3VCEcoverage. Cubic functions. Areas of study Unit 1 Functions and graphs Algebra

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1 Cubic functions VCEcoverage Areas of stud Unit Functions and graphs Algebra In this cha chapter A Polnomials B Epanding C Long division of cubic polnomials D Polnomial values E The remainder and factor theorems F Factorising cubic polnomials G Sum and difference of two cubes H Cubic equations I J Cubic graphs intercepts method Cubic graphs using translation K Domain, range, maimums and minimums L Modelling using technolog M Finite differences

2 0 Mathematical Methods Units and Polnomials A polnomial in, sometimes denoted P(), is an epression containing onl positive whole number powers of. The degree of a polnomial in is the highest power of in the epression. For eample: + is a polnomial of degree, or linear polnomial is a polnomial of degree, or quadratic polnomial is a polnomial of degree, or cubic polnomial. 0 is a polnomial of degree 0 (think of 0 as 0 0 ). Epressions containing a term similar to an of the following terms are not polnomials: --,,,, sin etc. For eample, the following are not polnomials sin + This chapter will deal mainl with polnomials of degree, called cubic polnomials, or cubics for short. In the epression P() = is the variable. 6 is the coefficient of. is the coefficient of. is the coefficient of. 6,, and + are all terms. The constant term is +. The degree of the polnomial is. An eample of where polnomials are useful is shown below. The surface area, S, of a plant hothouse of length L and height can be approimated b the polnomial S() = π + Lπ 4. You ma wish to use graphing software to further stud these polnomials.

3 Chapter Cubic functions Histor of mathematics ÉVARISTE GALOIS (5 Oct 8 - Ma 8) During his life... Jane Austen writes Sense and Sensibilit Shaka, the leader of the Zulus, attacks man neighbouring tribes. The electric telegraph is developed commerciall. Mar Shelle writes the stor Frankenstein. Galois was a French mathematician who worked on Group Theor and showed that polnomials of order 5 or above had no algebraic solutions. Galois came from a well educated famil and was initiall taught b his mother until he was ears old. In 8 Galois went to the Lcée of Louis-le-Grand where he did ver well in some areas, winning school awards. But he performed poorl in other areas and was asked to repeat a ear. In 87 Galois started to stud mathematics, finding a new area to inspire him. In 88 he tried to enrol at the École Poltechnique but failed the entrance eaminations. B 89 Galois was doing his own research. He published an article about continued fractions in the magazine, Annales de mathématiques. Again Galois tried to enter the École Poltechnique, but his father s suicide prevented him from studing properl and he failed again. Like man mathematicians of the time he entered the great mathematical contests. He sent his entr to another mathematician, Fourier, who unfortunatel died soon after Galois first contact. His paper on equations and their solutions was lost and never entered in the competition. Like his father, Galois was interested in politics and wrote frequent letters to newspapers attacking the government of the time. He was subsequentl epelled from school as a troublemaker. Galois joined the arm and again got into trouble. At a dinner celebration he was seen to have a knife in his hand and was thought to be making threats against the King, Louis-Phillipe. He was arrested, but at the trial showed that he was actuall offering his weapon to the King and threatening the King s enemies. He was jailed on another occasion for wearing the uniform of a group banned b the King. During a cholera epidemic, he was transferred from the prison to a hospital where he fell in love with Stephanie Felice du Motel, the daughter of a doctor. He started to write to her but she was not interested in him. Shortl after getting out of prison he became involved in a duel with Perscheu d Herbinville over his interest in her. Fights between gentlemen at this time were not settled immediatel, but arranged for the net da so that friends could be invited along to watch. Some believed that he spent his last night writing down all he knew about Group Theor. Galois was wounded in this duel and then abandoned b his opponent and his friends. Eventuall found b a peasant, he died in hospital the net da. Much of his life was controversial and there are man stories about him. Questions. What form of mathematics did Galois work on?. Wh did he fail to get into École Poltechnique?. Wh was he arrested the first time? 4. How did he die? How old was he when he died? Research Investigate the various methods of solving polnomials.

4 Mathematical Methods Units and A Polnomials State the degree of each of the following polnomials. a b c 8 + d e f --u u u 6 g e h g i.5f 6 800f State the variable for each polnomial in question. Which polnomial in question is: a linear? b quadratic? c cubic? 4 State whether each of the following is a polnomial (P) or not (N). a b 4p -- c d 4 4 e k + k k + 7 f 5r r 9 + g 4c 6 c h 8 + i sin + 5 Consider the polnomial P(w) = a What is the degree of the polnomial? b What is the variable? c What is the coefficient of? d What is the value of the constant term? e Which term has a coefficient of? 6 Consider the polnomial P(w) = 6w 7 + 7w 6 9. a What is the degree of the polnomial? b What is the variable? c What is the coefficient of w 6? d What is the coefficient of w? e What is the value of the constant term? f Which term has a coefficient of 6? 7 Consider the polnomial f() = a What is the degree of the polnomial? b What is the coefficient of 4? 8 A sports scientist determines the following equation for the velocit of a breaststroke swimmer during one complete stroke: v(t) = 6.876t 47.65t t 4 9.4t t t a What is the degree of the polnomial? b What is the variable? c How man terms are there? d Use a graphics calculator or graphing software to draw the graph of this polnomial. e Match what happens during one complete stroke with points on the graph. --

5 Chapter Cubic functions Epanding If we epand three linear factors, for eample, ( + )( + )( 7), we get a cubic polnomial (a polnomial of degree ) as the following worked eample shows. WORKED Eample Epand: a ( + )( ) b ( )( + 5)( + ). THINK WRITE a Write the epression. a ( + )( ) Epand the last two linear factors. = ( + 6) = ( 6) Multipl the brackets b. = 6 b Write the epression. b ( )( + 5)( + ) Epand the last two linear factors. = ( )( ) Multipl the second pair of brackets b, then b. = ( )( ) = Collect like terms. = remember remember When epanding three linear factors:. epand two factors first, then multipl b the remaining linear factor. collect like terms at each stage. ( + ) ma be written as ( + )( + )( + ). B Epanding WORKED Eample a Epand each of the following: a ( + 6)( + ) b ( 9)( + ) c ( )( + ) d ( + )( + ) e ( 4)( + 4) f 5( + 8)( + ) g ( + 4) h (7 ) i (5)( 6)( + 9) j 7( + 4) Epanding Mathcad WORKED Eample b a ( + 7)( + )( + ) b ( )( + 4)( 5) c ( )( 4)( + 8) d ( )( )( ) e ( + 6)( )( + ) f ( 7)( + 7)( + 5) g ( + )( + 5)( ) h ( + 5)( ) i ( + )( 7) j ( + )( )( + ) a ( )( + 7)( + 8) b ( + 5)( )( + 4) c (4 )( + )( ) d (5 + )( )( 4) e ( 6)( + 7)( + 5) f (7 4)( 4) g 9( )( + 8) h (6 + 5)( 7) i ( 4)( )(5 + 9) j (7 + )( + )( + 4) a ( + ) b ( + 5) c ( ) d ( ) e ( 6) f ( + 4) 4

6 4 Mathematical Methods Units and Long division of cubic polnomials The reverse of epanding is factorising (epressing a polnomial as a product of its linear factors). Before learning how to factorise cubics, ou must be familiar with long division of polnomials. You ma remember in earlier levels doing long division questions. Consider 745, or ) 745 The process used is as follows. into 7 goes times. Write at the top. ) 745 = 6 ) 745 Write down the 6. 6 Subtract to get. Bring down the 4 to form 4. into 4 goes 4. Write 4 at the top. 4 = Write down the. Subtract to get. Bring down the 5 to form 5. into 5 goes 8. Write 8 at the top. ) ) ) ) ) ) )

7 Chapter Cubic functions 5 8 = 4 Write down the 4. ) Subtract to get Answer: 48 remainder Divisor ) Quotient Dividend Remainder The same process can be used to divide polnomials b polnomial factors. Consider ( + + 0) ( ) or ) into goes times (consider onl the leading terms). Write at the top. ) ( ) = Write down the. ) Subtract. ( = 0, = 5 ) ) Note: Subtracting a negative is the same as changing the sign and adding. Bring down the. ) into 5 goes 5. Write +5 at the top. + 5 )

8 6 Mathematical Methods Units and 5 ( ) = 5 5 Write down the 5 5. Subtract. Note: 5 5 = 0, 5 = + Bring down the 0. into goes. Write + at the top. + 5 ) ) ) ) ( ) = 6 Write down the 6. Subtract to get 6. Answer: remainder ) ) Quotient Remainder

9 WORKED Eample Chapter Cubic functions 7 Perform the following long divisions and state the quotient and remainder. a ( ) ( + ) b ( 4 7 5) ( ) c ( ) ( 6) THINK WRITE + a Write the question in long division format. a + ) Q Perform the long division process R Write down the quotient and remainder. Quotient is +, remainder is. 0 Q b Write the question in long division format. b ) Perform the long division process R Write down the quotient and remainder. Quotient is 0, remainder is Q c Write the question in long division format. c 6 ) Perform the long division process. Write down the quotient and remainder Quotient is 6 R , remainder is 6. WORKED Eample State the quotient and remainder for ( 7 + ) ( + 5). THINK WRITE Q Write the question in long division + 5) format. Note that there is no term in + 5 this equation. Include 0 as a place 5 7 holder. 5 5 Perform the long division process Write down the quotient and remainder. 89 R Quotient is 5 + 8, remainder is 89.

