Task Complex Numbers: Roots of Quadratic Equatios. Add a ew Equatio scree: paf 2. Chage the Complex output to a+bi: LpNNNNwd 3. Select Polyomial ad set the Degree to 2: wq 4. Set a=, b=5 ad c=6: l5l6l 5. Solve the equatio: q Press d to edit the values of a ad b. Questios for discussio Whe are the roots of the quadratic real? Whe are the roots of the quadratic complex? Ca you fid values of b ad c so that the roots are complex ad the real part is 2? or? or? or p? Ca you fid a quadratic equatio with roots 2 ± 3i? Explai how you would fid a quadratic equatio with roots p ± qi (for ay p ad q)? Problem (Try the questio with pe ad paper first the check it o your calculator) The fuctio f(z) = z³ 0z² + 34z 40 has a root z = 3 + i. Hece fid the other two roots. Ivestigate the relatioship betwee the graphs of quadratic equatios ad their roots. Explai why the roots of a cubic with real coefficiets will always form a isosceles triagle i the Argad diagram.
Task 2 Matrices: Determiats ad iverse matrices. Add a ew Ru-Matrix scree: p 2. Fid the determiat of a 2 2 matrix. Determiat: Optio > Mat > Det: iwe Isert matrix: Math > Mat > 2 2: ddrqq Use the cursor keys to eter the values: N3B$N5l 3. Fid the iverse of the same 2 2 matrix. qn3b$n5$^l Questios for discussio What is the relatioship betwee the matrix, the determiat ad the iverse? What is the aswer whe a matrix is multiplied by its iverse? Are there ay matrices that do t have a iverse? Problem (Try the questio with pe ad paper first the check it o your calculator) For the matrices 3 2 A 4 ad 2 B 5 4 fid A, B ad (AB). For other 2 2 matrices, A ad B, ivestigate the relatioship betwee (AB). A, B ad Ivestigate the determiats ad iverse of matrices for stadard trasformatios: reflectio, rotatio ad stretches.
. Add a ew Graphs scree: p5 Task 3 Ratioal Fuctios 2. Eter the fuctio Y=(x+A)/((x+B)(x+C)) : jf+af kmjjf+agkjf+agkkl 3. Plot the curve usig Modify: y Use!/$ to vary the parameters ad N/B to chage which oe you are varyig. Questios for discussio What poit o the curve does the value of A give you? What is the relatioship betwee the shape of the curve ad the values of B ad C? Problem (Try the questio with pe ad paper first the check it o your calculator) Fid the values of A, B ad C for the followig curves: Fid coditios o A, B ad C so that the curve will have oe of these geeral shapes.
Task 4 Polar curves. Add a ew Graphs scree ad check the agle is set to radias: p5lpnnnnnnnnnnwd 2. Set the type to Polar (r=): ew 3. Eter the fuctio r=a +cos(θ) : af+jjfkl 4. Plot the curve usig Modify: y Use!/$ to vary A. Questios for discussio What is the maximum/miimum distace from the pole ad for what values of does this occur? How is this polar curve related to the Cartesia curve y A cos x? Problem (Try the questio with pe ad paper first the check it o your calculator) Plot the followig curves: r 2 cos r 2 r si 2 r 3 cos r 3 2si 2 For what values of does r take its maximum ad miimum values? How ca these be deduced from the polar equatio? For which parts of the graph does r take egative values? What are the coditios such that r a bcos ad r a bsi does t take egative values? The default settig is to plot values of from 0 to 2 (this ca be chaged with V- Widow). Are there ay curves for which this results i the same graph beig traced over agai? Are there ay graphs for which the graph is icomplete usig this rage?
