Graphs and polynomials

Transcription

1 1 1A The inomial theorem 1B Polnomials 1C Division of polnomials 1D Linear graphs 1E Quadratic graphs 1F Cuic graphs 1G Quartic graphs Graphs and polnomials AreAS of STud Graphs of polnomial functions including ais intercepts, stationar points and points of inflection, domain (including maimal domain), range and smmetr Review of algera of polnomials, solution of polnomial equations with real coefficients of degree n having up to and including n real solutions 1A The inomial theorem Digital doc 1 Quick Questions In Maths Quest 11 Mathematical Methods CAS we learned the following inomial epansions: ( + a) = + a + a ( + a) 3 = a + 3a + a 3 These are called inomial epansions ecause the epressions in the rackets contain two terms, i meaning. B continuing to multipl successivel a further ( + a), the following epansions would e otained: ( + a) 4 = ( a + 3a + a 3 )( + a) = a + 6 a + 4a 3 + a 3 ( + a) 5 = ( a + 6 a + 4a 3 + a 3 )( + a) = a a + 1 a 3 + 5a 4 + a 5 The coefficients associated with each term can e arranged in a triangular shape as shown: ( + a) 1 ( + a) ( + a) 1 1 ( + a) ( + a) ( + a) Graphs and polnomials 1

2 Notes 1. The first and last numers of each row are 1.. Each other numer is the sum of the two numers immediatel aove it. This triangle is known as Pascal s triangle. Each numer can also e otained using cominations, as follows. Row n n n Note: Cr r = =! n r! r! ( ) n Rememer that n C r is another wa of writing r. For eample, the epansion of ( + a) 6 can e written using cominations and then evaluated: ( + a) = 6 a a a + 3a3 a4 1a = a+ 154a + 3a3 + 15a4 + 6a5 + a6 Now the inomial theorem can e formall stated. n n n n ( a + ) n = ( a ) n + ( a ) n ( a) 1 n 1 + ( ) n 1 n a Notes 1. The indices alwas sum to n, that is, the powers of (a) and sum to n.. The power of a decreases from left to right while the power of increases. 3. The numer of terms in the epansion is alwas n + 1. n 4. The (r + 1)th term is r (a)n r r. The inomial theorem can also e stated using summation notation: a 6 n n n ( a + ) = ( a) r n n r r r =, where n r = means the sum of terms from r = to r = n. Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

3 Worked eample 1 Use the inomial theorem to epand ( 3) 4. Think Method 1: Technolog-free 1 Complete the inomial theorem epansion where a is the 1st term, is the nd term and n is the inde, using the appropriate row of Pascal s triangle to assist. WriTe 4 4 ( 3) 4 = ( ) 4( 3) ( ) 3( ) ( 1 + ) ) ( 3) ) ( 3 ) ( ( ) ( 3 ) Evaluate the cominations and the powers. = 1(16 4 ) + 4(8 3 )( 3) + 6(4 )(9) + 4()( 7) + 1(81) 3 Simplif. = Method : Technolog-enaled 1 On the Main page, ke in the equation ( 3) 4. Highlight the equation and tap: Interactive Transformation epand OK Write the result. ( 3) 4 = Worked eample Epand the inomial epression +. Think 1 Complete the inomial epansion where a =, = and n = 5, using row 5 of Pascal s triangle to assist. 5 WriTe = Evaluate the powers. 3 = Simplif = Graphs and polnomials 3

4 Worked eample 3 State the coefficient of i and ii 4 in ( 3 ) 8, without the use of technolog. Think i 1 The powers of the 1st term decrease and the powers of the nd term increase, 1,,... Use this to find which term gives a power of. Find the appropriate term using the inomial theorem. WriTe i, 1, The third term gives a power of. 8 Third term = 36 ( ) 3 Evaluate the term. = = State the coefficient. The coefficient of is ii 1 Find which term gives a power of 4. ii, 1,, 3, 4 The fifth term gives a power of 4. 8 Evaluate the term. Fifth term = 4 34 ( ) 4 = = Tutorial int-516 Worked eample 3 3 State the coefficient. The coefficient of the fifth term is 9 7. Worked eample 4 Find the fourth term in the epansion of ( ) 5. Think WriTe 1 Find the fourth term using the inomial theorem. 5 Fourth term = 3 () 3 Evaluate the term. = = 8 3 Worked eample 5 Find and evaluate the term that is independent of in the epansion of Think 1 Find how the powers of are generated in the epansion from left to right. WriTe Powers of are ( 3 ) 5 = 15, ( 3 ) 4 1 = 1, 3 ( 3 ) 3 1 = 5, ( 3 ) 1 =... that is, 15, 1, 5,. 4 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

5 Find the required term. The fourth term is independent of. 3 Evaluate. Fourth term = 5 3 ( ) = 1 = State the solution. The term that is independent of is the fourth term, 1. 3 Worked Eample 6 Find the coefficient of 4 in the epansion of ( + 3) ( ) 3 5. Think Write 1 4 terms will result when multipling from the first and second rackets respectivel: terms 1 and, terms and 3, terms 3 and 4 and terms 4 and 5. Write down the sum of these 4 products, using Pascal s triangle to assist. 4 terms = 3 [5() 4 ( )] + 3 (3)[1() 3 ( ) ] + 3(3) [1() ( ) 3 ] [5()( ) 4 ] 3 Evaluate. = = State the solution. The coefficient of 4 is 17. REMEMBER 1. Pascal s triangle: Binomial theorem: n n ( a + ) n = ( a ) n ( a ) n.. + n ( ) ( ) 1 + n a n 1 n 1 n a Notes 1. The powers of ( a) and sum to n.. There are n + 1 terms in the epansion. n 3. The ( r + 1)th term is a n rr r ( ). n Graphs and polnomials 5

6 eercise 1A Digital doc SkillSHEET 1.1 Binomial epansions 1B The inomial theorem 1 We 1 Use the inomial theorem to epand each of the following. a ( + 3) ( + 4) 5 c ( 1) 8 d ( + 3) 4 e (7 ) 4 f ( 3)5 We Epand each of the following inomial epansions. a c d We3 State the coefficient of i ii 3 and iii 4 in each of the following. a ( 7) 3 ( + 1) 5 c d 3 4 MC The coefficient of 3 5 in 3 is: A 135 B 45 C 75 D 45 E MC Which of the following does not have an 5 term when epanded? A ( + 6) 8 B C e D (8 3) 5 E 1 6 MC If 3 e a15 f 1 c5 + d = , then a + + c + d + e + f equals: 5 1 A 15 B 31 C 63 D 43 E 17 7 MC Which one of the following epressions is not equal to ( 3) 4? ( 3) A (3 ) 4 B ( 3)( 3) 3 C 3 ( ) D E We4 Find the fourth term in the epansion ( + 3) 6. 9 Find the third term in the epansion of 3 4, assuming ascending powers of. 6 1 We5 Find and evaluate the term that is independent of in the epansion of Find and evaluate the term independent of in the epansion of Find and evaluate the term that is independent of in the epansion of We6 Find the coefficient of p 4 in the epansion of (p + 3) 5 (p 5). 14 In the epansion of (a 1) n, the coefficient of the second term is 19. Find the value of n. Polnomials A polnomial in is an epression that consists of terms which have non-negative integer powers of onl. P() is a polnomial in if: P() = a n n + a n 1 n a + a 1 + a Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

7 where n is the degree (or highest power) of the polnomial and is a non-negative integer. The values of a n, a n 1,..., a, a 1 and a are called the coefficients of their respective power of terms. Worked Eample 7 Which of the following epressions are not polnomials? a c d e Think 1 a and d are polnomials ecause the are epressions with non-negative integer powers of onl. is not a polnomial as it has a power of 9, which is not an integer. 3 c is not a polnomial as it has a power of 1 ( ), which is not an integer, and it also has one term, 3, which is not a power of onl. Write, c and e are not polnomials. 4 e is not a polnomial ecause = and so has a power that is not a positive integer. Polnomials can e added and sutracted collecting like terms. Worked Eample 8 Given that P( ) = , Q( ) = and R ( ) = 4, find: a P ( ) + Q ( ) P ( ) R ( ). Think Write/displa Method 1: Technolog-free a 1 Add the polnomials. a P() + Q() = Collect like terms. = Sutract the polnomials. P() R() = ( 4) Remove rackets. = Collect like terms. = Graphs and polnomials 7

8 Method : Technolog-enaled a 1 On the Main page, define the polnomials p, q and r. Tpe in the first polnomial as shown, and tap: W p E Repeat the instructions for the polnomials q and r. To calculate P() + Q(), complete the entr line as: p + q Press E. To calculate P() R(), complete the entr line as: p r Press E. Evaluating polnomials A value for a polnomial, P(), can e found for a particular value of simpl sustituting the given value of into the polnomial epression and evaluating. That is, polnomial functions are evaluated in the same wa as an function. 8 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

