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# Parametric equations

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## Parametric equations

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.)

To 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 = t ^{2}, 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 2**

Plot 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:

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 sin^{2}θ + cos^{2}θ = 1 to eliminate the letter θ (see example 2).

In these examples we shall use the same parametric equations we used above.

**Example 1**

So, 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 2**

This is the Cartesian equation for the **ellipse**.

In 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 1**

This means that when t = -2, (for example), the gradient at (4, -4) is -0.5

**Example 2**