (-8, 4) is located by moving 8 units to the left of the origin along the x-axis, then moving up (in the + y direction) 4 units. The point (6, -4) is found by moving 6 units to the right of the origin, then four units downward.
The origin of the Cartesian system always has the coordinates (0, 0), and it's sometimes just labeled "O."
Cartesian graphs can be fancy, or they can be quick sketches; it depends on what you need. the axes can be labeled with units or other labels that are relative to the problem being illustrated. Here's one I like to hang in my classroom.
Lines on the Cartesian plane
Now we can connect points on the Cartesian plane to make figures. The simplest is a line. Here's an example.
One thing to notice is that on this line, there's a systematic relationship between the points.
Here's a blowup of part of the first quadrant (quadrant I) of that graph. Notice that to get from one point to the next, we simply move over one unit along the x-axis and up two units. This is always the case for lines.
There's always a simple rule like that for getting from one point to another.
The difference between two y-coordinates divided by the difference between two x-coordinates is constant, and is called the slope, usually given the label m.
$$m = \frac{y_2 - y_1}{x_2 - x_1},$$
where the subscripts, 1 and 2, just distinguish the two points we selected. Any two will do. Using the top two points, the slope of this line is
$$m = \frac{7 - 5}{3 - 2} = 2.$$
Special points — intercepts
Often we're interested in where graphs cross the Cartesian axes. These are referred to as intercepts. The x-intercept is where a graph crosses the x-axis (there may be more than one). These are often called roots; you'll learn a lot more about those later. The y-intercept is where a graph crosses the y-axis.
The x-intercept(s) will always have a coordinate like (x, 0), where x is some number, and can be zero. The y-intercept will aways be written like (0, y), where y is some number, and can be zero. The coordinate (0, 0) – the origin – may be both an x- and y-intercept.
Curved graphs can cross the x-axis many times, so they can have several x-intercepts or roots. Generally such a graph has only one y-intercept, but closed figures like circles and ellipses (see below) can have more than one y-intercept.
Estimating
This is a good time to notice that not every point on a graph passes through a nice grid point on the Cartesian plane. Sometimes when sketching the graph of a line or curve, we need to estimate the distance between marks on the axes in order to plot a point. That's OK. Graphs you draw by hand, even on graph paper, are just sketches. If you want a more accurate version of your figure, it's easy enough to use a computer or calculator.
Closed curves
We can plot all kinds of graphs on the Cartesian plane. Here are two more that you don't need to know how to graph just yet. They're just for you to see what kinds of things can be graphed using the Cartesian coordinate system.
The magenta curve is an ellipse and the green curve is a special ellipse called ... wait for it ... a circle.
Download blank Cartesian grids
You can download some nice blank Cartesian grids for your own practice here. Enjoy!
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