1YParametricEqns

Parametric Equations

Functions can be written in terms of a third variable called a __parameter__.

We commonly use t for the parameter (or sometimes θ).


 * Both x and y are expressed as a function of t.
 * Any given value of t will provide the coordinates of a single point on the graph.

Example ... ... x(t) = 2t

... ... y(t) = 4t + 1

substitute various values of t to find points on the graph. ... ... t = 0 gives (0, 1) ... ... t = 1 gives (2, 5) ... ... t = 2 gives (4, 9) ... ... ... etc.

These points can be plotted and joined to create the graph (see right).

Finding the Cartesian Equation

To find the cartesian equation of the graph, combine the two equations in some way so that t is eliminated.

In the example,, x = 2t can be rearranged to make t the subject, then substitute into y = 4t + 1.

math \\ . \qquad x=2t \qquad. \\ . \\ . \qquad t=\dfrac{x}{2} math

substitute into y(t) math \\ . \qquad y=4t+1 \qquad. \\ . \\ . \qquad y=4 \left( \dfrac{x}{2} \right) +1 \qquad .\\. \\ . \qquad y=2x+1 math

This is a straight line with gradient 2 and y-intercept of +1.

Restrictions on the values of t, will affect the domain and range of the cartesian equation.

Eg: if t Î [1, 5] then the domain of the cartesian equation will be x Î [2, 10] (because x = 2t)

Later in this course, we will use t for time. In this case, t is always t __>__ 0.

For general parametric equations, t Î R

**See also:** www.mathsonline.com.au Y12Extension --> Functions --> Parabolas --> Lessons 1, 2

Parametric Equations on the Calculator

Your calculator is normally set to Function mode. It can be changed to parametric mode in the TYPE menu (select ParamType).

The equation screen will then change to show xt1= and yt1= for each equation. You can now enter the parametric equations (access the letter t from the keyboard).



You may need to alter the domain of t. This can be done in the resize window. While in parametric mode, **tmin** and **tmax** will appear in the resize window underneath xmin, xmax, ymin and ymax.



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