Analytic geometry...

Diagram 1

"Analytic geometry is just a fancy name for graphing. You probably did plenty of it in algebra. It is a handy way to deal with equations.

In the first diagram, graphed a straight line, y=x/3+1. You can draw a graph by trying a few values of x and y. For example, I tried x=0 and I found that y=1. There is a little arrow pointing to that point, in the diagram. What do I get for x, when y=0 (the question mark in the diagram)? Well, I get x=-3. I can keep on graphing every point of our equation, a time-consuming process. A computer program may do this for many values of x and y, and draw what looks like a continuous, straight line. But I observe that y=x/3+1 is a typical equation of a straight line. Then I only need to draw two points (0,1) and (-3,0), and then I can draw the line through these two points (Euclid showed that two points determine a line). That line is the graph of our equation.

Our line has a slope of 1/3. The slope is the measure of how steep the line is. The graph goes up one for every three it goes to the right. On the freeway, we see signs that warn truck drivers of a 2% grade. This is a slope of 0.02 (which is 2%). That is not very steep for a line. But it is plenty steep for a freeway.

Here are the graphs of two slightly more complicated equations, a parabola (y=x²) and a hyperbola (y=1/x). We dealt with these in algebra. But, they caused us some problems, which you may not have noticed. It is difficult, using only algebra, to find the slope of these curves (at any given point). Why would we want to complicate our lives by finding the slope of a curve? Well, it is absolutely vital in physics and engineering. It is even of some interest to a skier.

This graph shows us something about functions (a function is a special kind of equation). This could be a graph of the equation x=y² (y=x² turned on its side). But such an equation is awkward for some uses. Using sqr() for the square root function (since my choice of symbols is limited), y=± sqr(x) is what we get, when we solve for y. For most values of x, we get two values of y. This is perfectly acceptible in analytic geometry. But there are situations where it is much simpler to have only one value of y, y=+ sqr(x). Such an equation is called a function. A function is like a machine in which you plug in an x and out pops a y (and only one y). If I had turned the y=x² graph on its side, I would end up with a graph of two functions, y=+ sqr(x) and y=- sqr(x), just as I can graph two straight lines on the same graph. Instead, I only graphed y=+ sqr(x), which is a function. In fact, when we write y=sqr(x), we mean the positive square root. If we want both positive and negative, we write y=± sqr(x).

So, what about slope (and area, which is also a difficult problem using algebra)? Well, now that we can graph these equations, we can learn a little calculus. And slopes and areas often become very easy to find, even with curves. But, I will deal with calculus in other articles. Anyway, calculus relies heavily on algebra, analytic geometry, and functions", I quoted...