10 8 Mathematical Methods Units and remember remember Long division of polnomials is similar to long division with numbers. The highest power term is the main one considered at each stage. Ke steps are:. How man?. Multipl and write the result underneath.. Subtract. (If necessar, change the sign and add.) 4. Bring down the net term. 5. Repeat the process until no pronumerals remain to be divided. 6. State the quotient and remainder. C Long division of cubic polnomials GC Mathcad program WORKED Eample Polnomial a division Polnomial division WORKED Eample b WORKED Eample c WORKED Eample Perform the following long divisions and state the quotient and remainder. a ( ) ( + ) b ( ) ( + ) c ( ) ( + ) d ( ) ( + 4) e ( ) ( + ) f ( ) ( + ) g ( ) ( + ) h ( ) ( + 8) i ( ) ( + ) j ( ) ( + 5) State the quotient and remainder for each of the following. a ( + 5 9) ( ) b ( ) ( ) c ( + 9 5) ( ) d ( ) ( ) e ( 5 + 8) ( ) f ( ) ( ) g ( ) ( 5) h ( ) ( 4) Divide the first polnomial b the second and state the quotient and remainder. a , + b , + c , d + 8 9, + 4 e , f , Divide the first polnomial b the second and state the quotient and remainder. a , b 6 + +, + 4 c , + 5 d 5 + 4, 7 e + 5 6, + f , 4 5 State the quotient and remainder for each of the following. a b c d State the quotient and remainder for each of the following. a ( + ) ( + ) b ( + 7) ( + ) c ( 5 + ) ( 4) d ( 7 + 8) ( ) e (5 + + ) ( + ) f ( + 8 4) ( + 5) g ( + ) ( ) h ( ) ( + )

11 Chapter Cubic functions 9 Polnomial values Consider the polnomial P() = The value of the polnomial when = is denoted b P() and is found b substituting = into the equation in place of. That is, P() = () 5() + () + P() = 7 5(9) + + P() = P() = 4. WORKED Eample If P() = + 4, find: a P() b P( ) c P(a) d P( + ). THINK WRITE a Write the epression. a P() = + 4 Replace each P() = () + () () 4 occurrence of with. Simplif. = + 4 = 4 4 b Write the epression. b P() = + 4 Replace each P( ) = ( ) + ( ) ( ) 4 occurrence of with. Simplif. = ( 8) + (4) = = 0 c Write the epression. c P() = + 4 Replace each P(a) = a + a a 4 occurrence of with a. No further simplification is possible, so stop here. d Write the epression. d P() = + 4 Replace each P( + ) = ( + ) + ( + ) ( + ) 4 occurrence of with ( + ). Epand the righthand side and collect like terms. = ( + )( + )( + ) + ( + )( + ) ( + ) 4 = ( + )( + + ) = ( ) + 6 = ( ) + 6 = =

12 0 Mathematical Methods Units and remember remember P(a) means the value of P() when is replaced b a and the polnomial is evaluated. D Polnomial values WORKED Eample 4 If P() = + + 0, find the following. a P(0) b P() c P() d P() e P( ) f P( ) g P( ) h P(a) i P(b) j P( + ) k P( ) l P( 4) Cop the following table. Column Column Column Column 4 Column 5 Column 6 Column 7 Column 8 Column 9 Mathcad Polnomial division a b P() P() P() P( ) P( ) Rem when divided b ( ) Rem when divided b ( ) Rem when divided b ( + ) Rem when divided b ( + ) c d EXCEL Spreadsheet WorkSHEET Cubic valuer. Complete columns to 5 of the table for each of the following polnomials. a P() = b P() = c P() = + 4 d P() = Find the remainder when each polnomial in question is divided b ( ) and complete column 6 of the table. 4 Find the remainder when each polnomial in question is divided b ( ) and complete column 7 of the table. 5 Find the remainder when each polnomial in question is divided b ( + ) and complete column 8 of the table. 6 Find the remainder when each polnomial in question is divided b ( + ) and complete column 9 of the table. 7 Cop and complete: a A quick wa of finding the remainder when P() is divided b ( + 8) is to calculate. b A quick wa of finding the remainder when P() is divided b ( 7) is to calculate. c A quick wa of finding the remainder when P() is divided b ( a) is to calculate.

13 Chapter Cubic functions The remainder and factor theorems The remainder theorem If ou completed the previous eercise, ou ma have noticed that: The remainder when P() is divided b ( a) is equal to P(a). That is, R = P(a). This is called the remainder theorem. We could have derived this result as follows. If is divided b 4, the quotient is, and the remainder is. That is, 4 = + -- and 4 = 4 +. Similarl, if P() = is divided b ( ), the quotient is and the remainder is 5. That is, ( ) ( ) = and ( ) = ( + + 7)( ) + 5. In general, if P() is divided b ( a), and the quotient is Q(), and the remainder is R, we can write R P() ( a) = Q() and ( a) P() = ( a)q() + R. Substituting = a into this last epression ields P(a) = (a a)q() + R = 0 Q() + R = R as before. The factor theorem The remainder when is divided b 4 is zero, since 4 is a factor of. Similarl, if the remainder (R) when P() is divided b ( a) is zero, then ( a) must be a factor of P(). Since R = P(a), all we need to do is to find a value of a that makes P(a) = 0, and we can sa that ( a) is a factor. If P(a) = 0, then ( a) is a factor of P(). This is called the factor theorem. Imagine P() could be factorised as follows: P() = ( a)q() where Q() is the other factor of P(). Then we have P(a) = (a a)q(a) = 0 Q(a) = 0. So it is no surprise if P(a) = 0, ( a) is a factor.

14 Mathematical Methods Units and WORKED Eample 5 Without actuall dividing, find the remainder when is divided b: a b + 6. THINK WRITE a Name the polnomial. a Let P() = The remainder when P() is divided b ( ) is equal to P(). R = P() = 7() () + 4 = 7 7(9) = b The remainder when P() is divided b ( + 6) is equal to P( 6). = 46 b R = P( 6) = ( 6) 7( 6) ( 6) + 4 = 6 7(6) = = 45 The remainder when + k + is divided b ( ) is equal to 0. Find the value of k. THINK WRITE Name the polnomial. Let P() = + k +. The remainder when P() is divided b ( ) is equal to P(). R = P() = + k() + 4 WORKED Eample 6 = 8 + 4k Since R = 0, 8 + 4k = 0 We are given R = 0. Put 8 + 4k = 0. Solve for k. 4k = k = Graphics Calculator tip! Calculating several values of a function at once. Enter the function such as f () = in Y.. Press nd [QUIT] to return to the home screen.. Enter Y on home screen (press VARS, select Y-VARS and :Function, and press ENTER ). 4. Tpe Y({,0,,,7}) and press ENTER to find the value of f ( ), f (0), f (), f () and f (7) in one hit. remember remember Remainder R = P(a) when P() is divided b a. If P(a) = 0, then ( a) is a factor of P().

