 We've been graphing in rectangular coordinates, but because I was trained the way that I was, I might slip from time to time and talk about Cartesian coordinates or Cartesian graphing. Well, where does that come from? A little explanation is in order. In 1637 the French mathematician Pierre de Fermat expressed a very important idea. Every algebraic equation in two variables corresponds to some geometric curve. From the same time, Rene Descartes expressed another idea. Every geometric curve can be described by an algebraic equation. Now what we're actually doing has more in common with what Fermat was describing than with what Descartes was describing. And so we should speak of Fermatian coordinates. But because Descartes was a famous philosopher, and Fermat was just a lawyer, we instead speak of Cartesian coordinates. So let's formalize Fermat's concept. The graph of an equation in x and y consists of all points hk, where x equals h, y equals k satisfies the equation. And what this means is that if x equals h, y equals k satisfies the equation, then the point hk is on the graph. If x equals h, y equals k does not satisfy the equation, the point hk is, wait for it, not on the graph. So for example, let's see if we can determine which, if any, of these points are the graph of y squared equals x cubed minus 3x plus 14. So let's take a look at our first point. We have x equals 0, y equals square root 14. And whether or not this point is on the graph is based on whether or not x equals 0, y equals square root of 14 satisfies the equation. So equals means replaceable, so let's replace x with 0, y with square root 14, and see if we get a true statement. And since the statement is true, 0, square root 14 is on the graph. How about our next point, 2, negative 4. We'll let x equals 2, y equals negative 4, and equals means replaceable, so we'll replace x with 2 and y with negative 4, and see if we get a true statement. And once again, we get a true statement, so we know that 2, negative 4, is on the graph. How about 4, negative 2. So here x is negative 4, y equals 2, we'll substitute those in, and this statement is true. I mean, false. Since this is a false statement, negative 4, 2 is not on the graph. Checking to see if a point is on the graph is relatively easy, it's a little bit harder to find points on the graph. So if I want to find two points on the graph, I need to find a pair of values, x equals h, y equals k, that satisfy the equation. And if it's useful to keep in mind, we already know how to solve equations in one variable, so let's transform this into an equation in one variable by choosing a value for the other. But choosing a value is hard. That's why we like living under political systems where voting doesn't matter. Well, maybe not. So let's make a choice. How about if x equals 1? If x equals 1, y must be a solution to this equation where every appearance of x has been replaced with a 1. And so that gives us this equation, but this equation looks hard to solve. Now the good thing about mathematics is that unlike politics, if you make a bad choice, you can unmake the choice. You can make a different choice and suffer no important consequences. So let's make a different choice. So we picked a value of x before, let's make a value of y. How about y equals 1? If y equals 1, then x must be a solution to the equation where every occurrence of y is replaced with 1. So x must be a solution to... Well this is a quadratic equation and we know how to solve it. This still looks like it's going to be hard to solve. Well, third time's the charm. Let's try another value and see if we get an easy equation. If we choose x equals 0, then our equation is going to become... And that's easy to solve. And so that gives us the solution x equals 0, y equals 2. And so we know that the 0.02 is on the graph. And if you're a politician, you can say, well we've done something, so let's go on to the next question. But if you're a good human being, or a mathematician, you might note that the problem involves more than just giving one solution. We actually want to find two points, we only found one point. So let's find another. So let's make another choice. Well how about if y is equal to 0? Then x has to be a solution to the equation. And again, this equation is also easy to solve. And we get two solutions, x equals 2, or x equals negative 2. And so that gives us two more points, x equals negative 2, y equals 0, x equals 2, y equals 0. Well, we actually did more than the problem asked us to do. We found three points on the graph. Let's talk about something like this. Now since this has the form of a formula for y in terms of x, we might just pick values of x and compute. So if we let x equals 0, substituting that in, we solve for y. So x equals 0, y equals 7, 0, 7 is a point on the graph. If we let x equals 1, substituting that in, we find that y is equal to 7 and 2 fifths. So x equals 1, y equals 7 and 2 fifths is also a point on the graph. So I know that most of you out there are wanting to deal with fractions, and you're really dying to pick a value of x, like 3, 7, or something like that. We might want to make this a little bit easier. And a useful thing to keep in mind is that if you multiply a fraction by its denominator, the fraction becomes an integer. And so that suggests we might want to choose x equal to 5, the denominator of our fraction. If we let x equals 5, then substituting that into our formula for y gives us 9. And so the point on our graph, x equals 5, y equals 9, that's the point 5, 9.