9 Worked Eample 9 For the polnomial P( ) = , find: a its degree P(1) c P( ). Think Write a The degree of the polnomial is the highest power of. 1 Sustitute the given value of into the polnomial epression. a The degree of P() is 4. P(1) = (1) 4 (1) 3 + 5(1) 6(1) + 4 Evaluate. = = 4 c 1 Sustitute the given value of into c P( ) = ( ) 4 ( ) 3 + 5( ) 6( ) + 4 the polnomial epression. Evaluate. = = 76 Worked Eample 1 If p( ) = a , p( 1) = 5 and p () = 65, find the values of a and. Think Write/displa Method 1: Technolog-free 1 Sustitute a given value of into the polnomial and equate it to the given answer. P( 1) = a( 1) 5 + ( 1) 4 3( 1) 3 + ( 1) 5 = 5 Simplif the equation. a = 5 a + 4 = 3 Make the suject of the equation and call this equation [1]. 4 Sustitute a given value of into the polnomial and equate it to the given answer. P() = a() 5 + () 4 3() 3 + () 5 = 65 = 4 a [1] 5 Simplif the equation. 3a = 65 3a + 13 = 65 3a + = 5 [] 6 Sustitute [1] into []. Sustituting = 4 a: 3a + (4 a) = 5 7 Solve this equation for a. 3a + 8 a = 5 3a = 6 a = 8 Sustitute the value of a into equation [1]. Sustituting a = into equation [1]: = 4 9 Find the value of. = 6 1 State the solution. Therefore, a = and = 6. Graphs and polnomials 9

10 Method : Technolog-enaled 1 On the Main page, define the polnomial tping the equation: a Highlight the equation and tap: Interactive Define Set: Func name: p Variale/s: OK {N Enter the information as shown at right and tap E. Write the answer. Given p() = a and solving p( 1) = 5 and p() = 65 gives a = and = 6. remember eercise 1B Digital doc Spreadsheet 43 Evaluating polnomials Digital doc SkillSHEET 1. Simultaneous equations 1. If P() = a n n + a n 1 n a + a 1 + a and n is a non-negative integer then P() is a polnomial of degree n and a n, a n 1,... a, a 1 are called coefficients and R.. A polnomial P() is evaluated in the same wa as an function. Polnomials 1 We 7 Which of the following are not polnomial epressions? i 3 ii iii iv v vi We8 Given that P() = , Q() = and R() = then find: a P() + Q() Q() R() c 3P() R() d P() Q() + 3R(). 3 We9 For each of the following polnomials, find: i its degree ii P() iii P() and iv P( 1). a P() = P() = c P() = d P() = MC If P() = , then P( ) is equal to: A 479 B 95 C 31 D 481 E 13 5 We 1 If P() = 7 + a , P(1) = 4 and P() = 163, find a and. 6 Find a and, given that f () = a , f (1) = and f () = 5. 7 For Q() = a 3 6 +, Q() = 45 and Q() = 7. Find a and. 8 Find a and if P() = a , 3P(1) = 4 and 3P( ) = 1. 9 MC a If P() = a and P(1) = 1, then a is equal to: A 1 B C D 3 E If f () = n and f () = 1, then n is equal to: A 4 B 6 C 7 D 5 E 1 1 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

11 1C division of polnomials When sketching cuic or higher order graphs, it is necessar to factorise the polnomials in order to find the -intercepts. As will e shown later in this section, division of polnomials can e used to factorise an epression. When one polnomial, P(), is divided another, D(), the result can e epressed as: P ( ) R ( ) = Q ( ) + D ( ) D ( ) where Q() is called the quotient, R() is called the remainder, and D() is called the divisor. Interactivit int-46 Division of polnomials Worked eample 11 Find the quotient, Q(), and the remainder, R(), when is divided the linear epression +. Think WriTe Tutorial int-517 Worked eample 11 Method 1: Technolog-free 1 Set out the long division with each polnomial in descending powers of. If one of the powers of is missing, include it with as the coefficient. Divide into 4 and write the result aove. ) Multipl the result 3 + and write the result 3 underneath. + ) Sutract and ring down the remaining terms to complete the epression. ) ) ( + ) Divide into 5 3 and write the result aove Continue this process to complete the long division. 7 The polnomial , at the top, is the quotient. 8 The result of the final sutraction, 4, is the remainder. ) ( + ) ( 5 1 ) ( 1 + 4) 4 8 ( 4 48) 4 The quotient, Q(), is The remainder, R(), is 4. Graphs and polnomials 11

12 Method : Technolog-enaled 1 On the Main page, tap: Interactive propfrac Complete the entr line as shown, and press E. Write the answer. Dividing gives a quotient, Q(), of and a remainder, R(), of 4. Note: P( ) = ( ) 4 3( ) 3 + ( ) 8 = = 4 The remainder when P() is divided ( + ) is P( ). This leads to the remainder theorem, which states: When P() is divided ( a), the remainder is P(a) or when P() is divided ( a + ), the remainder is P a Furthermore, if the remainder is zero, then ( a) is a factor of P(). This leads to the factor theorem, which states: If P( a) =, then ( a) is a factor of P() or if ( a + ) is a factor of P(), then P =. a Note: If ( a) is a factor of P() and a is an integer, then a must e a factor of the term independent of. For eample, if ( ) is a factor of P(), then the term independent of must e divisile. Therefore, ( ) could e a factor of 3 +, ut ( + 3) could not e a factor.. Worked Eample 1 Determine whether or not D( ) = ( 3 ) is a factor of P ( ) = Think Write Method 1: Technolog-free 1 Evaluate P(3). P (3) = (3) 3 4(3) 3(3) 8 = = 1 1 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

13 If P(3) = then ( 3) is a factor of P(), ut if P(), ( 3) is not a factor of P(). Method : Technolog-enaled 1 On the Main page, define p() tping the equation Highlight it and tap: Interactive Define Set: Func name: p Variale/s: OK Complete the entr line as: p(3) Press E. P (3) so ( 3) is not a factor of P(). Since p(3) = 1, ( - 3) is not a factor of P(). p() 3 so ( 3) is not a factor of P(). Worked Eample 13 a Factorise P( ) = Solve =. Think Write Method 1: Technolog-free a 1 Use the factor theorem to find a value for a where P(a) = and a is a factor of the numerical term. Tr a = 1, 1,,, 3, 3, 6, 6 until a factor is found. Divide P() the divisor ( + ) using long division. 3 Epress P() as a product of linear and quadratic factors. a P(1) = (1) 3 (1) 13(1) 6 = 18 P( 1) = ( 1) 3 ( 1) 13( 1) 6 = 4 P() = () 3 () 13() 6 = P( ) = ( ) 3 ( ) 13( ) 6 = So ( + ) is a factor. ) (3 + 4) ( 5 1) 3 6 ( 3 6) P() = ( + )( 5 3) 4 Factorise the quadratic, if possile. = ( + )( + 1)( 3) Graphs and polnomials 13

14 1 Rewrite the equation in factorised form, using the answer to part a. Use the Null Factor Law to state the solutions. Method : Technolog-enaled a 1 On the Main page, define p() a tping the equation Highlight it and tap: Interactive Define Set: Func name: p Variale/s: OK To factorise p(), complete the entr line as: factor(p()) Then press E = ( + )( + 1)( 3) = =, 1 or 3 Write the answer. Factorising p() = gives P ( ) = ( 3)( + )( + 1) 1 To solve p() =, complete the entr line as: solve(p() =, ) Then press E. Write the answer. Solving = gives 1 =, =, or = 3 REMEMBER P ( ) R ( ) 1. = Q ( ) + D ( ) D ( ) where Q() is called the quotient, R() is called the remainder, D() is called the divisor.. Remainder theorem: If P() is divided ( a), then the remainder is P(a). 3. Factor theorem: If P(a) =, then ( a) is a factor of P(). If (a + ) is a factor of P(), then P a =. 4. If ( a) is a factor of P(), then a must e a factor of the term independent of. 14 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

15 eercise 1C Digital doc Spreadsheet 96 Finding factors of polnomials 1d division of polnomials 1 We 11 Find the quotient, Q(), and the remainder, R(), when each of the following polnomials are divided the given linear epression. a 3 + 5, , + 3 c , 3 d , a For each corresponding polnomial in question 1, evaluate: i P(4) ii P( 3) iii P(3) 1 iv P( ) 3 Compare these values to R() in question 1 and comment on the result. 3 We 1 In each of the following determine whether or not D() is a factor of P(). a P() = , D() = 3 P() = , D() = + c P() = , D() = 4 d P() = , D() = MC Eamine the equation f () = a Which one of the following is a factor of f ()? A + 1 B C + D + 3 E 4 When factorised, f () is equal to: A ( + 1)( 3)( + 4) B ( + )( )( 3)( 1) C ( + )( 4)( + 3)( + 1) D ( 1)( + 1)( 3)( 4) E ( 1)( + )( + 3) 5 We 13a Factorise the following polnomials. a P() = P() = c P() = We13 Solve each of the following equations. d P() = a = = c = d = 7 If ( ) is a factor of 3 + a 6 4, then find a. 8 If ( 1) is a factor of 3 + a + 3, then find a. 9 Find the value of a if ( + 3) is a factor of 4 + a Find the value of a and if ( + 1) and ( ) are factors of a If ( 3) and ( + ) are factors of 3 + a + + 3, find the values of a and. linear graphs Linear graphs are polnomials of degree 1. Graphs of linear functions are straight lines and ma e sketched finding the intercepts. revision of properties of straight line graphs 1. The gradient of a straight line joining two points is: m = 1 1 Digital doc WorkSHEET 1.1 A ( 1, 1 ) B (, ) Graphs and polnomials 15