15 Chapter Cubic functions E The remainder and factor theorems WORKED Eample 5 WORKED Eample 6 Without actuall dividing, find the remainder when is divided b: a b + c d + 5 e 0 f k g + n h + c. Find the remainder when the first polnomial is divided b the second without performing long division. a , b 4 +, + c + +, + d 4 5, e , + 5 f + + 6, + g + + 8, 5 h, i + 8, + j +, 7 a The remainder when + k + is divided b ( + ) is 9. Find the value of k. b The remainder when + + m + 5 is divided b ( ) is 7. Find the value of m. c The remainder when + + n is divided b ( ) is. Find the value of n. d The remainder when a is divided b ( ) is. Find the value of a. e The remainder when b + is divided b ( + ) is 0. Find the value of b. f The remainder when is divided b ( c) is 5. Find a possible whole number value of c. g The remainder when + is divided b ( + d) is. Find the possible values of d. h The remainder when + a + b + is divided b ( 5) is 4. When the cubic polnomial is divided b ( + ), the remainder is. Find a and b. 4 Prove that each of the following are linear factors of b substituting values into the cubic function. ( + ), ( ), ( + 5) 5 Avoid division and show that the first polnomial is eactl divisible b the second (that is, the second polnomial is a factor of the first). a , b 7 + 7, 7 c , d + 9 8, + e + 9 7, + f , g + 9, 4 h , a multiple choice When is divided b ( + ), the remainder is: A 5 B C 0 D E 5 b Which of the following is a factor of ? A ( ) B ( ) C ( + ) D ( 5) E ( + 4) c When is divided b ( ), the remainder is: A B C D 0 E d Which of the following is a factor of ? A ( ) B ( + ) C ( 5) D ( + 5) E ( + 7) 7 Find one factor of each of the following cubic polnomials. a + b c + 8 d EXCEL Cubic valuer Spreadsheet

16 4 Mathematical Methods Units and Factorising cubic polnomials Using long division Once one factor of a polnomial has been found (using the factor theorem as in the previous section), long division ma be used to find other factors. In the case of a cubic polnomial, one, and possibl two other factors ma be found. WORKED Eample 7 Use long division to factorise the following. a b c SkillSHEET. THINK WRITE a Name the polnomial. a P() = Look for a value of such that P() = 0. For cubics containing a single, tr a factor of the constant term (4 in this case). Tr P(). P() 0, so ( ) is not a factor. Tr P(). P() 0, so ( ) is not a factor. Tr P( ). P() = = = 8 0 P() = = P( ) = ( ) 5 ( ) ( ) + 4 = = = 0 ( + ) is a factor. P( ) does equal 0, so ( + ) is a factor. 7 + Divide ( + ) into P() using long + ) division to find a quadratic factor Write P() as a product of the two P() = ( + )( 7 + ) factors found so far. Factorise the second bracket if possible. P() = ( + )( )( 4) Note: The discriminant of the quadratic ( 7 + ) is = b 4ac = ( 7) 4()() =, which is a perfect square, indicating the quadratic ma be factorised over Q (rational numbers).

17 Chapter Cubic functions 5 THINK WRITE b Name the polnomial. Note: there is no term, so include 0. b P() = P() = Look at the last term in P(), which is 0. This suggests it is worth tring P(5) or P( 5). Tr P( 5). P( 5) = 0 so ( + 5) is a factor. P( 5) = ( 5) 9 ( 5) + 0 = = 0 So ( + 5) is a factor Divide ( + 5) into P() using long + 5) division to find a quadratic factor Write P() as a product of the two P() = ( + 5)( 5 + 6) factors found so far. 5 Factorise the second bracket if possible. P() = ( + 5)( )( ) c Write the given polnomial. c Let P() = Take out a common factor of. (We could = ( + 4 ) take out + as the common factor, but taking out results in a positive leading term in the part still to be factorised.) Let Q() = ( + 4 ). Let Q() = ( + 4 ). (We have alread used P earlier.) 4 Evaluate Q(). Q() = 0, so ( ) is a factor. Q() = + 4 = 0 So ( ) is a factor Divide ( ) into Q() using long ) + 4 division to find a quadratic factor. 6 7 Write the original polnomial P() as a product of the factors found so far. In this case, it is not possible to further factorise P(). (The discriminant ( ) of the quadratic factor is negative, indicating no real or rational factors of this part.) P() = ( )( ) Note: In these eamples, P() ma have been factorised without long division, b finding all three values of which make P() = 0, and hence three factors, then checking that the three factors multipl to give P().

18 6 Mathematical Methods Units and Using short division The process of long division can be quite time (and space) consuming. An alternative is short division, which ma take a little longer to understand, but is quicker once mastered. Consider P() = Using the factor theorem, we can find that ( ) is a factor of P(). So, P() = ( )(? ). Actuall, we know more than this: as P() begins with and ends with +0, we could write P() = ( )( +? 0) The in the second pair of brackets produces the desired (the leading term in P()) when the brackets are multiplied. The 0 in the second pair of brackets produces +0 (the last term in P()) when the brackets are multiplied. Imagine epanding this version of P(). Multipling in the first bracket b in the second would produce, which is what we want, but multipling in the first bracket b in the second gives. Since P() = + + 0, we reall need +, not. That is, we need + more. To get this, the? must be, as when in the first bracket is multiplied b in the second bracket, + results. That is, we have deduced P() = ( ) ( + 0). Factorising the second bracket gives P() = ( )( 5)( + ) This procedure, which we will call short division, can be confusing at first, but with persistence can be a quick and eas method for factorising cubic polnomials. The following worked eample is a repeat of a previous one, but eplains the use of short, rather than long, division. WORKED Eample Use short division to factorise THINK WRITE Name the polnomial. Let P() = Look for a value of such that P() = 0. Tr P( ) P( ) = ( ) 5 ( ) ( ) + 4 = = = 0 So ( + ) is a factor. P( ) does equal 0, so ( + ) is a factor. Look again at the original P() = The first term in the brackets must be, and the last term must be. P() = ( + )( + ) Imagine the epansion of the P() = ( + )( 7 + ) epression in step. We have and, but require 5. We need an etra 7. We get this b inserting a 7 term in the second pair of brackets. Factorise the second bracket if possible. P() = ( + )( )( 4)

19 remember remember Chapter Cubic functions 7 To factorise a cubic polnomial:. let P() = the given polnomial. use the factor theorem to find a linear factor (tr factors of the constant term). use long or short division to find the quadratic factor 4. factorise the quadratic factor if possible. F Factorising cubic polnomials WORKED Eample Finish each division below, and hence factorise each dividend. 7a a + ) b + ) c + 9) d + ) e + ) f + 7) g + ) h + ) i + 5) j ) k ) l + 5) m + ) n + 6) + 6 WORKED Eample 7, 8 Factorise the following as full as possible. a + b + c d + 8 e f g + h 7 6 i + 4 j k l m 8 + n Factorise as full as possible. a b c + 8 d e f g h i j Factorise as full as possible. a 0 b 4 + c 6 4 d 8 e 6 6 f 7 g + + h + 0 i j * k Factorising EXCEL Polnomials zero search Mathcad Spreadsheet

20 8 Mathematical Methods Units and Sum and difference of two cubes Two special cases of cubic polnomials, called sum and difference of cubes are discussed in this section. As was the case with difference of squares in the quadratic functions chapter, there are shortcuts for factorising such epressions. Eamples of each are shown in the table below. Sum of cubes Difference of cubes b w 6 ( + ) (uv) Consider the following epansions. (a + b)(a ab + b ) and (a b)(a + ab + b ) = a a b + ab + ba ab + b = a + a b + ab ba ab b = a a b + ab + a b ab + b = a + a b + ab a b ab b = a + b = a b These epansions show that: a + b = (a + b)(a ab + b ) and a b = (a b)(a + ab + b ). That is, we have two formulas which ma be used to factorise sums and differences of cubes. Another wa of proving these results would be to perform the following long divisions: (a + b ) (a + b) and (a b ) (a b). WORKED Eample Factorise using the sum or difference of cubes formula: a 000 b ( + 6) + 6. THINK WRITE a Write the epression. a Recognise a difference of cubes. = (0) Identif a and b for use with the a =, b = formula a b = (a b)(a + ab + b ). Use the formula to factorise. = ( 0)[ + (0) + (0) ] Simplif. = ( 0)( )

