How to Graphically Interpret the Complex Roots of a Quadratic Equation


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1 Universit of Nersk  Linoln of Nersk  Linoln MAT Em Epositor Ppers Mth in the Middle Institute Prtnership How to Grphill Interpret the Comple Roots of Qudrti Eqution Crmen Melliger Universit of NerskLinoln Follow this nd dditionl works t: Prt of the Siene nd Mthemtis Edution Commons Melliger, Crmen, "How to Grphill Interpret the Comple Roots of Qudrti Eqution" (007). MAT Em Epositor Ppers. Pper 35. This Artile is rought to ou for free nd open ess the Mth in the Middle Institute Prtnership t of Nersk  Linoln. It hs een epted for inlusion in MAT Em Epositor Ppers n uthorized dministrtor of of Nersk  Linoln.
2 Mster of Arts in Tehing (MAT) Msters Em Crmen Melliger In prtil fulfillment of the requirements for the Mster of Arts in Tehing with Speiliztion in the Tehing of Middle Level Mthemtis in the Deprtment of Mthemtis. Dvid Fowler, Advisor Jul 007
3 How to Grphill Interpret the Comple Roots of Qudrti Eqution Crmen Melliger Jul 007
4 How to Grphill Interpret the Comple Roots of Qudrti Eqution As seondr mth teher I hve tught m students to find the roots of qudrti eqution in severl ws. One of these ws is to grphill look t the qudrti nd see were it rosses the is. For emple, the eqution of, s shown in Figure 1, hs roots t 1 nd. These re the two ples in whih the skethed grph rosses the is. Figure 1 The proess of uses the qudrti formul will lws find the rel roots of qudrti eqution. We ould hve lso used the qudrti formul to find the roots of the this eqution,. 1 ± ± ± We n think of the first term (½) s strting ple for finding the two roots. Then we see tht the roots re loted 3/ from the strting point in oth diretions. This leds us to roots of qudrti eqution tht does not ross the is. These roots re known s omple (imginr) roots. An emple of qudrti drwn on
5 oordinte plne with omple roots is shown in Figure 3. Notie tht the verte lies ove the is, nd the end ehvior on oth sides of the grph is pprohing positive infinit. The omple roots to n e found using the qudrti formul, ut it is enefiil to students to visulize grphil onnetion. Figure 3 1 GRAPHICAL INTERPRETATION OF COMPLEX ROOTS We know tht n qudrti n e represented. We lso know the roots of qudrti equtions n e derived from the wellknown qudrti formul:  ± If the roots re rel we visull interpret them to ross the is s shown in Figure. Figure 1 But, we re interested in grphill interpreting the roots of grph tht does not ross the is, s in Figure.
6 Figure 1 Let s use wht we grphill know out qudrtis with rel roots (Fig. ) to eplin wht we don t know grphill out qudrtis with omple roots (Fig. ). Now there re infinitel mn qudrtis with rel roots nd infinitel mn qudrtis with omple roots. But, when ompring one to nother, it would e helpful if the two qudrtis were relted in some w. If produes rel roots (old line in Figure ), refletion of this grph upwrd would ield new qudrti eqution tht would produe omple roots. (See Figure ) Figure To see how this is onfigured nltill, we will strt with the generl eqution of qudrti. (Rememer: The qudrti tht we re strting with (old line) is known to hve rel roots.) First we omplete the squre. Then to rete the flipped qudrti (whih is different eqution), negtive is pplied to. Then I simplified the eqution multipling Complete the squre ( ) 1
7 ) ( ) ( Completed the squre WE ARE CREATING A NEW QUADRATIC AT THIS POINT Reflet the Qudrti ) ( Refleted the Qudrti Simplif ) ( Simplified Now to e le to ompre these omple roots to the rel roots tht we strted with, plug the oeffiients of the eqution ove into the qudrti formul. ) ( ± Simplif ± ±
8 ± ± Beuse we re dding nd sutrting from it is unneessr to write the negtive in the seond denomintor. ± Simplified Sine we know tht we the roots re omple, we n show tht etrting negtive out of the rdil. Comple Roots of the Flipped Qudrti i ± Rel Roots of the Qudrti We Strted With ± Notie tht the rel roots nd the omple roots of different qudrti equtions ielded ver similr nswers. The re tull the sme eept for the i in the omple roots. The net step is to use figure out how to use these similrities to find the omple roots grphill. First, let s review how to grph the omple numer plne. Horizontl movement on the grph denotes the rel prt of the omple numer, while vertil movement represents the imginr prt of the omple numer. (See Figure 5)
9 Figure 5 Now, working in threespe, imgine tht these omple oordinte plne is the floor. It is represented the (i, ) oordinte plne in Figure 6. Figure 6 If we now use the (,) oordinte plne to drw qudrti with omple roots we ould get something tht looks like Figure 7. Notie the qudrti does not ross the is.
10 Figure 7 We urrentl n not see the omple roots grphill. But if we flip the qudrti horizontll over the verte, from the proof out we should get roots tht differ onl numer i. In order to grphill see the omple roots we need to rotte the refleted imge 90 degrees to ple the qudrti into the omple numer plne. Figure 8
11 Notie tht the two points indited n e found strting t units in the rel diretion nd i units in oth the positive nd negtive diretion. We re then le to grphill see the omple roots of qudrti eqution. Referenes Weeks, Audre. Conneting Comple Roots to Prol s Grph. June 0, Vest, Flod. The College Mthemtis Journl, Vol. 16, No.. (Sep., 1985), pp GeoGer demonstrtion reted Crmen Melliger t
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