 Back before the internet, cavemen used to sit around and do other activities. A popular pastime was connect the dots, which consisted of a set of numbered dots. By connecting the dots in order, you could form a surprising picture. In mathematical terms, each point in the plane is specified by three values, the x and y coordinates, and a third parameter that put an order on the point. And we can go a little further, it's sometimes convenient to express our coordinates as functions of another variable. The new variable is the parameter, and the coordinates are functions of the parameter. And this gives us the parametric equations for a curve or a surface. And by ordering the points on the curve based on the parameter, we can produce a curve. For example, let's sketch the parametric curve x of t, y of t, where some values of x of t and y of t are shown. Now to graph this, the x and y coordinates of the point come from the values of x of t and y of t. And so we'll plot the point and record the parameter value. So we see x of negative 2 is equal to 1, and y of negative 2 is equal to 1. And so this says that the curve goes through 1, 1 with parameter value t equals negative 2. So we'll plot the point and remind ourselves that this is where t is equal to negative 2. Similarly, x of negative 1 is negative 2, and y of negative 1 is 4. And so this says that the curve goes through the point negative 2, 4 with parameter value negative 1. And we'll plot the remaining points. And to sketch the curve, we'll connect the dots in order of the parameter values. And as with curve sketching, we'll start out by just using a connect the dots picture and not worry about curvature. Well, what if you're not given the table of values, but let's say we want to graph the parametric curve given by x of t, y of t, where x of t is t squared, and y of t is equal to t. So when you first learn to graph equations, you plot it points. And so we'll choose values of t and compute x of t and y of t. So if t is equal to 0, we get if t is equal to 1, we get if t is equal to 2, we get. Now, when you graphed curves, you also chose negative values of x. And so let's pick some negative values of t. So if t is equal to negative 1, we get if t is equal to negative 2. And again, we can plot these points. And again, we'll remind ourselves what the t values are. And connecting the dots in order of the t values produces the curve.