Graphs and polynomials

Transcription

1 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Graphs and polnomials VCEcoverage Areas of stud Units & Functions and graphs Algera In this chapter A The inomial theorem B Polnomials C Division of polnomials D Linear graphs E Quadratic graphs F Cuic graphs G Quartic graphs

2 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Maths Quest Mathematical Methods The inomial theorem In Maths Quest Mathematical Methods we learned the following inomial epansions: ( + a) = + a + a ( + a) = + a + a + a These are called inomial epansions ecause the epressions in the rackets contain terms, i meaning. B continuing to multipl successivel a further ( + a), the following epansions would e otained: ( + a) = ( + a + a + a )( + a) = + a + 6 a + a + a ( + a) 5 = ( + a + 6 a + a + a )( + a) = a + a + a + 5a + a 5 The coefficients associated with each term can e arranged in a triangular shape as shown: ( + a) ( + a) ( + a) ( + a) ( + a) 6 ( + a) Notes:. The first and last numers of each row are.. 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

3 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials Note: n r n n! = C r = ( n r)!r! Rememer that n C r is another wa of writing n. r For eample, the epansion of ( + a) 6 can e written using cominations and then evaluated: ( + a) 6 = 6 6 a a + 6 a + 6 a + 6 a + 6 a a = a + 5 a + a + 5 a + 6a 5 + a 6 Now the inomial theorem can e formall stated. (a + ) n = n (a) n + n (a) n n (a) n + n (a) n n n Notes:. 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.. The numer of terms in the epansion is alwas n +.. The (r + )th term is n (a) n r r. r The inomial theorem can also e stated using summation notation: ( a + ) n n = r ( a) n r n r = Eample r Use the inomial theorem to epand ( ). Complete the inomial theorem epansion where a is the st term, is the nd term and n is the inde, using the appropriate row of Pascal s triangle to assist. WRITE ( ) = () ( ) + () ( ) ( ) = + () ( ) + () ( ) + () ( ) Evaluate the cominations and the powers. = (6 ) + (8 )( ) + 6( )(9) + ()( 7) + (8) Simplif each term. = Continued over page

4 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Maths Quest Mathematical Methods Eample Epand the inomial epression Complete the inomial epansion where a = ----, = and n = 5, using row 5 of Pascal s triangle to assist. WRITE = = Evaluate the powers. = = Simplif each term. = Eample State the coefficient of i and ii in ( ) 8. WRITE ii The powers of the st term decrease and the powers of the nd term increase,,,... Use this to find which term gives a power of. ii,, The third term gives a power of. Find the appropriate term using the Third term = 8 6 ( ) inomial theorem. Evaluate the term. = 8 79 = 8 68 State the coefficient. The coefficient of is ii Find which term gives a power of. ii,,,, The fifth term gives a power of. Evaluate the term. Fifth term = 8 ( ) = = 9 7 State the coefficient. The coefficient of the fifth term is 9 7.

5 5_6_56_MQVMM - _t Page 5 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 5 Find the fourth term in the epansion of ( ) 5. Eample Find the fourth term using the inomial theorem. WRITE Fourth term = 5 () Evaluate the term. = 8 Eample = 8 Find and evaluate the term that is independent of in the epansion of Find how the powers of are generated in the epansion from left to right. WRITE Powers of are ( ) 5 = 5, ( ) ---- =, ( ) ---- =,... 5 ( ) ---- = that is, 5,, 5,. Find the required term. The fourth term is independent of. Evaluate. Fourth term = 5 ( ) ---- Fourth term = State the solution. 6 Fourth term = The term that is independent of is the fourth term,. Eample Find the coefficient of in the epansion of ( + ) ( ) 5. 6 terms will result when multipling from the first and second rackets respectivel: terms and, terms and, terms and and terms and 5. WRITE Continued over page

6 5_6_56_MQVMM - _t Page 6 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 6 Maths Quest Mathematical Methods Write down the sum of these products, using Pascal s triangle to assist. WRITE terms = [5() ( )] + ()[() ( ) ] + () [() ( ) ] + [5()( ) ] Evaluate. = = 7 State the solution. The coefficient of is 7. rememer rememer. Pascal s triangle: Binomial theorem: (a + ) n = n (a) n + n (a) n n (a) n + n (a) n n n Notes:. The powers of (a) and sum to n.. There are n + terms in the epansion.. The (r + )th term is n (a) n r r. r A The inomial theorem SkillSHEET Mathcad. Eample Binomial epansions Epanding Eample Use the inomial theorem to epand each of the following. a ( + ) ( + ) c ( + ) 5 d ( ) 8 e ( 5) f ( + ) g ( ) 5 h ( + ) 7 i (7 ) j ( ) 5 k ( + ) l ( ) 6 Epand each of the following inomial epansions. a c + --

7 5_6_56_MQVMM - _t Page 7 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 7 Eample d -- e f g h State the coefficient of i ii and iii in each of the following. a ( 7) ( + ) 5 c ( ) 8 d (5 ) 7 e -- + f g + -- h 5 -- i Binomial theorem GC Binomial theorem Mathcad program TI multiple choice The coefficient of in ( 7) 6 is: A 7 B 7 C 78 D 96 E 76 GCprogram Casio Binomial theorem 5 multiple choice The coefficient of in 5 -- is: A 5 B 5 C 75 D 5 E multiple choice Which of the following does not have an 5 term when epanded? A ( + 6) 8 B -- 7 C D (8 ) 5 E multiple choice If a, then a + + c + d + e + f equals: 5 c 5 e f = d A 5 B C 6 D E 7 5 Eample 8 multiple choice Which one of the following epressions is not equal to ( )? A ( ) B ( )( ) C ( ) ( ) D E Find the fourth term in the epansion ( + ) 6.