16 . The general equation of a straight line is: = m + c where m is the gradient and c is the value of the -intercept. 3. The equation of a straight line passing through the point ( 1, 1 ) and having a gradient of m is: 1 = m( 1 ) Gradient = m A ( 1, 1 ) 4. The intercept form of the equation of a straight line is: + =1 or + a = a a (, ) 5. Parallel lines have the same gradient. 6. The product of the gradients of two lines that are perpendicular equals 1. That is, m 1 m = 1 or m 1 = Worked Eample 14 Sketch the graph of the linear function 3 = 6 indicating the intercepts. 1 m (a, ) Think Write/Draw 1 Sustitute = into the equation. When =, 3 = 6 Solve the equation for to find the -intercept. = 3 Sustitute = into the equation. Therefore, the -intercept is. 4 Solve the equation for to find the -intercept. When =, 3 = 6 = 3 5 Draw a set of aes. Therefore, the -intercept is 3. 6 Indicate the -intercept and -intercept and rule a line through these points. 3 = 6 3 Worked Eample 15 Find the equation, in the form a + + c =, of each straight line descried elow. a The line with a gradient of and passing through ( 3, ) The line passing through (, 8) and (, ) c The line that passes through (3, 4) and is parallel to the line with equation 5 = d The line that passes through (1, 3) and is perpendicular to the line with equation + 3 = 16 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

17 Think WriTe a 1 Write the rule for the point gradient a 1 = m( 1 ) form of the equation of a straight line, 1 = m( 1 ). Sustitute the value of the gradient, ( ) = ( 3) m, and the coordinates of the point ( 1, 1 ), into the equation. 3 Epand the rackets. + = 6 4 Epress the equation in the form required. 1 Write the rule for the gradient, m, of a straight line, given points. Sustitute the values of ( 1, 1 ) and (, ) into the rule and evaluate the gradient. 3 Sustitute the values of m and ( 1, 1 ) into the rule for the point gradient form of the equation of a straight line. (Coordinates of either point given ma e used.) + 8 = or 8 = m = = = 6 = 3 1 = m( 1 ) 8 = 3( ) 4 Epand the rackets. 8 = 3 5 Epress the equation in the form = required. c 1 State the gradient of the given line, which is the same as the gradient of the parallel line. Write the rule for the point gradient form of the equation of a straight line. 3 Sustitute the values of m and the coordinates ( 1, 1 ) = (3, 4). c 5 = ecomes = + 5. The gradient of the parallel lines is. 1 = m( 1 ) 4 = ( 3) 4 Simplif and write in the required form. 4 = 6 = d 1 Find the gradient of the given line. d = + 3 The gradient of the line is. Find the gradient of the perpendicular line. The gradient of the perpendicular line is 1. 3 Write the rule for the point gradient form of the equation of a straight line. 4 Sustitute the values of m and the coordinates ( 1, 1 ) = (1, 3). 1 = m( 1 ) 3 = 1 ( 1) 5 Simplif and write in the required form. 6 = ( 1) + 5 = Graphs and polnomials 17

18 The domain and range of functions The domain of a function, = f (), is the set of values of for which the function is defined (that is, all -values that can e sustituted into f () and an answer found). The range of f () is the set of values of for which the function is defined. If the rule and the domain of a function are given, then the function is completel defined. For eample, = 4, f () = 4, or f : (, ] R, f () = 4 interval notation Restricted domains or ranges can e represented interval notation in three forms. 1. The closed interval.. The open interval. 3. The half-open interval. Worked eample 16 a a a [a, ] = { : a } (a, ) = { : a < < } [a, ) = { : a < } If the domain or range is unrestricted, it can e denoted as R or (, ). R + = (, ) R + {} = [, ) R = (, ) R {} = (, ] Sketch the graph of each of the following functions, stating the domain and range of each. a 4 = 8, [ 3, 3] f( ) = 1,, ( (, 1) Think a 1 Sustitute the smallest value of into the equation. Solve the equation for, to find an end point of the straight line. 3 State the coordinates of the end point. 4 Sustitute the largest value of into the equation. 5 Solve the equation for, to find the other end point of the line. 6 State the coordinates of the nd end point. 7 Plot the two points on a set of aes with closed circles (since oth points are included). WriTe/drAW/diSPlA a When = 3, 1 = 8 = = 1 ( 3, 1) is a closed end of the line. When = 3, 1 = 8 = 4 = (3, ) is the other closed end of the line. 3 ( 3, 1) 4 1 (3, ) 3 4 = 8, [ 3, 3] 18 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

19 8 Draw a straight line etween the two points. 9 Find the intercepts and mark them on the graph. 1 State the domain, which is given with the rule. 11 State the range from the graph. The range is [ 1, ]. Method 1: Technolog-free 1 There is no smallest value of, so sustitute the largest value of into the equation and find. State the coordinates of the upper end point. 3 Sustitute another value of within the domain into the equation (that is, a value of < 1, since (, 1)) and find. When =, = 4 When =, = The -intercept is and the -intercept is 4. The domain is [ 3, 3]. When = 1, = f ( 1) = 3 ( 1, 3) is an open end of the line. When =, = f ( ) = 5 4 State the coordinates of the point. (, 5) is another point on the line. 5 Plot the points on a set of aes and mark the point ( 1, 3) with an open circle. 6 Rule a straight line from ( 1, 3) to (, 5) and eond. An arrow should e placed on the other end to indicate that the line continues. 7 Note that there are no intercepts. 8 State the domain, which is given with the rule. f() = 1, (, 1) (, 5) 5 3 ( 1, 3) 1 The domain is (, 1). 9 State the range eamining the graph. The range is (3, ). Method : Technolog-enaled On the Graph & Ta page, complete the function entr line as: 1 = 1 < 1 Tick the 1 o and tap!. Graphs and polnomials 19

20 remember Linear graphs 1. Linear equations are polnomials of degree 1.. Gradient, m = General equation is a + + c = or = m + c where m = gradient and c = -intercept. 4. Equation if a point and the gradient are known: 1 = m( 1 ) 5. Equation if the intercepts are known: + =1 a 6. Parallel lines have the same gradient. 7. If m 1 and m are the gradients of perpendicular lines, then: m 1 m = 1 1 or m1 = m eercise 1d Digital doc SkillSHEET 1.3 Gradient Digital doc SkillSHEET 1.4 Using gradient to find the value of a parameter linear graphs 1 We 14 Sketch the graph of each of the following linear functions indicating the intercepts. a + 3 = = c = 1 We 15a Find the equation, in the form a + + c =, of each straight line descried elow. a The line with a gradient of 3 and passing through (, 1). The line with a gradient of and passing through ( 4, 3). 3 We 15 Find the equation, in the form a + + c =, of each straight line descried elow. a The line passing through ( 3, 4) and ( 1, 1). The line passing through (7, 5) and (, ). 4 MC Which one of the following points does not lie on the straight line with equation 3 6 =? A (, 6) B (, ) C (, 3) D (1, ) E (4, 9) 5 We 15c Consider the points A(, 5) and B(1, ). a Find if: i the gradient of the straight line AB is ii the equation of the straight line AB is = 7. Find the general equation of the straight line which passes through (4, 5) and is parallel to the line with equation =. c We 15d Find the equation in the form a + + c = that passes through (, 4) and is perpendicular to the line with equation + 1 =. Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

21 6 Match each of the following graphs with the appropriate rule elow. a c (, 4) 1 3 d e f 4 3 i = ii = 3 iii = iv 3 + = 6 v = vi = Digital doc SkillSHEET 1.5 Interval notation 7 State the range for each function graphed elow. a c (4, 3) Digital doc SkillSHEET 1.6 Domain and range for linear graphs d ( 5, ) ( 3, 3) (5, ) e 4 (6, 5) f ( 4, ) (5, 6) 8 We 16 Sketch the graph of each of the following functions, stating i the domain and ii the range of each. a = 4, [ 1, 1] 5 = 1, < 5 c =, [, 5) 9 Find the equation of the straight line which passes through the point (, 5) and is: a parallel to the line with equation = 3 perpendicular to the line with equation = 3 7. Write equations in the form a + + c =. 1 Find the equation of the straight line which passes through the point ( 3, 1) and is: a parallel to the line with equation 4 = 13 perpendicular to the line with equation 4 = MC If the straight lines 3 = and a + = 3 are parallel then a = : A 6 B C D 3 E 6 1 MC If the straight lines = and = are perpendicular, then is equal to: A 5 B 1 5 C 5 D 1 5 E 3 Graphs and polnomials 1

