Assignment #10Nicole MostellerEMAT 6680

Explorations of Parametric EquationsMy understanding of parametric equations is that the set of equations defines both thexcomponent and theycomponent of a relation that occurs on some defined interval oft. Take for example the parametric equations defined below . Thexcomponent is the cosine function on the interval of0 < t < 2P, and theycomponent is the sine function on the interval of0 < t < 2P. The graph of this parametric equations shows the relation inFigure 1.The unit circle. Its from this example that we arrive at a most common and useful trig identity: The equation of this circle is . Using the information from the parametric equation we see that . This trig identity will be used in the following investigation.Figure 1:

Question 3: For variousaandb, investigate for0 < t < 2P.

To begin this investigation, let's first look at the case whena = b.The case whenFigure 2:a = b, andaandbhave the following values. Notice that each of the circles that are produced from the parametric equations are still centered at the origin. Also each circle has a radius the value ofaandb.

Proof:For these parametric equations whena = b(let this value be r), the resulting graph is a circle with radius equal tor. Looking at this parametric equation as the combination of thexcomponent andycomponent, we see . To simplify each term, square them. . Using the distributive property, we have . Substituting the trig identity that we established earlier, . We have the equation for a circle with radiusr.

To continue the investigation, let's first look at the case whena and b are not equal. The easiest way to see the impact thataandbhave on the parametric equations is to change one value while maintaining the other as a constant.Figure 3shows with a variation in the value ofawhile keepingbconstantly1.The value ofFigure 3:avaries withb = 1.

By varying the value ofa, we see that the original circle has been stretched along thex-axis. We can anticipate a similar stretch only along they-axisby varying the value ofband allowingato remain constant. SeeFigure 4. Figure 4: The value ofbvaries whilearemains constant.

Interested to see why the parametric equations in the form of (when a and b are not equal) always give the graph of an ellipse?Click Here!for the proof.

To get a good idea what happens asaandbvaries,Click Here!for a Graphing Calculator animation.

Return to Nicole's Page