8 5_6_56_MQVMM - _t Page 8 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 8 Maths Quest Mathematical Methods Find the third term in the epansion of -- 9, assuming ascending powers of. Eample 5 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 Find and evaluate the term that is independent of in the epansion of Eample 6 5 Find the coefficient of p in the epansion of ( p + ) 5 (p 5). 6 Find the coefficient of m 5 in the epansion of ( m) 6 (m + ). 7 In the epansion of (a ) n, the coefficient of the second term is 9. 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 n a + a + a where n is the degree (or highest power) of the polnomial and is a non-negative integer. The values of a n, a n,..., a, a and a are called the coefficients of their respective power of terms. Eample Which of the following epressions are not polnomials? a c d e a and d are polnomials ecause the are epressions with non-negative integer powers of onl. WRITE

9 5_6_56_MQVMM - _t Page 9 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 9 is not a polnomial as it has a power of 9 --, which is not an integer. c is not a polnomial as it has a power of -- ( ), which is not an integer, and it also has one term,, which is not a power of onl. e is not a polnomial ecause ---- = and so has a power that is not a positive integer. WRITE, c and e are not polnomials. Polnomials can e added and sutracted collecting like terms. Eample Given that P() = 6 + +, Q() = and R() =, find: a P() + Q() P() R(). WRITE a Add the polnomials. a P() + Q() = P() + Q() = + 5 Collect like terms. P() + Q() = Sutract the polnomials. P() R() = ( ) Remove rackets. = Collect like terms. = 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. Eample 9 For the polnomial P() = , find: a its degree P() c P( ). a The degree of the polnomial is the highest power of. WRITE a The degree of P() is. Continued over page

10 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Maths Quest Mathematical Methods WRITE Sustitute the given value of into the polnomial epression. P() = () () + 5() 6() + Evaluate. = = c Sustitute the given value of into c P( ) = ( ) ( ) + 5( ) 6( ) + the polnomial epression. Evaluate. = = 76 If P() = a , P( ) = 5 and P() = 65, find the values of a and. Sustitute a given value of into the polnomial and equate it to the given answer. Simplif the equation Eample Make the suject of the equation and call this equation []. Sustitute a given value of into the polnomial and equate it to the given answer. Simplif the equation. WRITE P( ) = a( ) 5 + ( ) ( ) + ( ) 5 = 5 a = 5 a + = = a [] P() = a() 5 + () () + () 5 = 65 a = 65 a + = 65 a + = 5 [] Sustitute [] into []. Sustituting = a: a + ( a) = 5 Solve this equation for a. Sustitute the value of a into equation []. a + 8 a = 5 a = 6 a = Sustituting a = into equation []: = Find the value of. = 6 State the solution. Therefore, a = and = 6. Note: The simultaneous equations = a and a + = 5 could e solved using a graphics calculator. Rewrite the second equation as = 6a 6.

11 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 rememer rememer Chapter Graphs and polnomials. If P() = a n n + a n n a + a + a and n is a non-negative integer then P() is a polnomial of degree n and a n, a n,... a, a are called coefficients and R.. A polnomial P() is evaluated in the same wa as an function. B Polnomials Eample Which of the following are not polnomial epressions? 7 viii viii + + viii iiiv iiiv 6 + iivi ivii viii Eample 8 Eample 9 Given that P() = 8 + +, Q() = 5 and R() = then find: a P() + Q() Q() R() c P() R() d P() Q() + R() For each of the following polnomials, find: i its degree ii P() iii P() and iv P( ). a P() = P() = c P() = d P() = + 7 e P() = f P() = Evaluating polnomials Mathcad multiple choice If P() = , then P( ) is equal to: A 79 B 95 C D 8 E Evaluating polnomials EXCEL Spreadsheet Eample 5 If P() = 7 + a , P() = and P() = 6, find a and. 6 Find a and, given that f () = a + + 7, f () = and f () = 5. 7 For Q() = a 6 +, Q() = 5 and Q() = 7. Find a and. 8 Find a and if P() = a , P() = and P( ) =.. Simultaneous equations SkillSHEET 9 multiple choice a If P() = a + 5 and P() =, then a is equal to: A B C D E If f () = n + 5 and f () =, then n is equal to: A B 6 C 7 D 5 E

12 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Maths Quest Mathematical Methods Graphics Calculator tip! Finding several values of a function CASIO Finding several values of a function To evaluate several values of a function at once, tpe Y({-, -, -,,, }) (for eample) at the home screen, and press ENTER. 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: R ( ) P() = Q() D ( ) where Q() is called the quotient, R() is called the remainder, and D() is called the divisor. Find the quotient, Q(), and the remainder, R(), when + 8 is divided the linear epression +. 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 and write the result aove. Multipl the result + and write the result underneath. Sutract and ring down the remaining terms to complete the epression. WRITE + ) Eample + ) ) ) ( + )

13 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 5 Divide into 5 and write the result aove. Continue this process to complete the long division. The polnomial 5 +, at the top, is the quotient. The result of the final sutraction,, is the remainder. WRITE 6 + ) ( + ) The quotient is 5 +. The remainder is ( 5 ) + 8 ( + ) 8 ( 8) Note: P( ) = ( ) ( ) + ( ) 8 = = 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. ā - 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 =. ā - 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 +, ut ( + ) could not e a factor. Eample Determine whether or not D() = ( ) is a factor of P() = 8. WRITE Evaluate P(). P() = () () () 8 = = If P() = then ( ) is a factor of P(), ut if P(), ( ) is not a factor of P(). P() so ( ) is not a factor of P().