22 1e Quadratic graphs Quadratic functions are polnomials of degree. Graphs of quadratic functions are paraolas and ma e sketched finding the turning point and intercepts. revision of quadratic functions 1. The general form of the quadratic function is = a + + c, R.. The graph of a quadratic function is called a paraola and: (a) for a >, the graph has a minimum value () for a <, the graph has a maimum value (c) the -intercept is c (d) the equation of the ais of smmetr and the -value of the turning point is = (e) the -intercepts are found solving the equation a + + c =. 3. The equation a + + c = can e solved either: (a) factorising or ± 4ac () using the quadratic formula, =. a 4. The turning point can e found completing the square (see page 3). The turning point is located on the ais of smmetr, which is halfwa etween the -intercepts. The discriminant The value of ( 4ac), which is the value inside the square root sign in the quadratic formula, determines the numer of solutions to a quadratic equation or the numer of -intercepts on a quadratic graph. This value is called the discriminant. 1. If 4ac >, there are two solutions to the equation and there are two -intercepts on the graph.. If 4ac > and is a perfect square, the solutions are rational; otherwise the are irrational. 3. If 4ac =, the two solutions are equal and there is one -intercept on the graph; that is, the graph has a turning point on the -ais. 4. If 4ac <, there are no real solutions and there are no -intercepts on the graph. a Worked eample 17 Use the discriminant to determine the numer of -intercepts for the quadratic function f( ) = Think WriTe Tutorial int-518 Worked eample 17 1 Find the values of the quadratic coefficients a, and c using the general quadratic function, = a + + c. a =, = 3, c = 1 Evaluate the discriminant. 4ac = 3 4()( 1) = = 89 3 If the discriminant is greater than, there are two -intercepts. If it is not a perfect square, the solutions are irrational. 4ac > So there are two -intercepts, which are oth irrational. Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

23 Worked Eample 18 Sketch the graph of the function f( ) = 1 5, showing all intercepts. Give eact answers. Think 1 Evaluate f () to find the -intercept (or state the value of c). Write/draw f() = 1 5() () = 1 State the -intercept. The -intercept is 1. 3 Set f() = to find the -intercepts. f() = 1 5 = 4 Factorise the quadratic (or use the quadratic (4 + )(3 ) = formula). 5 Solve the equation using the Null Factor Law. 4 + = or 3 = = 4 or = 3 6 State the -intercepts. The -intercepts are 4 and 3. 7 Draw a set of aes and mark the intercepts or the coordinates of the points where the graph crosses the aes. 8 Sketch a paraola through the intercepts. 1 (, 1) f() = ( 4, ) (, ) 4 1 The -coordinate of the turning point of a quadratic function is eactl halfwa etween the two -intercepts, so for worked eample 18, = = ( or 1 1 ) 4. Sustitute = into the original equation to find the -coordinate of the turning point. The -coordinate of the turning point can also e found using the formula = a, where a + + c =. Finding turning points completing the square Consider the general quadratic equation: = a + + c B completing the square, this equation ma e manipulated into the form = a( ) + c where the turning point is (, c). This wa of writing the function is known as the power form or turning point form. The transformations associated with this form will e discussed more full in chapter. Worked Eample 19 For the function = ( + 3 ) 4, find: a the coordinates of the turning point Think the domain and range. Write Write the general formula. = a( ) + c Write the function. = ( + 3) 4 a 1 Identif the values of a, and c. a a =, = 3, c = 4 Graphs and polnomials 3

24 State the coordinates of the turning point (, c). The turning point is ( 3, 4). 1 Write the domain of the paraola. The domain is R. Write the range c (as a < ). The range is 4. Worked Eample The function graphed at right is of the form = + + c. Find: a the rule the domain c the range. Write the answers to and c in interval notation. ( 5, 1) ( 1, 6) Think Write a 1 Write the general rule for a quadratic in turning point form. Find the values of and c using the given turning point. 3 State the value of a (given). a = 1 a = a( ) + c Since the turning point is ( 1, 6): = 1, c = 6 4 Sustitute these values in the rule. So = 1( + 1) 6 5 Epand the rackets. = Simplif. = + 5 The rule is = Use the graph to find the domain. Look at all the values that can take. 5 State the domain in interval notation. Domain = [ 5, ) c 1 Use the graph to find the range. Look at all the values that can take. c 6 State the range in interval notation. Range = [ 6, ) Worked Eample 1 1 Sketch the graph of = ( 1) +, clearl showing the coordinates of the turning point and the intercepts with the aes. State its range. Think Write/Draw 1 Write the general equation of the paraola. = a( ) + c Identif the values of the variales. a = 1, = 1, c = 4 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

25 3 Write a rief statement on the transformation of the asic paraola. 4 State the shape of the paraola (that is, positive or negative). The graph of = is dilated in the direction the factor of 1 (that is, it is wider than the asic curve); it is translated 1 unit to the right and units up. a > ; the paraola is positive. 5 State the coordinates of the turning point (, c). The turning point is (1, ). 6 As oth a and c are positive, onl the -intercept needs to e determined. Find the -intercept making =. -intercept: = = 1 ( 1) + = 1 ( 1) + = Sketch the graph: Draw a set of aes and lael them. Plot the turning point and the -intercept. Sketch the graph of the positive paraola, so that it passes through the points previousl marked. = = ( 1) + 8 Since, that is the range. The range is [, ). Worked eample Sketch the graph of = 3+ 8, showing the turning point and all intercepts, rounding answers to decimal places where appropriate. Think WriTe/drAW Method 1: Technolog-free 1 Find when =. When =, = 3 State the -intercept. The -intercept is 3. 3 Let the quadratic equal zero. When =, = 4 Solve for using the quadratic formula. 8± 8 4( )() 3 = ( ) = = = = 8± ± 4 4± or + Graphs and polnomials 5

26 5 State the -intercepts, rounding to decimal places. The -intercepts are.35 and Use the formula for the -value of the turning point, = a = 8 ( ) = 7 To calculate the -coordinate of the turning point, sustitute = into the function. = - () + 8() + 3 = 11 8 State the turning point. The turning point is (, 11). 9 Draw a set of aes and mark the coordinates of the turning point and the points where the graph crosses the aes. 1 Sketch a paraola through these points. Method : Technolog-enaled On the Graph & Ta page, complete the function entr line as: 1 = Tick the 1 o and tap!. To calculate the intercepts and turning point, tap: Analsis G-Solve Root / -Intercept / Ma Note: Onl the local maimum value is shown. 1 (, 11) 9 f() = (, 3) (.35, ) (4.35, ) Note: Function notation includes the rule, the domain and the co-domain. For eample, f (): [, 1] R, where f () = 3, is a paraola with rule f () = 3 and domain [, 1]. The range is a suset of the co-domain, R. Worked Eample 3 The weight of a person t months after a gmnasium program is started is given the function: Wt t () = 3t + 8, where t [, 8] and W is in kilograms. Find: a the minimum weight of the person the maimum weight of the person. 6 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

27 Think Write 1 Complete the square to find the turning point. W t = 3t = [ t 6t+ 16] 1 = [ t 6t ] 1 = [( t 3) + 151] 1 = ( t 3) State the minimum turning point. The turning point is (3, 75.5). 3 Find the end point value for W when t =. When t =, W = 8 4 State its coordinates. One end point is (, 8). 5 Find the end point value of W when t = 8. When t = 8, W = 88 6 State its coordinates. The other end point is (8, 88). 7 On a set of aes, mark the end points and turning point. W (kg) 8 Sketch a paraola etween the end points. 9 9 Locate the maimum and minimum values of W on the graph. 8 7 (, 8) Maimum (8, 88) Minimum (3, 75.5) t (months) a State the minimum weight from the graph. a The minimum weight is 75.5 kg. State the maimum weight from the graph. The maimum weight is 88 kg. REMEMBER Quadratic graphs 1. Quadratic equations are polnomials of degree.. The general equation is = a + + c. 3. The quadratic formula is given the equation ± 4ac = a 4. The discriminant is 4ac and if: (a) 4ac >, there are two -intercepts. If 4ac is a perfect square, the intercepts are rational. () 4ac =, there is one -intercept, which is a turning point. (c) 4ac <, there are no -intercepts. 5. The turning point form of the quadratic graph or paraola is: = a( ) + c and the turning point is (, c). 6. The equation of the ais of smmetr of a paraola and the -value of the turning point is given the epression = a. 7. The ais of smmetr is halfwa etween the -intercepts. Graphs and polnomials 7

