A parametric equation is where the x and y coordinates are both written in terms of another letter. This is called a parameter and is usually given the letter t or θ. (θ is normally used when the parameter is an angle, and is measured from the positive x-axis.) Drawing the graphTo draw a parametric graph it is easiest to make a table and then plot the points: Example 1 Plot the graph of the following parametric equation: x = t2, y = 2t. The first thing to do is create a table which will tell you what x and y are for a selection of values of t: Now we can plot the points (4, -4), (1, -2), (0, 0)... etc to get the curve: Example 2Plot the graph of the following parametric equation: x=3sinθ, y=4cosθ As θ is used in the equation, we know this is an angle. Hence, we insert values of θ which are likely to give us a good range of points to plot on our graph: Finding the Cartesian equation There are two techniques for finding the Cartesian Equation from a Parametric equation, depending on whether the parameter is 't' or 'θ'. If the parameter is 't' then rewrite one equation as t =... and substitute this into the other equation (see example 1). If the parameter is θ, use a trigonometric relationship like sin2θ + cos2θ = 1 to eliminate the letter θ (see example 2). In these examples we shall use the same parametric equations we used above. Example 1So, to find the Cartesian equation use t = y/2 to get: Now we can just re-arrange to get the equation in terms of y: This is the equation of the parabola. Example 2This is the Cartesian equation for the ellipse. Finding the gradient of a parametric curveIn order to understand this you will need to have a good grasp of differentiation (see the differentiation topic). To find the gradient from a parametric equation we use the chain rule (which is explained in the differentiation topic): This is easiest to understand using our examples. Example 1This means that when t = -2, (for example), the gradient at (4, -4) is -0.5 Example 2