14 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Maths Quest Mathematical Methods Eample a Factorise P() = 6. Solve 6 =. WRITE a Use the factor theorem to find a value for a where P(a) = and a is a factor of the numerical term. Tr a =,,,,,, 6, 6 until a factor is found. a P() = () () () 6 = 8 P( ) = ( ) ( ) ( ) 6 = P() = () () () 6 = P( ) = ( ) ( ) ( ) 6 = So ( + ) is a factor. Divide P() the divisor ( + ) using long division. + ) 6 ( + ) ( 5 ) 6 ( 6) Epress P() as a product of linear and quadratic factors. P() = ( + )( 5 ) Factorise the quadratic, if possile. = ( + )( + )( ) Rewrite the equation in factorised form, using the answer to part a. 6 = ( + )( + )( ) = Use the Null Factor Law to state the solutions. =, -- or Note: These solutions can e checked drawing the graph of = 6 on a graphics calculator. The -intercepts should e the same as the solutions found in part.

15 5_6_56_MQVMM - _t Page 5 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 rememer rememer Chapter Graphs and polnomials 5 R ( ). P() = Q() 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).. Factor theorem: If P(a) =, then ( a) is a factor of P(). If (a + ) is a factor of P() then P =. ā -. If ( a) is a factor of P() then a must e a factor of the term independent of. C Division of polnomials GC program TI Eample Eample Find the quotient, Q(), and the remainder, R(), when each of the following polnomials are divided the given linear epression. a + 5, + + 7, c 5 + +, + d , + e 6 +, f + 6, g 6 +, + h , a For each corresponding polnomial in question, evaluate: i P() ii P() iii P( ) iv P( ) v P() vi P() vii P( -- ) viii P( -- ) Compare these values to R() in question and comment on the result. In each of the following determine whether or not D() is a factor of P(). a P() = , D() = P() = 5 8, D() = + c P() = , D() = + d P() = , D() = e P() = + + 5, D() = + 5 f P() = , D() = g P() = , D() = + multiple choice Eamine the equation f () = + 6. a Which one of the following is a factor of f ()? A + B C + D + E Division of polnomials GCprogram Casio Division of polnomials Division of polnomials Evaluating polnomials Finding factors of polnomials Mathcad Mathcad EXCEL Spreadsheet

16 5_6_56_MQVMM - _t Page 6 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 6 Maths Quest Mathematical Methods WorkSHEET. Eample a Eample When factorised, f () is equal to: A ( + )( )( + ) B ( + )( )( )( ) C ( + )( )( + )( + ) D ( )( + )( )( ) E ( )( + )( + ) 5 Factorise the following polnomials. a P() = + 8 P() = + c P() = + 6 d P() = e P() = + + f P() = g P() = h P() = + 6 Solve each of the following equations. a = = c + 8 = d + 8 = e + + = f + = 7 If ( ) is a factor of + a 6, then find a. 8 If ( ) is a factor of + a +, then find a. 9 Find the value of a if ( + ) is a factor of + a + 8. Find the value of a and if ( + ) and ( ) are factors of a +. If ( ) and ( + ) are factors of + a + +, find the values of a and. Linear graphs Linear graphs are polnomials of degree. Graphs of linear functions are straight lines and ma e sketched finding the intercepts. Revision of properties of straight line graphs. The gradient of a straight line joining two points is: m = The general equation of a straight line is: = m + c where m is the gradient and c is the value of the -intercept.. The equation of a straight line passing through the point (, ) and having a gradient of m is: = m( ) B (, ) A (, ) Gradient = m A (, ). The intercept form of the equation of a straight line is: = or + a = a a (, ) 5. Parallel lines have the same gradient. (a, )

17 5_6_56_MQVMM - _t Page 7 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 7 6. The product of the gradients of two lines that are perpendicular equals. That is, m m = or m = Eample Sketch the graph of the linear function = 6 indicating the intercepts. WRITE/DRAW Sustitute = into the equation. When =, = 6 Solve the equation for to find the -intercept. = Therefore, the -intercept is. Sustitute = into the equation. When =, = 6 Solve the equation for to find the -intercept. = Therefore, the -intercept is. 5 Draw a set of aes. 6 Indicate the -intercept and -intercept = 6 and rule a line through these points. m 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, =, or Interval notation f () =, f : (, ] R, f () = Restricted domains or ranges can e represented interval notation in three forms.. The closed interval.. The open interval.. The half-open interval. 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 {} = (, ]

18 5_6_56_MQVMM - _t Page 8 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 8 Maths Quest Mathematical Methods Eample 5 Find the equation, in the form a + + c =, of each straight line descried elow. a The line with a gradient of and passing through (, ) The line passing through (, 8) and (, ) c The line which passes through (, ) and is parallel to the line with equation = d The line which passes through (, ) and is perpendicular to the line with equation + = WRITE a Write the rule for the point gradient a = m( ) form of the equation of a straight line, = m( ). Sustitute the value of the gradient, ( ) = ( ) m, and the coordinates of the point (, ), into the equation. Epand the rackets. + = 6 Epress the equation in the form required. + 8 = or 8 = 5 Write the rule for the gradient, m, of a straight line, given points. Sustitute the values of m and (, ), into the rule and evaluate the gradient. Sustitute the value of m, and (, ), into the rule for the point gradient form of the equation of a straight line. (Coordinates of either point given ma e used.) Epand the rackets. Epress the equation in the form required. m = = = = 8 = ( ) 8 = + 8 = c 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. Sustitute the values of m and the coordinates (, ) = (, ). Simplif and write in the required form. c 5 = ecomes = + 5. The gradient of the parallel lines is. = m( ) = ( ) = 6 =