21 Chapter Cubic functions 9 THINK WRITE b Write the epression. b ( + 6) + 6 Take out a common factor of to produce a sum of cubes. = [( + 6) + 8] = [( + 6) + ] Identif a and b for use with the formula a = ( + 6), b = a + b = (a b)(a + ab + b ). 4 Appl the sum of cubes formula. = [( + 6) + ][( + 6) ( + 6)() + ] 5 Simplif the contents of the square = ( + 8)( ) brackets. = ( + 8)( ) remember remember Sum of two cubes: a + b = (a + b)(a ab + b ) Difference of two cubes: a b = (a b)(a + ab + b ) G Sum and difference of two cubes Identif a and b (as used in the above sum and difference of cubes epressions) in each of the following (do not factorise). a + 6 b 8 + z c ( + 5) + 7 d 64h e -- c e g f t u WORKED Eample 9a WORKED Eample 9b Factorise the following using the sum or difference of cubes formula. a 5 b j + k c 8 d 7 + e 64t 6u f g p h 7r 79 8 i (k) -- j s t + g 6 Factorise: a (a ) + a b ( + ) 8 c ( + ) + d (w 5) w e (m + p) + (m p) f 7 ( + ) g ( + 7) + ( ) h ( + ) + ( 4) i ( 4p) (p + ) j (5 9) (7 ) k l 54 m a + n 6( + ) + 6

22 0 Mathematical Methods Units and Cubic equations A cubic equation is tpicall an epression that ma be epressed in the form P() = 0, where P() is a cubic polnomial, and ma be solved b isolating or factorising P() and appling the null factor law. Isolating Equations of the form a( b) + c = 0 ma be solved b isolating as follows. a( b) = c ( b) c = -- a c b = -- a c = b + -- a Unlike a square root, a cube root can be onl positive or negative, but not both, for eample, 8 =, 8 =. WORKED Eample 0 Solve ( + ) + 9 = 0 b isolating. THINK WRITE Write the equation. ( + ) + 9 = 0 Subtract 9 from both sides. ( + ) = 9 Divide both sides b. ( + ) = 64 4 Take the cube root of both sides. + = 4 5 Subtract from both sides and simplif. = 4 = 6 Mathcad Solving cubic equations The mathcad file Solving cubic equations on the Maths Quest CD-Rom can be used for questions like that in worked eample 0. Factorising to solve cubic equations The Null Factor Law applies to cubic equations just as it does for quadratics. If P() = ( a)( b)( c) = 0, then solutions are: = a, = b and = c. If P() = k(l a)(m b)(n c) = 0, solutions are found b solving the following equations: l a = 0, m b = 0 and n c = 0.

23 WORKED Eample Chapter Cubic functions Solve: a = 9 b = 0 c = 0. THINK WRITE a Write the equation. a = 9 Rearrange so all terms are on the left. 9 = 0 Take out a common factor of. ( 9) = 0 4 Factorise the brackets using a difference ( + )( ) = 0 of squares. 5 Use the Null Factor Law to solve. = 0, + = 0 or = 0 = 0, = or = b Write the equation. b = 0 Take out a common factor of. ( 5) = 0 Factorise the brackets. ( 7)( + 5) = 0 4 Use the Null Factor Law to solve. = 0, 7 = 0 or + 5 = 0 = 0, = 7 or = 5 c Name the polnomial. c Let P() = Use the factor theorem to find a factor (search for a value a such that P(a) = 0). Consider factors of the constant term (that is, factors of 9 such as, ). The simplest value to tr is. P() = = 0 So ( ) is a factor Use long or short division to find ) another factor of P() P() = ( )( 9 + 9) Factorise the brackets. P() = ( )( )( ) Consider the factorised equation to solve. For ( )( )( ) = 0 Use the Null Factor Law to solve. = 0, = 0 or = 0 =, = -- or = remember remember To solve a cubic equation:. let P() =.... use the factor theorem (tr P() etc.) to find a factor of the form ( a). use long or short division to find a quadratic factor 4. factorise the quadratic factor if possible 5. let each linear factor equal zero and solve for in each case.

24 Mathematical Methods Units and H Cubic equations GC Mathcad Mathcad program Solving cubic equations Cubic roots Solving cubic equations WORKED Eample 0 WORKED Eample a, b WORKED Eample c Solve the following b isolating. a ( ) 50 = 0 b ( + ) + 8 = 0 c ( 4) 000 = 0 d ( + 7) 8 = 0 e ( 5) = 0 f ( + ) + = 0 g ( + ) 7 = 0 h 4( ) = 0 i ( + ) = 0 j -- (5 ) = 0 k ( 5) 4 = 4 l 4 -- ( + 8) = 04 Find all solutions of the following cubic equations. a ( )( )( 5) = 0 b ( + )( + 4)( + 7) = 0 c ( 5)( + )( 9) = 0 d ( 4)( + )( ) = 0 e ( + )( 4)( + 4) = 0 f ( )( + )( + ) = 0 g ( + 5)( 8) = 0 h ( ) = 0 i ( + )( ) = 0 j ( 9) = 0 k (6 ) ( + ) = 0 l ( + 7) = 0 m (5 6)( + ) = 0 n ( 4) (5 ) = 0 Solve: a 4 = 0 b 6 = 0 c 50 = 0 d + 8 = 0 e + 5 = 0 f = 0 g = 0 h + = 0 i 4 0 = 0 j = 0 k = 0 l + 6 = 7 m 9 = 0 + n + 6 = 4 4 Use the factor theorem and solve the following. a = 0 b = 0 c = 0 d = 0 e = 0 f = 0 g = 0 h = 0 i = 0 j = 0 k = 0 l = 0 m = 0 n = 0 5 multiple choice Which of the following is a solution to ? A 5 B 4 C D E 5 WorkSHEET. 6 multiple choice A solution of = 0 is = 5. How man other (distinct) solutions are there? A 0 B C D E 4

25 Chapter Cubic functions Solving cubic equations using graphs Graphs produced using technolog (as opposed to sketch graphs dealt with in later sections of this tet) ma be used to find solutions of otherwise unwield cubic equations. To solve an equation such as + 7 = 0, using a graphics calculator, consider the left hand side of the equation to be Y, and attempt to find where Y = 0, that is, find the intercepts of the graph of Y = + 7. The steps involved when using a graph to solve an equation are as follows: Ensure the equation is in the form f () = 0. ENTER f () in the Y= menu. Press GRAPH, and alter WINDOW settings or ZOOM until all -intercepts are visible. 4 Use nd [CALC] and :Zero to find an intercept (use the arrow kes as required to scroll before and after each intercept, pressing ENTER at each stage). 5 Repeat for each intercept. Use a graphics calculator or graphing software to find all solutions to the following equations = = = = = = = = 0

26 4 Mathematical Methods Units and Cubic graphs intercepts method A good sketch graph of a cubic function shows:. - and -intercepts,. the behaviour of the function at etreme values of, that is, as approaches infinit ( + ) and as approaches negative infinit ( ), and. the general location of turning points. Note that for cubic functions, humps are not smmetrical as the are for parabolas, but are skewed to one side. The graphs below show the two main tpes of cubic graph. Turning points A positive cubic A negative cubic Point point of inflection Sometimes the turning point occurs not on the peak or trough of a hump, but at a point called a point of inflection, where the graph changes from decreasing gradient to increasing gradient (or vice versa). This occurs when there is onl one -intercept. Cubic with a point of inflection Consider the general factorised cubic f () = ( a)( b)( c). The Null Factor Law tell us that f () = 0 when = a or = b or = c. The -intercept occurs when = 0, that is, the -intercept is f (0) = (0 a)(0 b)(0 c) = abc c b a abc

27 Chapter Cubic functions 5 WORKED Eample Sketch the following, showing all intercepts: a = ( )( )( + 5) b = ( 6) (4 ) c = ( ). THINK a Note that the function is alread factorised and that the graph is a positive cubic. 4 The -intercept occurs where = 0. Substitute = 0 into the equation. Use the Null Factor Law to find the -intercepts. (Put each bracket = 0 and solve a mini-equation.) Combine the above steps to sketch. WRITE a = ( )( )( + 5) -intercept: if = 0, = ( )( )(5) = 0 Point: (0, 0) -intercepts: if = 0, = 0, = 0 or + 5 = 0 =, = or = 5 Points: (, 0), (, 0), ( 5, 0) 0 5 b 4 The graph is a negative cubic (the in the last factor produces a negative coefficient if the RHS is epanded). Substitute = 0 to find the - intercept. Use the Null Factor Law to find the -intercepts. (Put each bracket = 0 and solve a mini-equation.) Combine all information and sketch the graph. Note the skimming of the -ais indicative of a repeated factor, in this case the ( 6) part of the epression. b = ( 6) (4 ) -intercept: if = 0, = ( 6) (4) = 44 Point: (0, 44) -intercepts: if = 0, 6 = 0or4 = 0 = 6or = 4 Points: (6, 0), (4, 0) Continued over page

