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.
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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, I have 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.
Addendum:
A short way to write a particular function of x is f(x) or g(x). F(x) is pronounced "f of x." Some of these functions are sin(x), log(x), 3x+2, or 3x²+2x-1. A function has been described as a machine or box. You push a value of x into the box, and out pops the value f(x) or y. Try y=3x+2. When x is 10, y is 32. We put 10 into the box and out pops 32.
The graph of one particular circle is 1=x²+y². This is not a function. Most x values produce two y values, and vice versa. So, it is normal to describe this as two functions: y=sqr(1-x²), and y=-sqr(1-x²). This simplifies the handling of a circle. For example, it easy to calculate the area of either of these semi-circles, and we can then double the result to get the area of the whole circle.