28 eercise 1e Digital doc Spreadsheet 13 Discriminant Quadratic graphs 1 We17 Use the discriminant to determine the numer of -intercepts for each of the following quadratic functions. a f () = f () = c f () = d f () = e f () = 1 6 f f () = We18 Sketch the graphs of each of the following functions, showing all intercepts. Give eact answers. a f () = f () = c f () = d f () = Find the turning point for each of the functions in question. Give eact answers. 4 We19 For each of the following functions find: i the coordinates of the turning point ii the domain iii the range. a = = ( 6) c = ( + ) d = ( + 3) 6 Digital doc Spreadsheet 17 Quadratic graphs Digital doc SkillSHEET 1.7 Domain and range for quadratic graphs 5 We Each of the functions graphed elow is of the form = + + c. For each function, give: i the rule ii the domain iii the range. Write the answers to and c in interval notation. a c (1, 9) ( 1, 6) 4 (1, ) (, 3) ( 4, 16) 6 We1 Sketch the graphs of the following, clearl showing the coordinates of the turning point and the intercepts with the aes. a = + 3 = ( 5)( 3) c = ( 3) 8 7 MC Consider the function with the rule = 3. a It has -intercepts: A (1, ) and (3, ) B ( 1, ) and (3, ) C (1, ) and ( 3, ) D (, ) and ( 1, ) E (, 1) and (, 3) It has a turning point with coordinates: A ( 1, ) B (, 3) C (1, 4) D ( 1, 4) E (1, ) 8 MC The function f () = ( + 3) + 4 has a range given : A (3, ) B (, 3] C [4, ) D (, 4] E R 9 MC The range of the function = ( 4), [, 6] is: A [, 16] B [4, 16] C [, 4] D (4, 1] E [, 16) 1 We Sketch the graph of each of the following functions, showing the turning point and all intercepts. Round answers to decimal places where appropriate. a f () = ( ) 4 f () = ( + 4) + 9 c = d = 4 6 Digital doc Spreadsheet 41 Function grapher Digital doc Spreadsheet 18 Quadratic graphs turning point form 8 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

29 11 Sketch the graph of each of the functions elow and state i the domain and ii the range of each function. a = +, [, ] = + 1, R + c f () = 3, [ 1, 6] d f () = , [ 5, 3) 1 We3 The volume of water in a tank, V m 3, over a 1 month period is given the function V(t ) = t 16 t + 4, where t is in months and t [, 1]. Find: a the minimum volume of water in the tank Maimum height the maimum volume of water in the tank. 13 A all thrown upwards from a tower attains a height aove the ground given the function h(t) = 1t 3t + 36, where t is the time in seconds and h is in metres. Find: a the maimum height aove the ground that the all reaches the time taken for the all to reach the ground c the domain and range of the function. Tower Ball h(t) = 1t 3t + 36 Ground Digital doc WorkSHEET 1. 1F 14 A section of a roller-coaster at an amusement park follows the path of a paraola. The function h(t) = t 1t + 48, t [, 11], models the height aove the ground of the front of one of the carriages, where t is the time in seconds and h is the height in metres. a Find the lowest point of this section of the ride. Find the time taken for the carriage to reach the lowest point. c Find the highest point aove the ground. d Find the domain and the range of the function. e Sketch the function. Cuic graphs Cuic functions are polnomials of degree 3. In this section, we will look at how graphs of cuic functions ma e sketched finding intercepts and recognising asic shapes. Forms of cuic functions Cuic functions ma take several forms. The three main forms are descried elow. General form The general form of a cuic function is = a c + d If a is positive (that is, a > ), the function is called a positive cuic. Several positive cuics appear elow. Graphs and polnomials 9

30 If a is negative (that is, a < ), the function is called a negative cuic. Several negative cuics appear elow. Basic form Some (ut certainl not all) cuic functions are transformations of the form = 3, which has a point of inflection at the origin. These ma e epressed in the power form = a( ) 3 + c where (, c) is the point of inflection. For eample, = ( 3) is the graph of = 3 translated +3 from the -ais, +5 in the direction and dilated a factor of from the -ais. This form, called asic form or power form, works in the same wa as a quadratic equation epressed in turning point form or power form: = a( ) + c where (, c) is the turning point and a is the dilation factor. The power form and its transformations will e discussed in more detail in chapter. = 3 = a( ) 3 + c (, c) Factor form Cuic functions of the tpe = a( )( c)( d) are said to e in factor form, where, c and d are the -intercepts. Often a cuic function in general form ma e factorised to epress it in factor form. = a( )( c)( d) where a > = ( + )( 1)( 3) 1 3 c d 3 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

31 repeated factors A twice onl repeated factor in a factorised cuic function indicates a turning point that just touches the -ais. a = ( a) ( ) Worked eample 4 For each of the following graphs, find the rule and epress it in factorised form. Assume that a = 1 or a = 1. a f() f() Tutorial int-519 Worked eample Think a 1 Find a deciding whether the graph is a positive or negative cuic. Use the -intercepts 4, and 3 to find the factors. 3 Epress f() as a product of a and its factors. WriTe a The graph is a positive cuic, so a = 1. The factors are ( + 4), and ( 3). f() = 1( + 4) ( 3) 4 Simplif. f() = ( + 4)( 3) 1 Find a deciding whether the graph is a positive or negative cuic. Use the -intercept, which is also a turning point, to find the repeated factor. 3 Use the other -intercept, 3, to find the other factor. 4 Epress f() as a product of a and its factors. The graph is a negative cuic, so a = 1. ( + ) is a factor. ( 3) is also a factor. f() = 1( + ) ( 3) 5 Simplif. f() = (3 )( + ) Worked eample 5 Sketch the graph of = 3 1 8, showing all intercepts. Think WriTe/drAW 1 Find when =. When =, = 8 Graphs and polnomials 31

32 State the -intercept. The -intercept is 8. 3 Let P() =. Let P() = Use the factor theorem to find a factor of the cuic P() = Use long division, or otherwise, to find the quadratic factor. P(1) = (1) 8 = 18 P( 1) = ( 1) 3 ( 1) 1( 1) 8 = so ( + 1) is a factor. B long division: 8 ) (3 + ) 1 8 ( ) 8 8 ( 8 8) = ( + 1)( 8) 6 Factorise the quadratic, if possile. = ( + 1)( 4)( + ) 7 Epress the cuic in factorised form and let it If ( + 1)( 4)( + ) = equal to find the -intercepts. 8 Solve for using the Null Factor Law. = 1, 4 or Alternativel, use a CAS calculator to solve for. On the Main page, complete the entr line as: solve( =, ) Press E. 9 State the -intercepts. The -intercepts are, 1, and 4. 1 Sketch the graph of the cuic. = Eam tip When sketching graphs, ensure that the are smooth, that relevant turning points and intercepts are laelled, and that the are drawn within the correct domain (end points should e shown using or ) using an appropriate scale. [Authors advice] 3 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

33 Worked Eample 6 Restricting the domain of cuic functions 1. If the domain is R then the range is also R.. To find the range if the domain is restricted, it is necessar to look at the end points and turning points, then find the highest and lowest -values. (6, 8) For eample: The range can not e stated for the diagram at right ecause the -coordinate of the local minimum is not known. Recall that cuic functions that do not have an turning points can have onl one -intercept. ( 4, 3) Coordinate of local minimum required Sketch the graph of 3 = 5, where (, 1 ], using the unrestricted function as a guide. State the domain and range, without the use of technolog. Think WRITE/DRAW 1 Decide whether it is a positive or negative cuic looking at the coefficient of 3. Negative cuic Find the -intercept/s. When =, 3 5 = ( + 5) = = ( + 5 ) The -intercept is. 3 Find the -intercept. When =, = () 3 5() = The -intercept is. 4 Find when has the value of the lower end point of the domain. 5 State the coordinates of this end point and decide whether it is open or closed. When =, = ( ) 3 5( ) = 18 The open end point is (, 18). 6 Find when has the value of the upper end point. When = 1, = ( 1) 3 5( 1) = 6 7 State the coordinates of this end point and decide whether it is open or closed. 8 Mark these points on a set of aes. (, 18) 9 Sketch the part of the cuic etween the end points. 1 Verif this graph using a graphics calculator. ( 1, 6) The closed end point is ( 1, 6). 11 State the domain, which is given with the rule. The domain is (, 1]. 1 From the graph, state the range. Note that the intercept is not included in the domain. The range is [6, 18). Graphs and polnomials 33

34 remember Cuic graphs 1. The general equation is = a c + d.. Basic shapes of cuic graphs: (a) If a > : Positive cuic Power form = a( ) 3 + c (, c) Factor form = a( )( c)( d) where a > Repeated factor a c d = ( a) ( ) () If a <, the reflections through the -ais of the tpes of graph in the aove figures are otained. eercise 1F Digital doc Spreadsheet 13 Cuic graphs factor form Cuic graphs 1 We 4 For each of the following graphs, find the rule and epress it in factorised form. Assume that a = 1 or a = 1. a Match each of the following graphs to the most appropriate rule elow. a c Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