28 6 Mathematical Methods Units and THINK WRITE c Positive cubic. c = ( ) 4 Substitute = 0 to find the -intercept. -intercept: if = 0, = ( ) Use the Null Factor Law to find the -intercept. (Put each bracket = 0 and solve a mini-equation.) Combine all information and sketch the graph. The cubed factor, ( ), indicates a point of inflection and onl one -intercept. = 8 -intercept: if = 0, = 0 = 8 If a cubic function is not in the form f () = ( a)( b)( c), we ma tr to factorise to find the -intercepts. We can use the factor theorem and division of polnomials to achieve this. WORKED Eample Sketch the graph of = showing all intercepts. THINK WRITE Write the equation, and name the = P () = polnomial, P (). Note the graph is a positive cubic. 4 5 Let = 0 to find the -intercept. Note: All terms involving are equal to zero. Factorise P () before finding -intercepts. ( ) is not a factor. Use long or short division to factorise P (). Here, short division has been used. -intercept: if = 0, = 0 Point: (0, 0) P () = = = 8 0 P ( )= ( ) ( ) + 7 ( ) + 0 = = 0 So ( + ) is a factor. P () = ( + )( + 0) = ( + )( 5)( 4)

29 Chapter Cubic functions 7 THINK 6 Write down the -intercepts (determined b putting each bracket = 0 and solving for ). 7 Use all available information to sketch the graph. WRITE -intercepts: if = 0, 5 =, --, 4 5 Points: (, 0), ( --, 0), (4, 0) We can obtain graphs like the one shown in worked eample b using technolog. For eample, a Mathcad file found on the Maths Quest CD-Rom produces the following: Cubic graphs general form Mathcad remember remember To sketch a cubic function of the form f () = A + B + C + D:. determine if the epression is a positive or 0 negative cubic (that is, if A is positive or = ( +)( 5)( 4) negative). find the -intercept (let = 0). factorise if necessar and/or possible, for eample, obtain an epression in the form f () = ( a)( b)( c) find the -intercepts (let factors of f () equal 0) 5. use all available information to sketch the graph, for eample,

30 8 Mathematical Methods Units and I Cubic graphs intercepts method EXCEL EXCEL Mathcad Spreadsheet Mathcad Cubic graphs factored form Cubic graphs factored form Cubic graphs general form Spreadsheet Cubic graphs general form WORKED Eample WORKED Eample Sketch the following, showing all intercepts. a = ( )( )( ) b = ( )( 5)( + ) c = ( + 6)( + )( 7) d = ( + 4)( + 9)( + ) e = ( + 8)( )( + ) f = ( 6)( )( + ) g = ( 5)( + 4)( ) h = ( + 7)( 5)( + 6) i = (4 )( + )( 4) j = ( + )( )( + ) k = ( ) ( 6) l = ( + )( + 5) Sketch the following (a miture of positive and negative cubics). a = ( )( + 5)( + ) b = ( )( + 7)( ) c = ( + 8)( 8)( + ) d = ( )( )( + 6) e = ( + )( ) f = ( + )( )( + ) g = ( + )( + 0)( + 5) h = ( 4) i = 4 ( + 8) j = (5 )( )( + 9) k = (6 ) ( + 7) l = (7 + ) Sketch each of the following as full as possible. a = + b = c = d = 4 + e = f = g = h = i = 5 j = k = l = m = n = o = 8 p = + 4 Sketch: a = b = + 6 c = d = e = 5 9 f = g = h = i = j = multiple choice Which of the following is a reasonable sketch of = ( + )( )( + )? A C E B D

31 Chapter Cubic functions 9 6 multiple choice The graph shown could be that of: A = ( + ) B = ( + ) C = ( )( + ) D = ( ) ( + ) E = ( ) 7 multiple choice The graph below has the equation: 8 6 A = ( + )( + )( + ) B = ( + )( )( + ) C = ( )( + )( + ) D = ( + )( + )( ) E = ( )( )( ) 8 multiple choice If a, b and c are positive numbers, the equation of the graph shown below is: A = ( a)( b)( c) B = ( + a)( b)( + c) C = ( + a)( + b)( c) D = ( + a)( + b)( + c) E = ( a)( + b)( c) b c a 9 multiple choice Which of the following has onl distinct -intercepts when graphed? A = ( + )( + ) B = ( + )( + )( + ) C = D = ( + )( + ) E = ( + )( ) Repeated factors Use graphing software such as Mathcad or Graphmatica to investigate the effect of a repeated factor on the graph of functions such as = ( ) ( + ) and = ( ). Stud the effect of repeated factors on the graphs of higher order functions such as = ( + ) ( ) and = ( + ) 4 ( ).

32 40 Mathematical Methods Units and Mathcad Cubic graphs basic form Cubic graphs using translation Remember the turning point form for quadratic graphs, which was related to dilations and translations of the basic parabola? The same understanding of translation can be used to sketch cubic functions. You ma wish to investigate graphs of = a( b) + c using a graphics calculator and tring various values for a, b and c. It can be shown that equations in the form = a( b) + c when graphed have the basic shape of = or =, with various dilations and translations applied. Such graphs have a point of inflection (POI) at (b, c). A point of inflection is where a graph levels off to have a zero gradient at one point with the same sign gradient either side. = = EXCEL Spreadsheet Cubic graphs basic form Point of inflection = a( b) + c Dilation factor (-stretch) -translation -translation (b, c) b c Point of inflection The effect of a (the -dilation factor) is illustrated below. = = = = = = Positive a (-dilation) Negative a (-dilation) Intercepts Intercepts ma be found b substituting = 0 (to find the -intercept) and = 0 (to find the -intercept).

33 WORKED Eample 4 Chapter Cubic functions 4 Sketch the graph of each of the following, showing the point of inflection and intercepts. a = ( ) + b = + 54 c = ( ) 6 THINK WRITE a Compare the equation with a = ( ) + = a( b) + c, having POI (b, c). Note the values that match, namel a =, b = and c =. State the POI (b, c). POI (, ) Find the -intercept. If = 0, = (0 ) + = ( 8) + = 4 Find the -intercept. If = 0, 0 = ( ) + ( ) = ( ) = = = 5 Note that the equation is for a positive cubic. Sketch, showing the point of inflection (POI) and intercepts. 6 (, ) b 4 Manipulate into = a( b) + c form. b = + 54 = ( 0) + 54 Note the graph is a negative cubic POI (0, 54) with POI (0, 54). Find the -intercept. If = 0, = (0) + 54 = 54 Find the -intercept. If = 0, 0 = + 54 = 54 = 7 = Continued over page

34 4 Mathematical Methods Units and THINK WRITE 5 Sketch, showing the POI and intercepts. (0, 54) c Manipulate into = a( b) + c form. c = ( ) 6 = [ ( -- )] 6 = [ 8( -- ) ] 6 = 6( ) -- 6 Note the graph is a positive cubic with POI ( --, 6) POI ( --, 6). Find the -intercept. If = 0, = ( 0) 6 = () 6 = 8 4 Find the -intercept. If = 0, 0 = ( ) 6 ( ) = 6 ( ) = 8 = = Sketch, showing POI and intercepts. 5 = -- 8 (, 6) remember remember Cubic with -dilation of a and POI at (b, c) = a( b) + c a > 0 a < 0 (b, c) (b, c) Positive a Negative a

35 Chapter Cubic functions 4 J Cubic graphs using translation WORKED Eample 4 Without sketching graphs for each of the following, state: i the -dilation factor ii the coordinates of the point of inflection. a = ( ) + b = ( + 5) c = ( 6) 8 d = 7( + 4) + e = ( 9) + 4 f = 7 g = ( + ) h = ( + ) -- i = -- ( ) + j = 4 k = 4 -- l = Sketch the graph of each of the following, showing the point of inflection and intercepts. a = ( ) + b = ( + ) + 8 c = 4( 4) Cubic d = 5( ) + 5 e = 8 f = graphs basic forms g = ( + ) + 7 h = ( + 5) -- i = -- ( ) 9 Sketch the following, showing the point of inflection. Intercepts are not required. a = (4 ) + b = (5 ) c = (4 ) 4 multiple choice The basic cubic graph = undergoes a -dilation of 6 and is translated right 4 units and down units. The equation for this graph is: A = 6( 4) B = ( 4) 6 C = 6( ) 4 D = 4( + 6) + E = 4( + ) j = -- ( + ) + 4 k = -- ( + ) l = m = 5 n = d = 5( ) + e = -- ( ) f = ( 4) g = (4 ) 5 + h = (9 5) i = -- (6 ) + 4 j = (5 ) multiple choice The graph of = 5( ) + 9 has a point of inflection at: A (5, ) B (5, 9) C (, 9) D (, 9) E (, 9) 6 Suggest a possible equation for each of the following, given that each is a cubic with -dilation or. a b c d Cubic graphs basic form EXCEL Mathcad Spreadsheet (, 5) (, 4) (, 0) (, ) 7 Write an equation for a cubic with: a -dilation 4, point of inflection (, ) b -dilation, point of inflection ( 5, ) c -dilation --, point of inflection (, ) d -dilation --, point of inflection (0, 4). 4