19 5_6_56_MQVMM - _t Page 9 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 9 WRITE d Find the gradient of the given line. d = + The gradient of the line is Find the gradient of the perpendicular line. The gradient of the perpendicular line is --. Write the rule for the point gradient = m( ) form of the equation of a straight line. Sustitute the values of m and the = -- ( ) coordinates (, ) = (, ). 5 Simplif and write in the required form. 6 = ( ) + 5 = Eample Sketch the graph of each of the following functions, stating the domain and range of each. a = 8, [, ] f () =, (, ) a Sustitute the smallest value of into the equation. Solve the equation for, to find an end point of the straight line. State the coordinates of the end point. 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). 8 Draw a straight line etween the two points. 9 6 Find the intercepts and mark them on the graph. WRITE/DRAW a When =, = 8 = = (, ) is a closed end of the line. When =, = 8 = = (, ) is the other closed end of the line. When =, = When =, = The -intercept is and the -intercept is. State the domain, which is given The domain is [, ]. with the rule. State the range from the graph. The range is [, ]. (, ) (, ) = 8, [, ] Continued over page

20 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Maths Quest Mathematical Methods 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. Sustitute another value of within the domain into the equation (that is, a value of <, since (, )) and find. State the coordinates of the point. Plot the points on a set of aes and mark the point (, ) with an open circle. Rule a straight line from (, ) to (, 5) and eond. An arrow ma e placed on the other end to indicate that the line continues. Note that there are no intercepts. State the domain, which is given with the rule. State the range eamining the graph. WRITE/DRAW When =, = f ( ) = (, ) is an open end of the line. When =, = f ( ) = 5 (, 5) is another point on the line. 5 f() =, (, ) (, 5) 5 6 (, ) rememer rememer The domain is (, ). The range is (, ). Linear graphs. Linear equations are polnomials of degree.. Gradient, m = General equation is a + + c = or = m + c where m = gradient and c = -intercept.. Equation if a point and gradient is known: = m( ) 5. Equation if the intercepts are known: = a 6. Parallel lines have the same gradient. 7. If m and m are the gradients of perpendicular lines, then: m m = or m = m

21 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials D Linear graphs Eample Eample 5a Eample 5 Eample 5c Eample 5d Sketch the graph of each of the following linear functions indicating the intercepts. a + = = 8 c 5 = d = e + = f = 5 Find the equation, in the form a + + c =, of each straight line descried elow. a The line with a gradient of and passing through (, ). The line with a gradient of and passing through (, ). c The line with a gradient of and passing through (, ). Find the equation, in the form a + + c =, of each straight line descried elow. a The line passing through (, ) and (, 6). The line passing through (, ) and (, ). c The line passing through (7, 5) and (, ). c Find the equation in the form a + + c = that passes through (, ) and is perpendicular to the line with equation + =. 6 Match each of the following graphs with the appropriate rule elow. a c Linear graphs GC Finding a linear equation Mathcad program TI GCprogram Casio Finding a linear equation multiple choice Which one of the following points does not lie on the straight line with equation. 6 =? A (, 6) B (, ) C (, ) D (, ) E (, 9) Gradient 5 Consider the points A(, 5) and B(, ). 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 (, 5) and is parallel to the line with equation + =.. Using gradient to find the value of a parameter SkillSHEET SkillSHEET (, ) d e f iii + + = iii = iii = iv + = 6 iv = vi =

22 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Maths Quest Mathematical Methods SkillSHEET SkillSHEET.5 Interval notation.6 Domain and range for linear graphs 7 State the range for each function graphed elow. a ( 5, ) c d (, ) (, ) (6, 5) (, ) (5, ) e f (5, 6) Eample 6 8 Sketch the graph of each of the following functions, stating i the domain and ii the range of each. a = 8 =, c + =, [, ] d 5 =, < 5 e + =, [ 8, ) f f () = 6, ( 6, ) g f () = 5 +, (, ] h 6 =, [, 5) 9 Find the equation of the straight line which passes through the point (, 5) and is: a parallel to the line with equation = perpendicular to the line with equation = 7. Write equations in the form a + + c =. Find the equation of the straight line which passes through the point (, ) and is: a parallel to the line with equation = perpendicular to the line with equation =. multiple choice If the straight lines = and a + = are parallel then a = : A 6 B C D E 6 multiple choice If the straight lines 5 + = and = are perpendicular, then is equal to: A 5 B -- C 5 D -- E 5 5

23 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials Quadratic graphs Quadriatic 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. 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 is = a (e) the -intercepts are found solving the equation a + + c =.. The equation a + + c = can e solved either: (a) factorising or ± () using the quadratic formula, ac = a. The turning point can e found completing the square (see page ). The turning point is located on the ais of smmetr, which is halfwa etween the -intercepts. The discriminant The value of ( ac), 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.. If ac >, there are two solutions to the equation and there are two -intercepts on the graph.. If ac > and is a perfect square, the solutions are rational; otherwise the are irrational.. If ac =, the two solutions are equal and there is one -intercept on the graph; that is, the graph has a turning point on the -ais.. If ac <, there are no real solutions and there are no -intercepts on the graph. Eample 7 Use the discriminant to determine the numer of -intercepts for the quadratic function f () = +. Find the values of the quadratic coefficients a, and c using the general quadratic function, = a + + c. Evaluate the discriminant. If the discriminant is greater than, there are two -intercepts. WRITE a =, =, c = ac = ()( ) = = 89 ac > So there are two -intercepts, which are oth irrational.