Task 5 Summatio of simple fiite series. Go ito Ru-Matrix mode: p 2. Add a summatio sig: ruw Math Σ( 3. Calculate 0 x : f$f$$0l x Ivestigate x for differet values of. x Questios for discussio Ca you fid a expressio i terms of for Ca you fid a expressio i terms of for x? x 2 x? x How would you fid a expressio i terms of for xx ( 2)? x Problem (Try the questio with pe ad paper first the check it o your calculator) Fid a expressio for ( x)( x3) i terms of ad hece fid x 0 ( x)( x3). x Ivestigate sums of the form, i.e. sums that do t start from. xm Ivestigate 3 x ad fid a relatioship betwee this ad x x. x
Task 6 Numerical solutios of equatios: Newto-Raphso. Add a ew Graphs scree: p5 2. Plot the graph, e.g y = x³ x 2, Y = x³ x 2: f^3$-f-2lu I this example you ca see that the root lies betwee x = ad x = 2. 3. Now go ito the Recursio mode: p8 3 a a 2 a 2 3a : rw-jw^3$-w-2kmj3ws- kl f( x ) 4. Add the recurrece relatio x x a f '( x ) 5. Usig SET (F5), set the Table Start to, Ed to 5 ad a 0 to 2: yl5l2ld 6. Geerate the Table: u I this example after 5 iteratios the approximatio is accurate to more tha 3 decimal places. 7. You ca check your aswer i Graphs mode usig G-Solv > Root: yq Try usig your calculator to fid the roots of other equatios usig the Newto-Raphso method.
Notes o usig the Modify fuctio Teacher guidace It is useful for studets to be familiar with this mode first. Whe i Modify mode the parameters ad the step size ca be chaged with the cursor keys or values ca be directly typed i. Whe i Modify mode the cursor keys are used to chage the parameters ad caot be used to move the axes. All movig of the axes ad zoomig is disabled i Modify mode. To move the axes or zoom press EXIT to come out of Modify mode. The axes ca the be set to the appropriate values. To re-eter modify mode press EXIT agai to retur to the list of fuctios the F5 to go back ito Modify mode. Task Complex Numbers: Roots of Quadratic Equatios It is useful to draw studets attetio to the liks with the relatioship betwee the roots of a quadratic ad x 2 (α+β)x + αβ. The quadratic z² 4z + 3 = 0 has a roots z = 2 ± 3i. Problem solutio: z³ 0z² + 34z 40 has a roots z = 3 + i, z = 3 + i ad z = 4. Task 2 Matrices: Determiats ad iverse matrices This task ca be used as a ivestigatio or cosolidatio exercise. It is importat to be familiar with how to eter matrices ad use the related fuctios before settig the task. Problem solutio: 0.4 0.2 A 0. 0.3, B 4 3 3, 5 2 3 3 (AB) 2 6 3 4 5 30 Task 3 Ratioal fuctios This task used the modify fuctio please see above. Problem solutios: x 3 y ( x4)( x) y x xx ( 3) For the further tasks it is importat to be strict with the studets about the positios of the asymptotes. As a extesio studets ca be ecouraged to look at other families of ratioal fuctios.
Task 4 Polar curves MEI Casio Tasks for Further Pure This task uses the modify fuctio please see the otes above. Studets should attempt to sketch the polar curves by had ad the check their curves with the calculator. It is importat the calculators are set i radias mode. I V-Widow the rage ad icremets for the curve are set with Tθmi, max ad ptch. It is usually best to reset this to INITIAL after chagig to radias as the pitch might be based o degrees. Some curves will eed the Tθmax value icreasig so that the whole curve is sketched. O the graph scree q will allow studets to trace aroud the curve. Studets might also fid it helpful to refer to the TABLE fuctios p7. Useful settigs for this are: Start: 0 Ed: 2π Step: π/2 Task 5 Summatio of simple fiite series This task ca be used as a ivestigatio, a cosolidatio exercise or as a meas of checkig aswers to summatios. Problem solutio: 3 2 3 ( x)( x 3) 3 2 6 x 0 x ( x)( x 3) 465 Task 6 Numerical solutios of equatios: Newto-Raphso This task is a set of istructios for how to implemet the Newto-Raphso method o the calculators. Studets are ecouraged to work through these istructios ad the try solvig some equatios of their ow. Studets should be asked to verify that the recurrece relatio i step 4 is correct. They should also cosider why a startig value of 2 has bee chose for a 0. NB The Start value is the first value of that the table displays. Studets ca also be ecouraged to try other values for a 0 ad cosider the effect this has. It is useful to have some additioal equatios for studets to be fidig the roots of oce they have completed this sheet.