36 44 Mathematical Methods Units and SkillSHEET. Domain, range, maimums and minimums The domain of a function is the set of - coordinates of points on its graph. The range is the set of -coordinates of points on the graph. Normall, the domain and range of a cubic function are the set of all real numbers, or R for short, as such graphs etend indefinitel in both positive and negative ais directions. The domain and range of a restricted cubic function ma be a smaller set of numbers. The restricted graph (above right) has a domain of -values between 5 and, denoted [ 5, ]. The range is [ 8, 0]. Square brackets are used to indicate that an end value is included, and correspond to a small coloured-in circle on the graph. If an end value is not included, a curved bracket is used. We show such points on a graph using a hollow circle. As can be seen below, the local maimum and minimum values of a cubic graph ma decide the range. These values ma be obtained using a graphics calculator. (5, 4) Range = [ 8, 0] Actual maimum (within given domain) Local maimum 0 Range = [ 8, 0) 5 8 Domain = [ 5, ] Domain = ( 5, ] = ( )( + )( ) ( 0.58,.9) = ( )( + )( ) (.58,.8) (, 0) Range = [ 0, 4] Range = [.8,.9] Function notation When we wish to conve information about the domain of a function, the following notation ma be used: f : [ 4, ] R where f () = ( )( + )( + 4) The name of the function The domain The rule for the function The range is within this set. Note: the range is not necessaril equal to R; the range is within R.

37 Chapter Cubic functions 45 Graphics Calculator tip! Turning points To find the coordinates of a local minimum or maimum turning point of a function entered as Y in the Y= menu,. Obtain a suitable view of the GRAPH.. Press nd [CALC] and sketch :minimum.. Scroll just to the left of the minimum point and press ENTER. 4. Scroll just to the right of the minimum point and press ENTER 5. Press ENTER again. A similar approach is used to find a local maimum. Just use 4:maimum at step above. WORKED Eample For the function f : [ 4, ] R where f () = ( )( + )( + 4): sketch the graph of f () showing intercepts and the coordinates of an local maimum or local minimum and state the range. THINK WRITE/DISPLAY Write the ke information from the question. f : [ 4, ] R where f () = ( )( + )( + 4) 4 5 Find the -intercept. If = 0, f () = ( )()(4) = 8. Use the Null Factor Law (or a If f () = 0, =, or 4. graphics calculator) to determine -intercepts. Use the maimum facilit of a graphics calculator to determine the coordinates of the local maimum. 5 Use a graphics calculator to determine the coordinates of the local minimum. Show the above information on a sketch. 6 (., 4.06) ( 4, 0) (, 0) ( 0., 8.) Use the graph to state the range. Range = [ 8., 4.06].

38 46 Mathematical Methods Units and The maimum and minimum of a graph within a certain domain are not necessaril the values of the local maimum or minimum. Sometimes an etreme value is simpl the -coordinate of an end point of a graph. WORKED Eample 6 Local maimum Absolute minimum Absolute maimum Local minimum Sketch f : [0, 7) R where f () = ( 5) showing intercepts, end points, the local maimum and minimum, and state the range. THINK WRITE Write the ke information from the f : [0, 7) R where f () = ( 5) question. Find the -intercept. If = 0, f () = (0)(0 5) ) = 0. Use the Null Factor Law (or a graphics If f () = 0, = 0, or 5. calculator) to determine -intercepts. 4 Use the value facilit of a graphics calculator to determine the -coordinate when = 7. It is clear that this value is greater than the local maimum. Using a graphics calculator we can establish that the absolute minimum and local minimum are at = 0. Show the above information on a sketch. 5 (0, 0) (7, 8) 6 Use the graph to state the range. Range = [0, 8). Domain and range will be discussed in more detail in chapter 6, Relations and functions. A method of finding maimums and minimums without a graphics calculator will be covered in the stud of calculus later in this book. remember remember The absolute maimum or minimum is either the -value at a local maimum or minimum or the -value at an end of the domain.

39 Chapter Cubic functions 47 K Domain, range, maimums and minimums A graphics calculator is required for this eercise. State the domain and range of the sections of graph shown in each case. SkillSHEET. a b c (, 5) (, 7) ( 4, ) ( 5, ) (, ) (4, 0) (, ) (4, ) (, 5) (, ) d (6, 0) e f (5, 5) (, ) (, 0) (4, ) (, 0) (, 0) (, 8) (4, 5) (, ) (, 9) WORKED Eample 5, 6 For each of the following, sketch the graph (showing local maimums and minimums, and intercepts) and state the range. a f : [, 4] R where f () = ( )( 4)( + ) b f : [ 5, ] R where f () = ( + )( + 5)( ) c f : [, ) R where f () = ( ) ( ) d f : (, 0] R where f () = ( + )( + ) e f : [ 8, ) R where f () = ( )( + )( + 7) f f : [0, 4] R where f () = g f : [ 4,.44] R where f () = h f : (,.) R where f () = i f : [, 5.] R where f () = j f : (, -- ) R where f () = multiple choice The range of the function shown below is: A [ 4.06, 8] B [ 4.06, 8.09] C (0, 8] D (, ] E (, 4] ( 0.786, 8.09) 6 (4, 8) (.0, 4.06)

40 48 Mathematical Methods Units and 4 multiple choice Point A on the curve is: A an intercept B a local minimum C an absolute minimum D a local maimum E an absolute maimum 5 A roller-coaster ride is modelled b the function f () = 0.00( 0)( + 0)( 40). a What is the height above ground level of the track at = 50? b How far apart verticall are points A and B? A 0 0 A B ground level 40 6 The course of a river as marked on a map follows the curve defined b the function f () = Find the coordinates of the southern-most point on the river between = 0 and =. N

41 Chapter Cubic functions 49 Modelling Scientists, economists, doctors and biologists often wish to find an equation which closel matches, or models a set of data. For eample, the wombat population of a particular island ma var as recorded in the following table. Year of stud () Wombat population (W) The graph shows these data, with a possible model for the wombat population superimposed. We will eamine polnomial models up to degree, that is, models of the form: = a + a + a + a 0 where a 0, a, a and a are constants. Wombat population = Year Modelling using technolog Several technological options are available to assist in obtaining models for data, including the graphics calculator, spreadsheets and computer algebra sstems such as Mathcad. Man of these applications use a method involving minimising the sum of the squares of the vertical distances of the data points from the graph of the function known as the least squares method = Sum of squares of these lengths is minimised.

42 50 Mathematical Methods Units and Modelling using a graphics calculator To find a model or regression function using a graphics calculator, enter the data in L and L, then press nd [CALC], and choose the tpe of regression required. A STAT PLOT can be set up, with the regression line or curve superimposed b entering the regression equation as Y. The following eample emplos cubic regression, but the general approach is the same for all tpes. WORKED Eample 7 Fit a cubic model to the following data using a graphics calculator THINK WRITE/DISPLAY Press STAT and select :Edit. Enter the values as L, and the values as L. Press STAT and choose CALC and 6:CubicReg, and enter L,L,Y. To enter L, press nd [L]. To enter L, press nd [L]. To enter Y, press VARS, select Y-VARS, and :Function and press ENTER. Set up a STAT PLOT as shown. 4 Press GRAPH to view the data and regression curve.