24 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Maths Quest Mathematical Methods Sketch the graph of the function f () = 5, showing all intercepts. Give eact answers. 5 6 Evaluate f () to find the -intercept (or state the value of c). WRITE/DRAW f () = 5() () = State the -intercept. The -intercept is. Set f () = to find the -intercepts. f () = 5 = Factorise the quadratic (or use the ( + )( ) = quadratic formula). Solve the equation using the Null + = or = Factor Law. + = or = State the -intercepts. The -intercepts are and --. Draw a set of aes and mark the intercepts or the coordinates of the points where the graph crosses the aes. (, ) f() = 5 Sketch a paraola through the intercepts. (, ) (, ) 7 8 Eample 8 -- The -coordinate of the turning point of a quadratic function is eactl halfwa + -- etween the two -intercepts, so for worked eample 8, = = -- (or -- ). 5 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 = -----, where a + + c =. a 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( h) + k where the turning point is (h, k). This wa of writing the function is known as turning point form.

25 5_6_56_MQVMM - _t Page 5 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 5 Eample For the function = ( + ), find: ii the coordinates of the turning point ii the domain and range. WRITE Write the general formula. = a( h) + k Write the function. = ( + ) ii Identif the values of a, h and k. ii a =, h =, k = State the coordinates of the turning The turning point is (, ). point (h, k). ii Write the domain of the paraola. ii The domain is R. Write the range k (as a < ). The range is. 9 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, 5) (, 6) WRITE a Write the general rule for a quadratic a = a ( h) + k in turning point form. Find the values of h and k using the given turning point. Since the turning point is (, 6): h =, k = 6 State the value of a (given). a = Sustitute these values in the rule. So = ( + ) 6 5 Epand the rackets. = Simplif. = + 5 The rule is = + 5. Use the graph to find the domain. 5 Look at all the values that can take. State the domain in interval notation. Domain = [ 5, ) c Use the graph to find the range. c 6 Look at all the values that can take. State the range in interval notation. Range = [ 6, )

26 5_6_56_MQVMM - _t Page 6 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 6 Maths Quest Mathematical Methods Eample Sketch the graph of = -- ( ) +, clearl showing the coordinates of the turning point and the intercepts with the aes. WRITE/DRAW Write the general equation of the paraola. = a( h) + k Identif the values of the variales. a = --, h =, k = Write a rief statement on the transformation The graph of = is dilated in the direc- of the asic paraola. tion the factor of -- (that is, it is wider than the asic curve); it is translated unit to the right and units up. State the shape of the paraola (that is, a > ; the paraola is positive. positive or negative). 5 State the coordinates of the turning The turning point is (, ). point (h, k). 6 As oth a and k are positive, onl the - -intercept: = intercept needs to e determined. Find = -- ( ) + the -intercept making =. = -- ( ) + 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. = -- + = 7 -- = ( ) + Eample Sketch the graph of = + 8, showing the turning point and all intercepts, rounding answers to decimal places where appropriate. WRITE/DRAW Find when =. When =, = State the -intercept. The -intercept is. Let the quadratic equal zero. When =, + 8 =

27 5_6_56_MQVMM - _t Page 7 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 7 WRITE/DRAW Solve for using the quadratic formula. 8 ± 8 = ( ) ( ) ( ) 8 ± 88 = ± = ± = = or State the -intercepts, rounding to decimal places. Write the original rule using decreasing powers of. The -intercepts are.5 and.5. = Complete the square. = ( -- ) = [( + ) -- ] = [( ) ] = ( ) + State the turning point. The turning point is (, ). 9 Draw a set of aes and mark the coordinates of the turning point and the points where the graph crosses the aes. Sketch a paraola through these points. (, ) f() = (, ) (.5, ) (.5, ) 5 In general, the turning point of a quadratic function is required if the range needs to e determined. However, the -intercepts and -intercept are not required in determining the range of quadratic functions. Sketch graphs are also useful. Intercepts and turning points can e found using a graphics calculator. This is useful for multiple-choice questions, questions that are allocated onl one mark, and questions that do not require algeraic methods.

28 5_6_56_MQVMM - _t Page 8 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 8 Maths Quest Mathematical Methods Eample The weight of a person t months after a gmnasium program is started is given t the function: W(t) = --- t + 8, where t [, 8] and W is in kilograms. Find: a the minimum weight of the person the maimum weight of the person. WRITE Complete the square to find the turning point. t W = --- t + 8 = -- [t 6t + 6] = -- [t 6t ] = -- [(t ) + 5] = -- (t ) State the minimum turning point. The turning point is (, 75.5). Find the end point value for W when t =. When t =, W = 8 State its coordinates. One end point is (, 8). Find the end point value of W when t = 8. When t = 8, W = 88 State its coordinates. The other end point is (8, 88). W (kg) On a set of aes, mark the end points Maimum (8, 88) and turning point. 9 8 (, 8) Sketch a paraola etween the end 7 Minimum (, 75.5) points. Locate the maimum and minimum values of W on the graph. 8 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.

