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 = 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:

Table

Now we can plot the points (4, -4), (1, -2), (0, 0)... etc to get the curve:

Curve

Example 2

Plot the graph of the following parametric equation:

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:

Example 2

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 1

Parametric equations

So, to find the Cartesian equation use t = y/2 to get:

Equation for x

Now we can just re-arrange to get the equation in terms of y:

Cartesian form

This is the equation of the parabola.

Example 2

Parametric equations

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):

Chain rule

This is easiest to understand using our examples.

Example 1

Example 1

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

Example 2

Example 2

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