43 Chapter Cubic functions 5 Modelling using a spreadsheet Microsoft Ecel has a built-in trendline function, outlined in the following eample. Fit a cubic model to the data in worked eample 7 using a spreadsheet. THINK 4 5 WORKED Eample 8 Enter columns of data, with optional headings as shown. Click on the chart wizard toolbar icon, and follow the necessar steps to create a scatter plot. Click on the chart, and select Chart and Add Trendline from the top menu. Choose the degree of polnomial required (in this case for cubic). The equation should be displaed on the chart. If not, select this option b double clicking on the trendline and choosing Options and Displa equation on chart. WRITE/DISPLAY Modelling using Mathcad (a computer algebra sstem) Your Maths Quest CD contains the program Mathcad Eplorer which ma be used to open the Maths Quest files Linear modelling, Quadratic modelling and Cubic modelling (also on the CD). These files ma be simpl edited to create curves and equations to model data. WORKED Eample 9 Fit a cubic model to the following data in worked eample 7 using the Mathcad file Cubic modelling. THINK WRITE/DISPLAY Overtpe the and values in each f() ellow matri with our new values. 600 Scroll down the worksheet to view the equation and graph. data 400 j If necessar, change the ais scales b 00 clicking on the graph and entering new values in the placeholders on each ais data j Cubic modelling EXCEL Modelling Mathcad Spreadsheet

44 5 Mathematical Methods Units and remember remember An equation to model data ma be obtained using the following:. Graphics calculator: Enter data as lists, and find the regression equation (linear, quadratic etc.).. Spreadsheet: Use the chart wizard and fit a trendline to the plotted points.. Mathcad (computer algebra sstem): Use the Maths Quest files Linear modelling, Quadratic modelling and Cubic modelling. L Modelling using technolog EXCEL Spreadsheet Mathcad Modelling Linear Modelling WORKED Eample 7,8,9 Use a graphics calculator, spreadsheet or Mathcad file to answer the questions in this eercise. In each case, find an equation to model the data, and draw a rough sketch of the graph. Find a linear model for each of the following sets of data. a b c d EXCEL Spreadsheet Modelling Find a quadratic model for each of the following sets of data. a Mathcad Quadratic Modelling b c d

45 Chapter Cubic functions 5 Find the cubic model for each of the following sets of data. a b c EXCEL Modelling Cubic Modelling Spreadsheet Mathcad d For the following data set, find and sketch: a a linear model b a quadratic model c cubic model. 5 Which of the models in question 4 fits best? 6 Use the model from question a to predict the value of when = 0. 7 Use the model from question a to predict the value of to the nearest unit when = Use the model from question a to predict the value of to the nearest unit when =. 9 The value of shares in the compan Mathsco is plotted b a sharemarket analst over a -month period as shown. a b c Month 0 J Share price J A S Find and sketch a quadratic model for the data. Use our model to predict the share price months later. Wh is such a prediction fraught with danger? 4 O 5 N D 7 J 8 F 9 M 0 A M

46 54 Mathematical Methods Units and 0 The population of a colon of ellow bellied sap-suckers on an isolated island is studied over a number of ears. The population at the start of each ear is shown in the table below. Year Population Find and sketch a cubic model for the population, and use it to estimate the population at the start of ear. * Investigate the improvement provided b higher order polnomial models for the data in question 0. Career profile ASHLEY HANNON Profession: Manager 5 mm Packaging Qualifications: Bachelor of Engineering in Mechanical Engineering Emploer: Kodak Australia Compan website: There are a number of Kodak sites M interests lie in machiner and manufacturing. As a manager, m job involves assessing departmental performance around productivit, qualit, service deliver and safet. It is m responsibilit to ensure appropriate corrective action if necessar. M duties also include prioritising m team s activities, sponsoring improvement teams and planning for marketing requirements. M job involves the development of statistical control charts, machine performance modelling, capital project budgeting and analsis measures (such as first pass ield and defects per million opportunities). I use UMP software to develop mathematical models that emulate production processes. This is useful for better control of processes and in budgeting for production labour and material costs. I also process capabilit (in terms of sigma analsis). I believe modern manufacturing could not function without the regular use of mathematics, particularl as the environment that we work in becomes more competitive. We use mathematics and statistics to ensure product qualit, to control our costs and to statisticall control our processes. Questions. What does Ashle use mathematics and statistics for?. How do mathematical models assist Ashle in his work?. What other engineering, besides mechanical engineering, can ou major in?

47 Chapter Cubic functions 55 Fitting a model eactl Use the regression capabilit of a graphics calculator to find a linear model (equation) for the following data, and plot the equation and the data together. a b 4 5 c Use the regression capabilit of a graphics calculator to find a quadratic model (equation) for the following data, and plot the equation and the data together. a b c Use the regression capabilit of a graphics calculator to find a cubic model (equation) for the following data, and plot the equation and the data together. a b c When can ou be certain (without plotting or calculating) that: a a linear model will go through ever data point? b a quadratic model will go through ever data point? c a cubic model will go through ever data point? 5 What can ou sa about data sets in general, and the likelihood of a polnomial model fitting each point eactl if the highest power of the polnomial is n? EXCEL Modelling Spreadsheet

48 56 Mathematical Methods Units and Finite differences If pairs of data values in a set obe a polnomial equation, that equation or model ma be found using the method of finite differences. Consider a difference table for a general polnomial of the form = a + a + a + a 0. We begin the difference table b evaluating the polnomial for values of 0,, etc. The differences between successive -values (see table) are called the first differences. The differences between successive first differences are called second differences. The differences between successive second differences called the third differences. We will call the first shaded cell (nearest the top of the table) stepped cell, the second shaded cell stepped cell and so on. (= a + a + a + a 0 ) First differences Second differences Third differences 0 a 0 a + a + a a + a + a + a 0 6a + a 7a + a + a 6a 8a + 4a + a + a 0 a + a 9a + 5a + a 6a 7a + 9a + a + a 0 8a + a 7a + 7a + a 6a 4 64a + 6a + 4a + a 0 4a + a 6a + 9a + a 5 5a + 5a + 5a + a 0 If a 0, the above polnomial equation represents a cubic model, and the third differences are identical (all equal to 6a ). If a = 0, a 0 and the polnomial reduces to = a + a + a 0, that is, a quadratic model, and the second differences become identical (all equal to a ). If a = 0 and a = 0, the polnomial becomes = a + a 0, that is, a linear model, and the first differences are identical (all equal to a ).. Stepped cell = a 0. Stepped cell = a + a + a. Stepped cell = a + 6a 4. Stepped cell 4 = 6a

49 Chapter Cubic functions 57 WORKED Eample 0 Complete a finite difference table based on the data below, and use it to determine the equation for in terms of THINK Place the data in columns as shown, allowing space for difference columns. Calculate the first differences and place them in the net column. The first differences are not constant, so we need to find the second differences. Calculate these and place them in the net column. The second differences are constant, so our table is complete. Showing the third differences is optional. Recall the stepped cell equations, and equate them to the shaded cells as shown: Stepped cell = a 0 Stepped cell = a + a + a Stepped cell = a + 6a Stepped cell 4 = 6a Here, an * is used to denote solved values. [] gives a 0 = and [4] ields a = 0. Substitute this information into [] and []. WRITE Substitute a = 0 into [5] to find a. a + = a = * Use the asterisked values to build the equation: = a + a + a + a 0. 0 Differences st nd rd a 0 = * [] a + a + a = [] a + 6a = 6 [] 6a = 0 [4] So a = 0* Sub a = 0 into []: a + a + 0 = a + a = [5] Sub a = 0 into []: a = 6 a = 6 a = * Sub a = into [5]: = a + a + a + a 0 becomes = (0) + () + ( ) + ( ) =

50 58 Mathematical Methods Units and The stepped equations work onl if the finite differences table begins with = 0, and increases in steps of. It ma be necessar on occasions to adjust the table to achieve this, as the following eample shows. WORKED Eample Complete a finite difference table based on these data and use it to determine the equation for in terms of THINK 4 Construct a difference table, leaving room for = 0. Calculate and fill in the first differences where possible. Note the first differences are constant, so last two columns are optional. Working backwards, the first stepped cell must be in order for the difference between it and the net cell to be 8. That is: (net cell) (first stepped cell) = 8. WRITE Differences st nd rd Recall the stepped cell equations, and equate them to the shaded cells as shown: Stepped cell = a 0 Stepped cell = a + a + a Stepped cell = a + 6a Stepped cell 4 = 6a Here, an * is used to denote solved values. [] gives a 0 =, [4] ields a = 0, and hence [] ields a = 0. Substitute this information into []. Use the asterisked values to build the equation: = a + a + a + a 0. a 0 = * [] a + a + a = 8 [] a + 6a = 0 [] 6a = 0 [4] So a = 0* and a = 0* Sub a = 0 and a = 0 into []: a = 8 a = 8* = a + a + a + a 0 becomes = (0) + (0) + (8) + ( ) = 8

51 remember remember Chapter Cubic functions 59 To determine the polnomial equation using the method of finite differences:. set up a table and find the first, second and third differences. equate the stepped cell equations to the appropriate shaded or circled cells in the table. find the coefficients a 0, a, a, a for the equation = a + a + a + a 0. Stepped cell : a 0 Stepped cell : a + a + a Stepped cell : a + 6a Stepped cell 4: 6a M Finite differences WORKED Eample 0 Complete a finite difference table, and use it to determine the equation for in terms of for each of the following data sets. a b c WORKED Eample d e f g

52 60 Mathematical Methods Units and h i j k l Triangular numbers ma be illustrated as shown at right. If is the number of dots on the base of each diagram, and is the total number of dots: a complete the table below b find an equation linking and (base dots) (total dots) c find the total number of cans in the supermarket displa shown at right using the equation found in b, and check our answer b counting cans. The diagonals in polgons of various tpes are shown below in red. Find the relationship between the number of dots () and the number of diagonals (n). (Hint: Continue patterns in a difference table so that it is completed back to = 0.) = = = = 4 4 If n is the number of different squares that can be found within a square grid of edge length, find an equation for n in terms of, and use this equation to find the number of different squares on a chessboard. = n = n = = 5 = n =?