29 5_6_56_MQVMM - _t Page 9 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 rememer rememer Chapter Graphs and polnomials 9 Quadratic graphs. Quadratic equations are polnomials of degree.. General equation is = a + + c.. The quadratic formula is given the equation ± ac = a. The discriminant is ac and if: (a) ac >, there are two -intercepts. If ac is a perfect square, the intercepts are rational. () ac =, there is one -intercept, which is a turning point (c) ac <, there are no -intercepts. 5. The turning point form of the quadratic graph or paraola is: = a( h) + k and the turning point is (h, k). 6. The ais of smmetr of a paraola is given the epression The ais of smmetr is halfwa etween the -intercepts. a Eample 7 Eample 8 E Quadratic graphs Use the discriminant to determine the numer of -intercepts for each of the following quadratic functions. a f () = + f () = c f () = d f () = + 7 Discriminant e f () = 6 f f () = Sketch the graphs of each of the following functions, showing all intercepts. Give eact answers. a f () = f () = c f () = 5 + d f () = 6 e f () = + f f () = + 5 g f () = 6 h f () = Discriminant Quadratic graphs Mathcad EXCEL Spreadsheet Mathcad EXCEL Spreadsheet Eample 9 Find the turning point for each of the functions in question. Give eact answers. For each of the following functions find: i the coordinates of the turning point ii the domain iii the range. a = -- = c = ( 6) d = ( + ) e = ( + ) 6 f = ( ) g = ( + ) 5 Quadratic graphs.7 Domain and range for quadratic graphs SkillSHEET

30 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Maths Quest Mathematical Methods Eample 5 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 (, ) (, ) d e f (, 6) (, 6) (5, ) (, 9) (, ) (, 6) Mathcad Function grapher Eample 6 Sketch the graphs of the following, clearl showing the coordinates of the turning point and the intercepts with the aes. a = + = c = ( ) d = ( + ) + e = ( ) f = ( ) 8 EXCEL Spreadsheet Function grapher multiple choice Consider the function with the rule =. a It has -intercepts: A (, ) and (, ) B (, ) and (, ) C (, ) and (, ) D (, ) and (, ) E (, ) and (, ) It has a turning point with coordinates: A (, ) B (, ) C (, ) D (, ) E (, ) multiple choice The function f () = ( + ) + has a range given : A (, ) B (, ] C [, ) D (, ] E R multiple choice The range of the function = ( ), [, 6] is: A [, 6] B [, 6] C [, ] D (, ] E [, 6) EXCEL Spreadsheet Mathcad Eample Quadratic graphs turning point form Quadratic graphs turning point form 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 () = ( ) f () = ( ) c f () = ( + ) + d f () = ( + ) + 9 e = + + f = 6 Sketch the graph of each of the functions elow and state i the domain and ii the range of each function. a = = c = +, [, ] d = +, R + e f () =, [, 6] f f () = g f () = 5 + 6, [ 5, ) h f () = 5 5 +, (, ]

31 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials Eample The volume of water in a tank, V m, over a month period is given the function V(t) = t 6t +, where t is in months and t [, ]. Find: a the minimum volume of water in the tank the maimum volume of water in the tank. A all thrown upwards from a tower attains a height aove the ground given the function h(t) = t t + 6, 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 Maimum height h(t) = t t + 6 Ground A section of a roller-coaster at an amusement park follows the path of a paraola. The function h(t) = t t + 8, t [, ], 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 c d e Find the lowest point of this section of the ride. Find the time taken for the carriage to reach the lowest point. Find the highest point aove the ground. Find the domain and the range of the function. Sketch the function. WorkSHEET. Cuic graphs Cuic functions are polnomials of degree. 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

32 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Maths Quest Mathematical Methods If a is positive (that is, a > ), the function is called a positive cuic. Several positive cuics appear elow. If a is negative (that is, a < ), the function is called a negative cuic. Several negative cuics appear elow. MM& fig. You ma wish to investigate in more detail the tpe of equation required to produce each of the aove graphs. Basic form Some (ut certainl not all) cuic functions are transformations of the form =, which has a point of inflection at the origin. These ma e epressed in the form = a( h) + k where (h, k) is the point of inflection. For eample, = ( ) + 5 is the graph of = translated + in the direction, +5 in the direction and dilated a factor of in the direction. This form, called asic form, works in the same wa as a quadratic equation epressed in turning point form: = a( h) + k where (h, k) is the turning point and a is the dilation factor. Basic form and its transformations will e discussed in more detail in chapter. = = a( h) + k (h, k)

33 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 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 > = ( + )( )( ) c d Repeated factors A twice onl repeated factor in a factorised cuic function indicates a turning point that just touches the -ais. Verif this for several cases using a graphics calculator. a = ( a) ( ) Eample For each of the following graphs, find the rule and epress it in factorised form. Assume that a = or a =. a f() f() WRITE a Find a deciding whether the a The graph is a positive cuic, so a =. graph is a positive or negative cuic. Use the -intercepts, and to The factors are ( + ), and ( ). find the factors. Epress f () as a product of a and its f () = ( + )( ) factors. Simplif. f () = ( + )( ) Continued over page

34 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Maths Quest Mathematical Methods WRITE 5 Find a deciding whether the graph is The graph is a negative cuic, so a =. a positive or negative cuic. Use the -intercept, which is also a ( + ) is a factor. turning point, to find the repeated factor. Use the other -intercept,, to find the ( ) is also a factor. other factor. Epress f () as a product of a and its f () = ( + ) ( ) factors. Simplif. f () = ( )( + ) Sketch the graph of = 8, showing all intercepts. WRITE/DRAW Find when =. When =, = 8 State the -intercept. The -intercept is 8. Let P() =. Let P() = 8 Use the factor theorem to find a factor of the cuic P() = 8. P() = () 8 = 8 P( ) = ( ) ( ) ( ) 8 = so ( + ) is a factor Eample 5 Use long division, or otherwise, to find the quadratic factor. B long division: 8 + ) = ( + )( 8) Factorise the quadratic, if possile. = ( + )( )( + ) Epress the cuic in factorised form If ( + )( )( + ) = and let it equal to find the -intercepts. Solve for using the Null Factor Law. =, or State the -intercepts. The -intercepts are,, and.