35 d e f g h i = ( 3) 3 ii = ( + 3)(1 )( 4) iii = ( + 4)( + )(1 ) iv = ( + ) (5 ) v = ( + 3)( 1)( 4) vi = ( + 4)( + )( 1) vii = (3 ) 3 viii = ( + ) ( 5) 3 We 5 Sketch the graph of each of the following, showing all intercepts. a = = c = d = Verif our answers using a calculator. 4 MC a Full factorised, is equal to: A ( + 3) 3 B ( + ) 3 C ( ) 3 D ( 3) 3 E ( + )( ) The graph of = is: A B C Digital doc Spreadsheet 14 Cuic graphs D E 5 MC The function graphed in the figure could have the following rule: A = ( ) 3 + B = ( + ) 3 + C = ( ) 3 + D = ( + ) 3 1 E = ( ) 3 (, ) Graphs and polnomials 35

36 Digital doc Spreadsheet 15 Cuic graphs = a( ) 3 + c form 6 MC The graph of f () = 5( + 1) 3 3 is est represented : A B ( 1, 3) ( 1, 3) C (1, 3) D E ( 1, 3) (1, 3) 7 MC The graph of f () = ( 1) ( + 3) is est represented : A B (, 6) (, 6) ( 1, ) (3, ) ( 3, ) (1, ) C D ( 3, ) (1, ) ( 3, ) (1, ) (, 6) (, 6) E ( 1, ) (3, ) (, 6) 36 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

37 8 MC The graph shown is est represented the equation: A = ( a) 3 + B = ( a) 3 + C = (a ) 3 + D = ( + a) 3 + E = ( + a) 3 + (a, ) (, c) 9 MC If a < and, c > then the graph shown is est represented the equation: A = ( ac + a ) ( c ) B = ( ac + a ) ( c ) C = ( ac a ) ( + c ) d = ( ac + a ) ( )( c ) E = ( ac a ) ( c ) a c 1 WE 6 Sketch the graph of each of the following restricted functions, using the unrestricted function as a guide. State i the domain and ii the range in each case. a f () = , [, ) f () = , [, 1] c f () = , (, 1] d f () = 3 3, [ 1, ) e f () = 3 +, [, 1) (, 3] f f () = 3, ( 1, 1) [, 3) 11 The function f () = 3 + a + 64 has -intercepts (, ) and (4, ). Find the values of a and. 1 The functions = 3 + a + 1 and = 6 + (a + ) 4 3 oth have ( 1, ) as an -intercept. Find the values of a and. 13 The cross-section of a glass vessel that is 6 cm high can e modelled the cuic function f () and its reflection through the -ais, g(), as shown at right. a Find the values of a, and c, and hence state the rule g() f() = a( + ) 3 + c of f (). Find the rule for g() and state its domain and range. c What is the width of the vessel when the height is cm? (4, 6) (3, 3) (, ) 14 The distance of a group of hikers, d km, from their starting point t hours after setting off on a hike can e modelled the function with the rule: d(t) = at ( t) The hikers are 3 km from the start after hours and return to the starting point after 5 hours. a Find the values of a and. Hence, give the rule for d(t ) stating its domain and range. Graphs and polnomials 37

38 c Sketch the graph of d(t ). d Find to the nearest 1 metres the maimum distance of the hikers from their starting point and the time, to the nearest minute, that it occurs. 1G Quartic graphs Quartic functions are polnomials of degree 4. The general form of a quartic is: = a c + d + e When sketching the graphs of quartic functions, all aes intercepts can e found factorisation and a sign diagram used to check the shape. If a sign diagram is not sufficient and the asic shape is not recognised, then a graphics calculator could e used to estalish the shape of the graph. Basic shapes of quartic graphs Positive quartics (a > ) 1. = a 4. = a 4 + c, c 3. = a ( )( c) 4. = a( ) ( c) c c The repeated factor shows there is a turning point at the origin. The factors ( ) ( c) show -intercepts at = and = c. The repeated factors ( ) and ( c) show the graph touches the -ais at = and = c. 5. = a( )( c) 3 6. = a( )( c)( d )( e) c c d e The cued factor ( c) 3 shows the graph as a point of inflection at = c. The factors show intercepts at =, c, d and e. 38 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

39 negative quartics (a < ) If a <, that is, each of the aove rules is multiplied 1, then the graphs are reflected through the -ais. For eample, the graph of = 4 (at right) is a reflection, through the -ais, of the graph of = 4. Similarl, the graph of = 4 + = ( 4 ) is a reflection through the -ais of the graph of = 4. Note: The aove graphs can e translated horizontall or verticall ut this is considered in chapter. To find the -intercepts of a quartic function, let = and solve the equation for. Repeated factors touch the -ais as the do for cuic and quadratic functions. = 4 = 4 + Worked eample 7 Sketch the graph of = , showing all intercepts. Think WriTe/drAW Tutorial int-5 Worked eample 7 Method 1: Technolog-free 1 Find the -intercept. When =, = 1 The -intercept is 1. Let = P(). Let P() = Find two linear factors of the quartic epression, if possile, using the factor theorem. P(1) = (1) 4 (1) 3 7(1) + 5(1) + 1 = 8 P( 1) = ( 1) 4 ( 1) 3 7( 1) + 5( 1) + 1 = ( + 1) is a factor. P() = () 4 () 3 7() + 5() + 1 = ( ) is a factor. 4 Find the product of the two linear factors. ( + 1)( ) = 5 Use long division to divide the quartic the quadratic factor (or use another method). ) ( 3 ) ( ) 6 Epress the quartic in factorised form. = ( + 1)( )( 5) 7 Factorise the quadratic factor, 5, using difference of perfect squares. = ( + 1)( )( + 5)( 5) 8 To find the -intercepts, set equal to zero. Let = ( + 1)( )( + 5)( 5) = 9 Solve for using the Null Factor Law. 1 State the -intercepts. = 1,, ± 5 The -intercepts are 1,, 5 and 5. 5 Graphs and polnomials 39

40 11 Sketch the graph of the quartic. (, 1) ( 1, ) ( 5, ) (, ) ( 5, ) Method : Technolog-enaled 1 On the Main page, define p() tping the equation Highlight it and tap: Interactive Define Set: Func name: p Variale/s: OK To find the intercepts, complete the entr lines as: p() solve(p()=, ) Press E after each entr. To sketch the graph of p(), in the Graph & Ta page complete the entr line as: 1 = p() Then tick the 1 o and tap!. Worked eample 8 Sketch the graphs of each of the following equations, showing the coordinates of all intercepts. Use a CAS calculator to find the coordinates of the turning points, rounding to decimal places as appropriate. a = ( 1)( + ) = ( + 3) ( 1) Tutorial int-79 Worked eample 8 Think WriTe/drAW a 1 State the function. a = ( 1)( + ) Find the -intercept. When =, = The -intercept is. 4 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

41 3 Find the -intercepts. When =, = ( 1)( + ) =,, 1 4 State the -intercepts, noting where the graph touches and where it cuts the -ais. 5 State the coordinates of the turning points. 6 Sketch the graph of the quartic, using a CAS calculator to assist. The graph touches the -ais at =. The other -intercepts are and 1. The minimum turning points are ( 1.44,.83) and (.69,.4). The maimum turning point is (, ). (, ) (, ) (1, ) (.69,.4) ( 1.44,.83) 1 State the function. = ( + 3) ( 1) Find the -intercept. When =, = (3) ( 1) = 9 The -intercept is 9. 3 Find the -intercepts. When =, = ( + 3) ( 1) 4 State the points where the graph touches the -ais from the repeated factors. 5 State the coordinates of the turning points. 6 Sketch the graph of the quartic, using a graphics calculator to assist. = 3, 1 The graph touches the -ais at = 3 and = 1. The maimum turning points are ( 3, ) and (1, ), and the minimum turning point is ( 1, 16). ( 3, ) (1, ) (, 9) ( 1, 16) Graphs and polnomials 41

42 Worked Eample 9 Determine the equation of the graph shown Think Write/displa Method 1: Technolog-free 1 State the -intercepts. The -intercepts are 3, 1, 1,. Write the equation using factor form with a dilation factor of a. = a( + 3)( + 1)( 1)( ) 3 State the -intercept. The -intercept is 3. 4 Sustitute the coordinates of the point where the graph crosses the -ais into the equation. (, 3) 3 = a( + 3)( + 1)( 1)( ) 5 Solve the equation to find a. 3 = a 6 a = 1 6 Write the equation. 1 = ( 1)( )( + 3)( + 1) Method : Technolog-enaled 1 On the Main page, complete the entr line as: a( + 3)( + 1)( 1)( ) W. Tap: Interactive Advanced solve Set: Equation: = 3 Variale: a OK Complete the entr lines as: solve( = 3, a) = a = 1, and press E after each entr. Write the equation. ( )( 1)( + 1)( + 3) = 4 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