53 Chapter Cubic functions 6 summar Epanding When epanding three linear factors:. epand two factors first, then multipl the result b the remaining linear factor. collect like terms at each stage. ( + ) ma be written as ( + )( + )( + ). Long division of cubic polnomials Long division of polnomials is similar to long division with numbers. The highest power term is the main one considered at each stage. Ke steps are:. How man?. Multipl and write the result underneath.. Subtract. 4. Bring down the net term. 5. Repeat until no pronumerals remain to be divided. 6. State the quotient and the remainder. Polnomial values P (a) means the value of P () when is replaced b a and the polnomial is evaluated. The remainder and factor theorems Remainder R = P (a), when P() is divided b a. If P (a) = 0, then ( a) is a factor of P (). Factorising cubic polnomials To factorise a cubic polnomial:. let P () = the given polnomial. use the factor theorem to find a linear factor. use long or short division to find the quadratic factor 4. factorise the quadratic factor if possible. Sum and difference of cubes a + b = (a + b)(a ab + b ) a b = (a b)(a + ab + b ) Cubic equations To solve a cubic equation:. let P () =.... use the factor theorem (tr P () etc.) to find a factor of the form ( a). use long or short division to find a quadratic factor 4. factorise the quadratic factor if possible 5. let each linear factor equal zero and solve for in each case. Cubic graphs intercepts method To sketch a cubic function of the form f () = A + B + C + D:. determine if the epression is a positive or negative cubic (that is, if A is positive or negative). find the -intercept (let = 0)

54 6 Mathematical Methods Units and. factorise if necessar or possible, for eample, obtain an epression in the form f () = ( a)( b)( c) 4. find the -intercepts (let factors of f () equal 0) 5. use all available information to sketch the graph, for eample, Cubic graphs using translation Cubic with -dilation of a and POI at (b, c) = a( b) + c a > a < 0 = ( +)( 5)( 4) 4 (b, c) (b, c) Positive a Negative a Domain, range, maimums and minimums The absolute maimum or minimum is either the -value at a local maimum or minimum, or the -value at an end of the domain. Modelling using technolog An equation to model data ma be obtained using the following:. Graphics calculator: Enter data as lists, and find the regression equation (linear, quadratic etc.).. Spreadsheet: Use the chart wizard and fit a trendline to the plotted points.. Mathcad (computer algebra sstem): Use the Maths Quest files Linear modelling, Quadratic modelling and Cubic modelling. Finite differences To use the method of finite differences, set up a table as shown, and find differences b subtracting successive values (value previous value). Circle or shade the stepped cells. First differences Second differences 0 Stepped cell Stepped cell Stepped cell Etc. 4 Previous value 5 Value Etc. Value previous value Third differences Stepped cell 4 Use the following equations to determine the polnomial model s coefficients:. Stepped cell = a 0 }. Stepped cell = a + a + a Equation of the polnomial model is. Stepped cell = a + 6a = a + a + a + a 0 4. Stepped cell 4 = 6a

55 Chapter Cubic functions 6 CHAPTER review Multiple choice Which of the following is not a polnomial? A B a 4 + 4a + a + C + + D E 5 The epansion of ( + 5)( + )( 6) is: A 0 B C D 0 E is the epansion of: A ( + ) B ( + )( ) C ( )( + ) D ( )( + )( + ) E ( + )( + )( ) A B B Questions 4 and 5 refer to the following long division ) The quotient is: A 9 B 4 C + 4 D + + E The remainder is: A 9 B C 4 D E If P () = + 7 +, then P ( ) equals: A 4 B C 9 D 7 E 5 7 The remainder when 7 is divided b is: A 6 B C 6 D 7 E 8 8 Which of the following is a factor of ? A ( 4) B ( ) C ( + ) D ( + ) E ( ) factorises to: A ( ) ( + 8) B ( + ) ( + 8) C ( + ) D ( + )( + )( + 4) E ( )( + )( + 4) C C D E E F

56 64 Mathematical Methods Units and G H 0 64 factorises to: A (4 )( ) B (4 )(6 4 + ) C (4 )( ) D (4 + )(6 8 + ) E (4 + )(6 4 + ) Which of the following is the solution to ( 4) = 6? A 6 B C D 4 E 6 H I Which of the following is a solution to ( )( + 5)(7 )? A B -- C -- D -- E The equation for this graph could be: A = ( 5)( + )( + ) B = ( )( )( + 5) C = ( )( + )( + 5) D = ( )( + )( 5) E = (5 )( + )( + ) J 4 Which of the following shows the graph of = ( + 5)? A B C D E (5, ) ( 5, ) ( 5, ) (5, ) ( 5, ) K K L Questions 5 and 6 refer to the following graph. 5 The domain of the graph is: A ( 08., 5] B (.47, 4.8] C ( 08., 60.70] D ( 7, 7] E ( 6, 7] 6 The range of the graph is: A [ 08., 5] B [.47, 4.8] C [ 08., 60.70] D [7, 5) E [0, 5) 7 The regression equation of the graph at right (which passes through the origin) is: A = B = C = + 9 D = E = ( 7, 5) (4.8, 60.70) 5 7 (.47, 08.)

57 Chapter Cubic functions 65 8 The data below obe which tpe of relationship? M A Linear B Quadratic C Cubic D Quartic E None of the above Short answer Consider the polnomial f () = a What is the degree of f ()? b What is the coefficient of 4? c What is the constant term? Epand: a ( ) ( + 0) b ( + 6)( )( + 5) c ( 7) d (5 )( + )( + ). Find the quotient and remainder when the first polnomial is divided b the second in each case. a + 6, + b + 7, c , + 4 If P () = + + 4, find: a P () b P ( 4) c P (a). 5 Without dividing, find the remainder when is divided b. 6 Show that + is a factor of Factorise Factorise: a 5 b ( ) + ( + ). 9 Solve: a 5( + 5) + 5 = 0 b ( + )( ) = 0 c = 0. 0 Sketch: a = ( )( + ) b = c = +. 8 Sketch = -- ( + ) + 8. Find the range of f:[ 5, ] R, where f () = ( + )( )( + 5). Find and sketch a cubic model for the following data set Complete a finite difference table, and use it to determine the equation for in terms of for the following data set. A B C D E E F G H I J K L M

58 66 Mathematical Methods Units and M 5 The following series of diagrams show the maimum number of regions produced b drawing chords in a circle. Find a relationship between the number of chords () and the maimum number of regions (r). = 0 r = = r = = r = 4 = r = 7 Analsis A diagram of a proposed water slide based on a cubic function appears at right. Find: a The height, h, of the top of the slide. b The coordinates of point A (where the slide enters the water). c d The length, L, of the ladder. The height, h, of the mini-hump to the nearest centimetre. ( 5, 0) L h = ( ) A h test ourself CHAPTER An innovative local council decides to put a map of the district on a web site. Part of the map involves two ke features the Cubic River and the Linear Highwa. A mathematicall able web site designer has found the following equations for these features: Cubic River: = Linear Highwa: = a Sketch the river and highwa, showing - and -ais intercepts. b c Find the coordinates of the points of intersection of the highwa and the river. A fun-run organiser wishes to arrange checkpoints at the closest points of intersection. Find the distance between the proposed checkpoints.