35 5_6_56_MQVMM - _t Page 5 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 5 WRITE/DRAW Sketch the graph of the cuic. = 8 8 Verif the graph and intercepts using a graphics calculator. Restricting the domain of cuic functions. 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. For eample: The range could not e stated here unless the -coordinate of the local minimum is known. Recall that cuic functions that do not have an turning points can have onl one -intercept. (, ) (6, 8) Coordinate of local minimum required Eample 6 Sketch the graph of = 5, where (, ], using the unrestricted function as a guide. State the domain and range. WRITE/DRAW Decide whether it is a positive or negative cuic looking at the coefficient of. Negative cuic Find the -intercept/s. When =, 5 = ( + 5) = = ( + 5 ) The -intercept is. Find the -intercept. When =, = () 5() = The -intercept is. Find when has the value of the lower end point of the domain. When =, = ( ) 5( ) = 8 Continued over page

36 5_6_56_MQVMM - _t Page 6 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 6 Maths Quest Mathematical Methods 5 State the coordinates of this end point and decide whether it is open or closed. 6 Find when has the value of the upper end point. 7 State the coordinates of this end point and decide whether it is open or closed. Mark these points on a set of aes. 9 Sketch the part of the cuic etween the end points. Verif this graph using a graphics calculator. WRITE/DRAW 8 (, 8) The open end point is (, 8). When =, = ( ) 5( ) = 6 The closed end point is (, 6). (, 6) State the domain, which is given with the rule. From the graph, state the range. Note that the intercept is not included in the domain. The domain is (, ]. The range is [6, 8). rememer rememer Cuic graphs. The general equation is = a + + c + d.. Basic shapes of cuic graphs: Positive cuic Negative cuic Basic form = a( h) + k (h, k) 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.

37 5_6_56_MQVMM - _t Page 7 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 7 F Cuic graphs Eample For each of the following graphs, find the rule and epress it in factorised form. Assume that a = or a =. Cuic graphs factor a form Mathcad 6 5 Cuic graphs factor form EXCEL Spreadsheet Match each of the following graphs to the most appropriate rule elow. a c 5 d e f 5 g h vii = ( ) viii = ( + )( )( ) iiii = ( )( + )( ) iiiv = ( + ) (5 ) iiv = ( + )( )( ) iivi = ( + )( + )( ) vii = ( ) viii = ( + ) ( 5) Cuic graphs Mathcad Eample 5 Sketch the graph of each of the following, showing all intercepts. a = + = c = + 6 d = e = + 8 f = g = 8 Verif our answers using a graphics calculator. Cuic graphs EXCEL Spreadsheet

38 5_6_56_MQVMM - _t Page 8 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 8 Maths Quest Mathematical Methods multiple choice a Full factorised, is equal to: A ( + ) B ( + ) C ( ) D ( ) E ( + )( ) The graph of = is: A B C D E Mathcad Cuic graphs = a( ) + c form 5 multiple choice The function graphed in the figure could have the following rule: A = ( ) + B = ( + ) + C = ( ) + D = ( + ) E = ( ) (, ) EXCEL Spreadsheet 6 multiple choice Cuic graphs = a( ) + c form The graph of f () = 5( + ) is est represented : A B C (, ) (, ) (, ) D E (, ) (, )

39 5_6_56_MQVMM - _t Page 9 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 9 7 multiple choice The graph of f () = ( ) ( + ) is est represented : A B (, 6) (, 6) (, ) (, ) (, ) (, ) C D (, ) (, ) (, ) (, ) (, 6) (, 6) E (, ) (, ) (, 6) 8 multiple choice The graph shown is est represented the equation: A = ( a) + B = ( a) + C = (a ) + D = ( + a) + E = ( + a) + (a, ) (, c)

40 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Maths Quest Mathematical Methods Eample 6 9 multiple choice If a < and, c > then the graph shown is est represented the equation: A = ( + a) ( c) a c B = ( + a) (c ) a c C = ( a) ( + c) a c D = ( + a) ( ) (c ) a c a c E = ( + a) ( c) a c 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 () = + + 8, [, ) f () = 5 +, [, ] c f () = , (, ] d f () = + 6, [, ] e f () = 5, (, ) f f () =, [, ) g f () = +, [, ) (, ] h f () =, (, ) [, ) Verif our answers using a graphics calculator. The function f () = + a + 6 has -intercepts (, ) and (, ). Find the values of a and. The functions = + a + and = 6 + (a + ) oth have (, ) as an -intercept. Find the values of a and. 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 elow. g() f() = a( + ) + c (, 6) (, ) (, ) a c Find the values of a, and c, and hence state the rule of f (). Find the rule for g() and state its domain and range. What is the width of the vessel when the height is.75 cm?