43 Worked Eample 3 Sketch the graph of = 4, ( 1, 1 ], using the unrestricted function as a guide. State the domain and the range in each case. Think Write/Draw 1 State the function. = 4, ( 1, 1] Find the -intercept. When =, = () 4 () 3 State the -intercept. The -intercept is. 4 Find the -intercepts. When = 4 4 = 5 Factorise the quartic epression. ( + ) = 6 Solve for. = is the onl solution (as + ). 7 State the -intercepts. The onl -intercept is. 8 Find when is one end point of the domain. When = 1, = ( 1) 4 ( 1) 9 State the coordinates and whether it is an open or closed point. = = 3 ( 1, 3) is an open end point. 1 Find when is the other end point of the domain. When = 1, = (1) 4 (1) = 3 11 State the coordinates and whether it is an open or closed point. 1 Sketch the graph of the quartic, using knowledge of asic shapes or a CAS calculator to assist, over the domain. (1, 3) is a closed end point. (, ) ( 1, 3) (1, 3) = 4 13 State the domain, which is given with the rule. The domain is ( 1, 1]. 14 From the graph, state the range. The range is [ 3, ]. Graphs and polnomials 43

44 REMEMBER Quartic graphs 1. General equation is = a c + d + e.. Basic shapes of quartic graphs: (a) If a > : = a 4 = a 4 + c, c = a ( )( c) c = a( ) ( c) c = a( )( c) 3 c = a( )( c)( d)( e) c d e () If a <, then the reflection through the -ais of the tpes of graph in the figures aove is otained. 44 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

45 eercise 1G Digital doc SkillSHEET 1.8 Solving quartic equations Digital doc Spreadsheet 15 Quartic graphs factor form Quartic graphs 1 We 7 Sketch the graph of each of the following, showing all intercepts. a = ( )( + 3)( 4)( + 1) = c = d = e = We 8 Sketch the graph of each of the following equations, showing the coordinates of all intercepts. Use a calculator to find the coordinates of the turning points, rounding to decimal places as appropriate. a = ( )( 3) = ( + 1) ( 1) c = ( 1) ( + 1)( + 3) d = ( + ) 3 (1 ) 3 MC Consider the function f () = a When factorised, f () is equal to: A ( + )( )( 1)( + 4) B ( 1)( 4)( + 4) C ( + 3)( )( 1)( + 1) D ( ) 3 ( + ) E ( ) ( + ) The graph of f () is est represented : A B C D 16 E 4 c If the domain of f () is restricted to [, ], then the range is: A [, 16] B [, 1] C [, 1] D R + E [, ) d If the range of f () is restricted to (, 5) then the maimal domain is: A [, 3) B (, 3) C ( 3, ) D ( 3, 3) E ( 3, 4) e If the domain of f () is restricted to ( 1, ), then the range is: A (, 16) B (, 4) C ( 1, 9) D (9, 16) E [9, ) f If the domain of f () is restricted to [, ), then the range is: A R B R + C [, ) D [, 16) E [, ) Graphs and polnomials 45

46 4 We 9 Determine the equation of each of the following graphs. a We 3 Sketch the graph of each of the following restricted functions, using the unrestricted function as a guide. State i the domain and ii the range in each case. a = ( )( 4)( + 3), [, 3] = , (, 1] c = ( ) ( + 1), (, ] d = 4 4, [ 3, ] 6 The function f () = 4 + a has -intercepts (, ) and ( 3, ). Find the values of a and. 7 The function f () = 4 + a has -intercepts (1, ) and ( 3, ). Find the values of a and. 8 The functions = (a ) 4 3 and = (a + 5) oth have an -intercept of 1. Find the value of a and. Digital doc Investigation Quartics and eond 46 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

47 Summar Pascal s triangle Binomial theorem n n ( a + ) n = ( a ) n ( a ) n.. + n ( ) ( ) 1 + n a n 1 n 1 n a Notes 1. Indices add to n.. There are n + 1 terms in the epansion. n 3. The ( r + 1)th term is a n rr r ( ). Polnomials If P() = a n n + a n 1 n a + a 1 + a and n is a non-negative integer then P() is a polnomial of degree n and a n, a n 1,..., a, a 1 are called coefficients and R. Remainder theorem: If P() is divided ( a), then the remainder is P(a). If P() is divided (a + ) then the remainder is P a. Factor theorem: 1. If P(a) =, then ( a) is a factor of P() or if (a + ) is a factor of P(), then P a =. If ( a) is a factor of P() then a must e a factor of the term independent of. Linear graphs Linear equations are polnomials of degree 1. General equation is a + + c = or = m + c where m = gradient c = -intercept The gradient m = 1 1 Equation if a point and the gradient is known: 1 = m( 1 ) n Graphs and polnomials 47

48 Parallel lines have the same gradient. If m 1 and m are the gradients of perpendicular lines, then: m 1 m = 1 or m1 = 1 m Quadratic graphs Quadratic equations are polnomials of degree. General equation is = a + + c ± 4ac Quadratic formula is = a Discriminant = 4ac and 1. if 4ac >, there are -intercepts (and if 4ac is a perfect square, the intercepts are rational). if 4ac =, there is 1 -intercept 3. if 4ac <, there are no -intercepts. The power form or turning point form of the quadratic is: = a( ) + c and the turning point is (, c). The equation of the ais of smmetr and the -value of the turning point of a paraola is a. The ais of smmetr is halfwa etween the -intercepts. Cuic graphs Cuic equations are polnomials of degree 3. General equation is = a c + d Basic shapes of cuic graphs: 1. If a > : Positive cuic Basic form = a( ) 3 + c (, c) Factor form = a( )( c)( d) where a > Repeated factor a c d = ( a) ( ). If a <, then the reflections through the -ais of the tpes of graph in the aove figures are otained. Quartic graphs Quartic equations are polnomials of degree 4. General equation is = a c + d + e 48 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

49 Basic shapes of quartic graphs: 1. If a > : c = a 4 = a( - ) ( - c) c = a 4 + c, c = a( - )( - c) 3 c c d e = a ( - )( - c) = a( - )( - c)( - d)( - e). If a <, then reflection through the -ais of the tpes of graph aove is otained. Note: It is possile to translate the cuic and quartic graphs shown in the cuic graphs and quartic graphs sections aove. Functions A function is full defined if the rule and domain are given. The domain of a function is the set of values of for which the function is defined. The range of a function is the set of values of for which the function is full defined. Restricted domains can e represented interval notation: [a, ] = {: a } (a, ) = {: a < < } [a, ) = {: a < } Graphs and polnomials 49

50 chapter review Short answer 1 Epand each of the following: a ( 3) 5 8 If a factor of P() = 7 + a is ( 1), find the values of a and. 3 Factorise each of the following epressions: a Find the equation of each of the straight lines descried elow. a The line which passes through the points ( 5, 6) and (1, 1). The line which is perpendicular to the line with equation + 1 = and passes through the point (3, 3). 5 Sketch the graph of = 8, laelling the turning point and all intercepts. State its domain and range. 6 Sketch the graph of = , [ 3, ). State the range of this function. 7 a If ( + 3) is a factor of f () = a 18 and g() = a + 75, then find the values of a and. Sketch the graph of f () laelling all intercepts. 8 Sketch the graph of f () = Multiple choice 1 When epanded, (1 ) 5 is equal to: A B C D E The coefficient of 5 in the epansion of is: A 496 B C 496 D E Assuming descending powers of, the fifth term of 1 1 the epansion of 3 + is: A B C D E 5 4 Which of the following epressions is not a polnomial? A B C D E The value of P( 3) in the polnomial, P() = , is: A 31 B 139 C 191 D 6 E 1 6 The degree of the polnomial ( ) ( ) when epanded is: A 4 B 8 C 1 D 16 E 1 7 The remainder when is divided ( + 3) is: A 71 B 51 C 171 D 3 E 18 8 For which one of the following polnomial epressions is ( ) not a factor? A B C D E Which one of the following is a factor of ? A ( ) B ( + 3) C ( + 1) D ( 4) E ( 3) 1 The rule for the graph shown is: A = B 4 = C 4 = D + 4 = E 4 + = 4 5 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

51 Questions 11 and 1 refer to the graph elow, which has a gradient of. 11 The value of must e: A 5 B 3 C 1 D 1 E 4 1 The -intercept is: A (, 3) B (, ) C (, 1 ) D (, 1) E ( 1, ) ( 3, 5) 13 If =, then the value of the discriminant is: A 76 B 44 C 44 (, ) D 3 E 76 Questions 14 and 15 refer to the function with the rule: = where ( 6, ). 14 Which one of the following graphs could represent this function? A ( 6, 14) (, 14) D E 6 ( 6, ) ( 6, 6) (, 3.6) 1 (, 1) 1 15 The range of this function is: A ( 18, 14) B ( 1, 14) C [ 18, 14) D [ 18, 14] E ( 14, 1) 16 The graph of = 3 3 could e: A B 1 1 C D B ( 6, 14) (, 14) 1 E C (, 14) (6, 14) Which of the following intercepts does the graph of f () = have? A ( 1, ), (, ), (3, ) and (, 6) B (, ), (, ), (3, ) and (, 6) C ( 1, ), (, ), (3, ) and (, 6) D (, ), ( 1, ), (3, ) and (, 6) E ( 1, ), ( 3, ), (, ) and (, 6) Graphs and polnomials 51

