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  Concept about Parametric Equations

Parametric Equations:

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

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

Path of Falling Object:

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

x(t) = r t

Here,

v0 = initial vertical velocity

y0 = initial height

r = horizontal velocity

t = time in seconds

Note that both of such functions, that is, the vertical height and horizontal distance are functions of time. Therefore, to fully explain the path of an object, we require two equations. One for vertical component and one for the horizontal component. Both of such functions are the functions of a third variable, t.

This provides us parametric equations. The parameter is simply an independent variable in the function.

Sketching a Plane Curve:

The plane curve outcomes whenever the ordered pairs (x(t), y(t)) are graphed for all the values of t on some interval.

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

Orientation or Direction:

Whenever sketching a plane curve, the ‘direction of increasing t’ or ‘orientation’ of the curve is pointed out by little arrows indicating the direction in which the curve is moving and when the value of the parameter t rises.

Graphing Calculator:

The graphing calculator does a fantastic job of graphing parametric equations. You should, though, tell the calculator that you wish to graph parametric equations as opposed to the regular functions. To do this, place your calculator to the parametric mode by pressing [MODE] and selecting the [PAR] option. Make sure to reset your calculator back to [FUN] for the Function mode whenever you are done with the parametric equations. While you are in Mode menu, you might wish to set your calculator to [RADIAN] mode rather than [DEGREE] mode. They are employed for trigonometric functions, that we won't be using, however they do affect the way the zoom keys work.

After setting up your calculator for the parametric mode, note that whenever you press the Y= key, you no longer have a y1=. You now contain a pair of equations, a x and an y which are both functions of t. Simply enter the parametric equations in for x and y. Note that the key you have been employing for X is as well marked T. In parametric mode, the T will automatically appear rather than X.

Window Settings:

You will now contain three additional window choices which you did not have before. Tmin, Tmax, and Tstep. Tmin is the minimum value for the parameter which you wish to use. Unless you have excellent reason not to (similar to the domain states t >= 0), be sure to employ negative values for Tmin. Tmax is the maximum value for the parameter which you wish to use. Unless you have excellent reason not to, employ a positive value for Tmax. In another word, make sure T can take on both negative and positive values. Tstep is the change in T, and must be reasonable for the range of T values you have specified.

TMin = -5, TMax = 5, and TStep = 0.1 are generally good beginning values. If you recognize that the graph does not show up, you might need to change such values.

Note: The Zoom Standard will reset the settings on T. When you do a zoom standard, your T will go between 0 and 2 pi (that is, in radian mode) by pi/24 and 0 to 360 (in degree mode) by 7.5. Neither of such contains negative values and might not show the whole graph.

The direction of rising t is the direction; the calculator draws the curve in. Point out this with directional arrows all along the curve.

Eliminating the Parameter:

The other way to sketch a plane curve is to eliminate or remove the parameter. The steps to eliminating the parameter are very simple.

a) Solve one of parametric equations for t.

b) Replace for t to the other parametric equation.

In step a, you must solve for t in simpler equation. Simpler to solve does not always mean smaller exponent. When you have a t2 and a t3, solve for t in t3 (when possible). By doing so, you avoid a plus or minus situation whenever you take the square root of t.

It might not always be essential to completely resolve for t. This is valuable whenever one of the terms appears in other equations.

Illustration:

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

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

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

Illustration:

Let consider the system of equations x = et and y = e3t.

When you were to solve this by using the steps listed above, you would take x = et equation and solve it for t to obtain t = ln x.

Then replace that to the y = e3t equation to obtain y = e3ln x. By using the properties of logarithms, you would move 3 to be theexponent on x and then e and ln functions inverse out, leaving you with y = x3.

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

The other note regarding this problem. As x and y are both exponential functions, the range on each of them are positive reals.

Though, that is lost whenever you simplify to y = x3. Be sure to stick a domain restriction on x to make it similar as the original.

This restriction would be x > 0.

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