8.3 - Parametric Equations

In the past, we have been working with rectangular equations, that is equations involving only x and y so that they could be graphed on the Cartesian (rectangular) coordinate system.

We also had an example of the height of a freely falling body as a function of time in seconds t. That function was a quadratic function. If the object is not dropped or thrown straight up into the air, there will also be a horizontal component of its position. The horizontal component is a simple distance function (d=rt).

Path of Falling Object

y(t) = -16t2 - v0t + y0

x(t) = r t

v0 = initial vertical velocity

y0 = initial height

r = horizontal velocity

t = time in seconds

Notice that both of those functions, the vertical height and the horizontal distance are functions of time. So, to fully describe the path of an object, we need two equations. One for the vertical component and one for the horizontal component. Both of these functions are functions of a third variable, t.

This gives us parametric equations. A parameter is simply the independent variable in a function.

Sketching a Plane Curve

A plane curve results when the ordered pairs ( x(t), y(t) ) are graphed for all values of t on some interval.

One way to sketch the plane curve is to make a table of values. The parameter t has several values listed for it and the corresponding values for x(t) and y(t) are computed. Then the ordered pairs are plotted and the curve is drawn in between the plotted pairs.

Orientation or Direction

When sketching a plane curve, the "direction of increasing t" or "orientation" of the curve is indicated by little arrows indicating which direction the curve is moving in when the value of the parameter t increases.

Graphing Calculator

The graphing calculator does a wonderful job of graphing parametric equations. You must, however, tell the calculator that you want to graph parametric equations as opposed to regular functions. To do this, put your calculator into the parametric mode by hitting [MODE] and choosing the [PAR] option. Be sure to reset your calculator back to [FUN] for Function mode when you are done with parametric equations. While you're in the Mode menu, you may wish to set your calculator to [RADIAN] mode instead of [DEGREE] mode. They are used for trigonometric functions, which we won't be using, but they do affect the way the zoom keys work.

After setting up your calculator for parametric mode, notice that when you hit the Y= key, you no longer have a y1=. You now have a pair of equations, a x and an y that are both functions of t. Simply enter the parametric equations in for x and y. Notice that the key you have been using for X is also marked T. In parametric mode, a T will automatically appear instead of the X.

Window Settings

You will now have three additional window choices that you did not have before. Tmin, Tmax, and Tstep. Tmin is the smallest value for the parameter that you wish to use. Unless you have good reason not to (like the domain says t >= 0), be sure to use negative values for Tmin. Tmax is the largest value for the parameter that you wish to use. Unless you have good reason not to, use a positive value for Tmax. In other words, make sure T can take on both positive and negative values. Tstep is the change in T, and should be reasonable for the range of T values you've specified.

TMin = -5, TMax = 5, and TStep = 0.1 are usually good starting values. If you find out that the graph doesn't show up, you may need to change those values.

Warning! The Zoom Standard will reset the settings on the T. If you do a zoom standard, your T will go between 0 and 2 pi (in radian mode) by pi/24 and 0 to 360 (in degree mode) by 7.5. Neither of these contain negative values and may not show the entire graph.

The direction of increasing t is the direction the calculator draws the curve in. Indicate this with directional arrows along the curve.

Eliminating the Parameter

Another way to sketch a plane curve is to eliminate the parameter. The steps to eliminating the parameter are simple.

  1. Solve one of the parametric equations for t.
  2. Substitute for t into the other parametric equation.

In step 1, you should solve for t in the simpler equation. Easier to solve doesn't always mean smaller exponent. If you have an t2 and a t3, solve for the t in the t3 (if possible). By doing so, you avoid a plus/minus situation when you take the square root of t.

It may not always be necessary to completely solve for t. This is valueable when one of the terms appears in the other equations.

Example 1

Eliminate the parameter from x = 3t2 - 4 and y = 2t.

The y is definitely the easier function to solve for t, and when you do that, you get t = y/2.

Subsitute that into the x equation for t and you get x = 3(y/2)2 - 4. Simplify to get x = 3/4 y2 - 4.

Example 2

Consider the system of equations x = et and y = e3t.

If you were to solve this using the steps listed above, you would take the x = et equation and solve it for t to get t = ln x. Then substitute that into the y = e3t equation to get y = e3ln x. Using properties of logarithms, you would move the 3 to be the exponent on the x and then the e and ln functions inverse out, leaving you with y = x3.

Now, consider this. y = e3t = (et)3. Since x = et, replace the et by x. y = (x)3 or just y = x3. There was no need to go all the way down to t.

Another note about this problem. Since x and y are both exponential functions, the range on each of them are the positive reals. However, that is lost when you simplify to y = x3. Be sure to stick a domain restriction on x to make it the same as the original. That restriction would be x > 0.