We now recall a highly useful construct called the Cartesian or coordinate plane. It was invented by Rene Descartes in the 17th century. And yet, it’s strangely familiar to us today in view of GPS coordinates. What is it about? Well, we have a flat infinite plane. And we place on this plane a horizontal axis, as we did for the real numbers. We choose where the 0 will be. And then through that point we place another copy of the real line, but this time vertically. These two axes together will allow us to precisely describe the position of any point on the plane. Take this one, for example. How do we determine exactly where it lies and describe it to someone else?
If we take a vertical line through the point, it will intersect our original horizontal axis in one point. As we know, that defines a real number. Let’s call it x. Similarly, if we take a horizontal line through the point, it will intersect the vertical axis in a point, which will in turn define a number called y, let’s say. Well that pair of numbers– x, y– is called the pair of coordinates of the point we started with. Notice that we have placed them in parentheses and they describe what we call an ordered pair– the order of x and y matters. Now, in this example I’ve chosen, you can see that the coordinates will both turn out to be positive numbers.
If I had a point in the lower left quadrant, for example, I could proceed exactly the same way to find the coordinates. But in that case, the x would be negative and the y would be negative as well. Similarly, if I looked in the upper left quadrant, I would get a negative x but a positive y. And in the lower right quadrant, a positive x and a negative y. The usual terminology for these four quadrants is in this order– first, second, third, and fourth. The point at the intersection of my two axes is called the origin. Its coordinates are obviously 0, 0.
More terminology– the horizontal axis for obvious reasons is referred to as the x-axis and the vertical axis is referred to as the y-axis.
Sometimes you call x and y the abscissa and the ordinate. Let’s now talk about the graph of a function– a most interesting concept. I give myself a function f and some notation. We often write y equals f of x, y being a substitute for the values of f. When you do this, x is called the independent variable and y the dependent variable. For evident reasons, I think. You normally choose x first and then the value of x determines the dependent variable y through the function f. The graph of the function f means all points whose Cartesian coordinates are of the form x, f of x.
And it is a very useful tool to provide a graphic image– literally– a graphic image of the function. Let’s take an example– the function f of before. I give myself coordinate axes. I’m going to indicate what the scale is on these axes for x and for y. You’ll notice the scale is different, but that’s all right. This is very commonly the case. Let us call the graph of the function capital G. Here’s a way to generate points that are on the graph– I choose a value of x, and I evaluate f, and that will give me the y. For example, f of 0 is equal 1.
That means, by definition, that the point 0, 1 is on the graph of the function. Let’s put it into the picture. There it is. Similarly, f of 1 is equal to 3. That tells me the point 1, 3 is on the graph. I add that to the plot that I’m making. f of 2 is 5– gives me this point. And f of 3 is 7– gives me yet another. I can also take negative values of x. f of minus 1 is minus 1, which means the point minus 1, minus 1 is on the graph. Let’s add it to the picture. When we have enough points, we can comfortably draw the graph.
And you know, it looks a lot like a straight line. In fact, it is a straight line. It’s a straight line because f is an affine function. What does affine mean? A function f is affine if its defining formula is of the form mx plus b for two parameters m and b. It’s a general fact that the graph of an affine function is a line. The parameter m is called the slope of the line. And in fact, it measures how steep the line might be and whether it’s going up or down. If m is positive, for example, the line goes up as we move to the right. If m is negative, it goes down.
And if m is 0, then you have a flat line, a constant function. The parameter b is called the y-intercept of the line for the following simple reason– if you look at the point where the line intersects the y-axis, it describes the value y equals b. We’ll have more on graphs in the next segment.