10.5 Parametric Equations

This section was all about parametric equations

Most equations just use x and y variables, but parametric equations use a third variable, which is often t for time and is called a parameter.  You can then write x and y as functions of the parameter to get two parametric equations.

Graphing Parametric Equations:

Here's an example of a parametric equation and its graph:

x = t - 2

y = t + 1

To graph this, you can pick values of t and plug them into the equations to calculate the coordinate points of the graph.  For example, when t = -2, the x value is -4 and the y value is -1 so the coordinate is (-4, -1).  This is what the graph should look like:

Eliminating the Parameter:

Once you've graphed the parametric equations, you can rewrite it as an equation using only x and y.  The equation for the line above is y = x + 3.

You can also eliminate the parameter algebraically without graphing to get, as the book says, a rectangular equation, which is just a normal equation with two variables.  You do this by solving for the parameter in one of the equations, then plugging that into the other equation.

x = t -2
t = x +2

y = t + 1
y = x + 2 + 1
y = x + 3


Finding Parametric Equations:

 

You can  find parametric equations from a rectangular equation.

ex.  x² + y² = 1

There are many different ways to rewrite this equation parametrically.  One of them is...

x = sinθ

y = cosθ

Calculator:

You can graph parametric equations on your calculator by going to MODE and selecting PAR, then you just type the equations in and graph.

That's basically it.
-Olivia R