52 18 The rule for the graph shown elow could e: A f () = ( 1) ( + 3) B f () = ( + 1)( 3) C f () = ( + 1) (3 ) f() D f () = ( 1)( + 3) E f () = ( 3)( + 1) C The rule for the graph shown elow could e: A f () = ( + ) 3 B f () = ( ) C f () = ( ) D f () = ( ) 3 f() E f () = ( ) D E The graph of = ( + 3) ( 1)( 3) is est represented : A B The graph with equation = is translated 3 units down and units to the right. The resulting graph has the equation: A = ( - 3) + B = ( - ) + 3 C = ( - ) - 3 D = ( + ) - 3 E = ( + ) + 3 [ VCAA 6] Etended response 1 An empt parfait glass has een left on a tale with the rim just touching a wall. Ants are marching in a line down into the parfait glass and then up the other side, following the path of a paraola. The egin their journe where the glass touches the wall, 18 cm aove the tale. a The stem of the glass is 4 cm long and the diameter of the top of the glass is 5 cm. Find the rule for the quadratic function that descries the shape of the glass. State the domain and range of the function. c If there is fruit juice in the ottom of the glass to a depth of 1 cm, find the coordinates of the point where the ants first touch the juice. Round answers to the nearest whole numer. d Using function notation, write the rule for the surface of the crosssection of the juice in the glass. 5 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

53 A rogue satellite has its distance from Earth, d thousand kilometres, modelled a cuic function of time, t das after launch. After 1 da it reaches a maimum distance from Earth of 4 kilometres, then after das it is kilometres awa. It effectivel returns to Earth after 3 das, then moves further and further awa. a What is the satellite s initial distance from Earth? Sketch the graph of d versus t for the first 6 das of travel. c Epress d as a function of t. d The moon is approimatel 4 kilometres from Earth. Which is closer to Earth after 8 das, the satellite or the moon? B how far? e The satellite is programmed to self-destruct. This happens when it is 49 kilometres from Earth. What is the life span of the satellite? f State the domain and range of d(t). 3 A ridge spans a narrow canal as shown in the diagram at right. a Find the equation of a paraola that models the shape of the archwa. Show that a arge 3 m wide and carring cargo with a total height of 1.7 m (with a rectangular cross-section) cannot fit under the arch. c How much cargo (in height, correct to 1 decimal place) must e removed for the arge to fit under the ridge? 4 In the town of Newtonia there is an annual 1 m race (for the Polnomial Cup) 1 1 for mini roots that have een programmed with mathematical formulas Surface of canal Professor Lienitz. There is a lot of etting on the race as the professor keeps the formulas secret and is known to favour surprise winners. The three contestants were programmed as follows, where is the distance from the start line in metres and t is the time in minutes: Line = t Quadder =.t(t - 5.1) Curic =.t(t - 5.1)(t - 9.1) Using a CAS calculator, descrie the motions of the three contestants, specificall: a the direction the travelled in and how fast the were moving where and when the changed direction c where and when the passed or met each other d who won the race and how much. Sketch the graphs of their movements on the same set of aes, laelling all relevant points. You will need an etra graph to get a close-up of the finish. 5 The diagram at right shows a main road passing through O, A, C and E. The road crosses a river at point O and 3 kilometres further along the road at point C. Between O and C, the furthest the river is from the road is 8.54 kilometres, at a point D,.5 kilometres east of a north south line through O. Point A is 1 kilometre east of point O. If point O is taken as the origin and the road as the -ais, then the path of the river can e modelled a quartic function, as shown in lue. a Give the coordinates of C and D. Find the rule for the quartic function, f (). c How far is the river from the main road along the track AB? d A canoeing race, of at least 17 kilometres in length, along the river is eing organised. It is suggested that the race could start at O and finish at C. Is this course satisfactor? Wh? N 3 1 Barge W E River S O A C B D Main road E Graphs and polnomials 53

54 6 Willie Wonkie, of Willie Wonkie s Construction Compan, makes a sketch of the smmetrical W for a large neon sign as shown elow. The - and -aes represent the supporting crosspieces. The width of the W along the -ais is 6 metres and the point on the vertical support is 1 metres aove the horizontal support. The W 4 can e modelled a quartic function, with all -intercepts eactl evenl spaced. a Find the rule for the letter W. If the top of the W is 8 metres wide, find the coordinates of the highest points of the letter. c State the domain of the function. d Use a graphics calculator to find the coordinates of the lowest points of the W, giving values correct to 3 decimal places. Hence find the range of the function. e In order to test the strength of his design, Willie Wonkie moves the horizontal crosspiece so that it just touches the lowest points of the W. Find the new rule that descries the W now. f State the domain and range of the new function. Note: The following questions use differentiation of polnomials. 7 A plane cruising at 1 m is coming in to land at an airport at sea level, as can e seen in the diagram elow. 1 km Plane s flight path 1 m Airport 5 km If the plane descends smoothl and makes no changes in direction, show that a possile model would e = a ( - ). a Find the equation if the plane egins its descent when 5 km horizontall from the airport. What is the altitude of the plane when it is km horizontall from the airport? c How accurate do ou think this model is? 54 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad

55 8 The diagram elow shows a smmetrical skateoard ramp with horizontal platforms at A and B, and vertical supporting struts at C and D. A E F B 3 m C G D m m 4 m 4 m a Write an equation for a quartic function that models the ramp, assuming a smooth connection at A and B. Show that the right half of the ramp can e modelled a cuic equation = a( ) ( 4) and find its equation evaluating a and. c The right-hand side can also e modelled two smoothl connected paraolas. i If the strut DF is 1 m long, find the equation of the lower paraola passing through F. ii Find the equation of the upper paraola if it meets the lower one at F, and show that the connection is not smooth (that is, their gradients are not equal at the point where the meet). iii Show that the two paraolas meet smoothl at (3,.75) provided the lower paraola passes through F. d Which model is the closest to the actual ramp if the strut is reall 1.6 m long? Digital doc Test Yourself Graphs and polnomials 55

56 ACTiviTieS Chapter opener Digital doc 1 Quick Questions: Warm up with ten quick questions on graphs and polnomials. (page 1) 1A The inomial theorem Tutorial We 3 int-516: Watch a worked eample on inomial epansion. (page 4) Digital doc SkillSHEET 1.1: Practise inomial epansions using Pascal s triangle. (page 6) 1B Polnomials Digital docs Spreadsheet 43: Investigate evaluating polnomials. (page 1) SkillSHEET 1.: Practise solving simultaneous equations. (page 1) 1C Division of polnomials Interactivit Division of polnomials int-46: Consolidate our understanding of the division of polnomials and rational functions. (page 11) Tutorial We 11 int-517: Watch a worked eample on the division of polnomials. (page 11) Digital docs Spreadsheet 96: Investigate finding factors of polnomials. (page 15) WorkSHEET 1.1: Binomial epansion, division of polnomials and solving and factorising polnomial equations (page 15) 1D Linear graphs Digital docs SkillSHEET 1.3: Practise calculating the gradient of parallel and perpendicular lines. (page ) SkillSHEET 1.4: Practise using the gradient to find the value of a parameter. (page ) SkillSHEET 1.5: Practise using interval notation. (page 1) SkillSHEET 1.6: Practise finding the domain and range for linear graphs. (page 1) 1E Quadratic graphs Tutorial We 17 int-518: Watch a worked eample on using the discriminant. (page ) Digital docs Spreadsheet 13: Investigate the value of the disciminant. (page 8) Spreadsheet 17: Investigate quadratic graphs. (page 8) SkillSHEET 1.7: Practise recognising domain and range for quadratic graphs. (page 8) Spreadsheet 41: Investigate graphs of functions. (page 8) Spreadsheet 18: Investigate quadratic graphs in turning point form. (page 8) WorkSHEET 1.: Calculate gradients, aial intercepts and values of the discriminant, sketch graphs of polnomials, and determine equations for graphs. (page 9) 1F Cuic graphs Tutorial We 4 int-519: Watch a worked eample on determining the rule of a cuic. (page 31) Digital docs Spreadsheet 13: Investigate cuic graphs in factor form. (page 34) Spreadsheet 14: Investigate cuic graphs. (page 35) Spreadsheet 15: Investigate cuic graphs of the form = a( ) 3 + c. (page 36) 1G Quartic graphs Tutorials We 7 int-5: Watch a worked eample on sketching the graph of a quartic. (page 39) We 8 int-79: Watch a worked eample on finding the turning points of a quartic using a CAS calculator. (page 4) Digital docs SkillSHEET 1.8: Practise solving quartic equations. (page 45) Spreadsheet 15: Investigate quartic graphs in factor form. (page 45) Investigation: Quartics and eond (page 46) Chapter review Digital doc Test Yourself : Take the end-of-chapter test to test our progress. (page 55) To access ebookplus activities, log on to 56 Maths Quest 1 Mathematical Methods CAS for the Casio ClassPad