Exploring prmetric representtion with the TI-84 Plus CE grphing clcultor Richrd Prr Executive Director Rice University School Mthemtics Project rprr@rice.edu Alice Fisher Director of Director of Technology Applictions nd Integrtion Rice University School Mthemtics Project fisher@rice.edu If you look up prmetric equtions in the index of most Pre-Clculus ooks, you will proly see one reference locted deep in the middle of the chpter on vectors. With the use of technology, however, prmetric equtions cn e n integrl prt of most of the Pre-Clculus curriculum. We hope to shre few ides of where I use prmetric equtions in my own clsses. Functions nd Trnsformtions We introduce prmetric equtions in the first unit tught in our Pre-Clculus curriculum, immeditely fter teching the definition of function. The use of prmetric equtions gretly enhnces the understnding of domin nd rnge. We strt with the students doing explortions on domin nd rnge, using prescried vlues for the prmeter t. We use this ctivity not only to re-enforce the concepts of domin nd rnge ut we lso use the clcultor s t-step to help re-enforce students understnding of the difference etween continuous nd discrete functions. It is lso gret reintroduction to trnsformtions. For exmple, grph the following prmetric functions for t in the sme window: x1 () t t y1 () t t x( t) t 3 y() t t x3() t t y t 3( ) t 3 Finding the functionl nottion for ech of these curves, we cn relte functionl trnsformtion concepts to equtions defined prmetriclly. This cn e especilly helpful when trying to distinguish etween the trnsformtions y f ( x) nd y f ( x). By using the uilt in domin nd rnge restrictions tht occur y the choice of vlues of t, it is esier to show tht lthough for certin functions trnsformtions my hve the sme grph, they re ffecting the grph in different wys. Grph the following prmetric functions for t in the sme window x1 () t t x( t) t x3() t t y1 () t t y() t t y3( t).5t
Inverses We would feel lost trying to tech the concept of the inverse of function without using prmetric representtion. After strting with functionl representtions of one-to-one functions we move to using prmetric equtions for oth one-to one nd non one-to-one functions. The ide of flipping the x nd y equtions to crete the inverse prmetriclly mkes sense to the students. We let the students discover the ide of restricting domin of function so tht it hs functionl inverse, nd wht mkes good restriction. This is especilly importnt when teching the inverses of the trigonometric functions. Grph the following in the sme window for 4 t 4. x1 () t t y1 () t t x() t t y() t t The grphs represent the function inverse is function. f ( x) x nd its inverse. Now restrict t so tht the Introduction to Trigonometry After defining the x nd y coordintes of the unit circle in terms of sine nd cosine, it is esy to develop the ide of prmetriclly defined unit circle. It is nice to show the unwrpping of the circle to crete the sine grph. Grph the following for 0t using t-step of. Use window of 4 1.5 x 6.5 nd.67 y.67. x1 ( t) cos( t) y1( t) sin( t) x() t t y( t) sin( t)
Conic Sections The Pythgoren trigonometric identities llow for esy prmetric representtion of ellipses nd hyperols. Prols re most esily represented without the use of trigonometry. Ellipses A comprison of the Pythgoren identity: cos t sin t 1 nd stndrd form for the eqution of n ellipse : x h ( y k) 1 llows for two simple sustitutions : cos t ( x h) nd sin t ( y k) Solving these two equtions for x nd y yields pir of prmetric equtions: x cos t h y sin t k ( x 5) ( y 1) Re-express 1 using prmetric equtions nd grph. Use degree 9 4 mode nd grph in n pproprite squre window for 0 t 360 A few personl comments re importnt t this point: I chose sustitutions I did to reinforce the use of x nd y coordintes of unit circle to represent sine nd cosine respectively. In using this method I m de-emphsizing the ide tht corresponds to the mjor xis, etc. I focus on the ide tht is stretch in the x eqution nd therefore horizontl stretch. Likewise, is verticl stretch. I d just s soon not use the letters nd t ll, ut focus on the mjor xis eing the xis with the lrgest stretch. Some students see contrdiction in the trnsformtion in prmetric representtion when compred to the Crtesin representtion. By re-writing x 3cos t 5in the form x 5 3cost, I try to show tht there is relly no contrdiction.
Hyperols By using the Pythgoren identity: sec t tn t 1 nd stndrd form for hyperol : ( x h) ( y k) 1 ( y k) ( x h) or 1 One cn derive the following pirs of prmetric equtions to represent hyperols: x sec t h x tn t h y tn t k y sec t k (horizontl trnsverse xis) (verticl trnsverse xis) In hyperol, unlike n ellipse, it mkes difference which trigonometric function corresponds with which vrile. Using the sme window settings s efore, re-express the eqution prmetriclly to grph the hyperol ( y ) ( x ) 1 1. 5 36 Prols Prols re most esily grphed prmetriclly without the use of trigonometric functions. All non-rotted prols cn either e written in the form y f ( x) or x f ( y). Prmetriclly, prols tht cn e written y f ( x) cn e grphed using x t nd y f ( t), likewise prols tht cn e represented s x f ( y) cn e grphed prmetriclly using x f ( t) nd y t. In this cse the t-step of the window must e djusted to include negtive vlues for t or the entire prol will not pper. Extensions Exploring rotted conic sections is n extension of this work. To do this view pir of xt () prmetric equtions s x 1 vector mtrix; yt (), cos θ sin θ then left-multiply this mtrix y rottion mtrix; [ sin θ cos θ ].
The resultnt x1 mtrix; cos θ x(t) sin θ y(t) [ sin θ x(t) + cos θ y(t) ] represents new pir of prmetric equtions tht rotte the conic q degrees counterclockwise. In the sme window grph the hyperol from the previous exmple nd the sme hyperol rotted 45 counter-clockwise. Vectors Prmetric representtion llows for the explortion of two dimensionl motion prolems, especilly those relted with projectile motion. By using the prmetric equtions: x ( v cos ) t o 1 y t ( v sin ) t o s o (Where v o is initil velocity, q is the ngle of projection, is ccelertion due to grvity nd s o is initil height t projection.) One cn explore the effects of chnging the vrious prmeters. Mny clcultor gmes hve een developed tht use these ides in situtions such s throwing sketll or jvelin, or hitting sell. Find n pproprite grphing window nd model the motion of n oject projected with n initil velocity of 50m/s t n ngle of 30 from the horizontl from height of 3 m. Assume tht ccelertion due to grvity is -9.8m/s.