41 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 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 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. c Sketch the graph of d(t). d Find to the nearest metres the maimum distance of the hikers from their starting point and the time, to the nearest minute, that it occurs. Quartic graphs Quartic functions are polnomials of degree. 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 > ). = a. = a + c, c. = a ( )( c). = a( ) ( 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. c

42 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Maths Quest Mathematical Methods 5. = a( )( c) 6. = a( )( c)( d)( e) c c d e The cued factor ( c) show the graph as a point of inflection at = c. The factors show intercepts at =, c, d and e. Negative quartics If a <, that is, each of the aove rules is multiplied, then the graphs are reflected through the -ais. For eample, the graph of = (at right) is a reflection, through the -ais, of the graph of = =. Similarl, the graph of = + = ( ) is a reflection through the -ais of the graph of =. Note: The aove graphs can e translated horizontall or verticall ut this is considered in chapter, Other graphs and modelling. 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. Eample 7 Sketch the graph of = , showing all intercepts. WRITE/DRAW Find the -intercept. When =, = The -intercept is. Let = P(). Let P() = Find two linear factors of the quartic epression, if possile, using the factor theorem. P() = () () 7() + 5() + = 8 P( ) = ( ) ( ) 7( ) + 5( ) + = ( + ) is a factor. P() = () () 7() + 5() + = ( ) is a factor.

43 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials WRITE/DRAW 5 Find the product of the two linear factors. Use long division to divide the quartic the quadratic factor (or use another method). ( + )( ) = 5 55 ) ( ) ( ) 6 Epress the quartic in factorised form. = ( + )( )( 5) 7 Factorise the quadratic factor, 5, = ( + )( )( + 5 )( 5 ) using difference of perfect squares. 8 To find the -intercepts, set equal to zero. Let = ( + )( )( + 5 )( 5 ) = 9 Solve for using the Null Factor Law. =,, ± 5 State the -intercepts. The -intercepts are,, 5 and 5. Sketch the graph of the quartic. (, ) Check the graph using a graphics calculator. (, ) ( 5, ) (, ) ( 5, ) Eample 8 Sketch the graphs of each of the following equations, showing the coordinates of all intercepts. Use a graphics calculator to find the coordinates of the turning points, rounding to decimal places as appropriate. a = ( )( + ) = ( + ) ( ) WRITE/DRAW a State the function. a = ( )( + ) Find the -intercept. When =, = The -intercept is Find the -intercepts. When =, ( )( + ) = =,, State the -intercepts, noting where The graph touches the -ais at =. the graph touches and where it cuts The other -intercepts are and. the -ais. Continued over page

44 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Maths Quest Mathematical Methods WRITE/DRAW 5 6 State the coordinates of the turning points. Sketch the graph of the quartic, using a graphics calculator to assist. The minimum turning points are (.,.8) and (.69,.). The maimum turning point is (, ). Fig..9 to go here (, ) (, ) (, ) (.69,.) (.,.8) State the function. = ( + ) ( ) 5 6 Find the -intercept. When =, = () ( ) = 9 The -intercept is 9. Find the -intercepts. When =, = ( + ) ( ) State the points where the graph touches the -ais from the repeated factors. State the coordinates of the turning points. Sketch the graph of the quartic, using a graphics calculator to assist. =, The graph touches the -ais at = and =. The maimum turning points are (, ) and (, ), and the minimum turning point is (, 6). (, ) (, ) (, 9) (, 6)

45 5_6_56_MQVMM - _t Page 5 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 5 Eample 9 Determine the equation of the graph shown. - WRITE State the -intercepts. The -intercepts are,,,. Write the equation using factor form = a( + )( + )( )( ) with a dilation factor of a. State the -intercept. The -intercept is. Sustitute the coordinates of the point (, ) = a( + )( + )( )( ) where the graph crosses the -ais into the equation. 5 Solve the equation to find a. = a 6 a = 6 Write the equation. = -- ( )( )( + )( + ) Eample Sketch the graph of each of the following restricted functions, using the unrestricted function as a guide. State the domain and the range in each case. a = ( + ) ( ), (, ] =, (, ] WRITE/DRAW a State the function. a = ( + ) ( ), (, ] Find the -intercept. When =, = () ( ) = State the -intercept. The -intercept is. Find the -intercept. When =, ( + ) ( ) = 5 Solve for. = or 6 State the -intercepts. The -intercepts are and. 7 Sketch the graph over the domain (, ], using knowledge of asic shapes or a graphics calculator to (, ) assist. (The cued factor indicates a (, ) point of inflection.) (, ) -- = ( + ) ( ) Continued over page

46 5_6_56_MQVMM - _t Page 6 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 6 Maths Quest Mathematical Methods WRITE/DRAW 8 9 State the domain, which is given with the rule. From the graph, state the range. The domain is (, ]. The range is [, ). State the function. =, (, ] Find the -intercept. When =, = () () = State the -intercept. The -intercept is. Find the -intercepts. When = = Factorise the quartic epression. ( + ) = Solve for. = is the onl solution (as + ). State the -intercepts. The onl -intercept is. 8 9 Find when is one end point of the domain. State the coordinates and whether it is an open or closed point. When =, = ( ) ( ) = (, ) is an open end point. Find when is the other end point of the domain. When =, = () () = State the coordinates and whether it is an open or closed point. (, ) is a closed end point. Sketch the graph of the quartic, using knowledge of asic shapes or a graphics calculator to assist, over the domain. (, ) (, ) (, ) = State the domain, which is given with the rule. The domain is (, ]. From the graph, state the range. The range is [, ].

47 5_6_56_MQVMM - _t Page 7 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 rememer rememer Quartic graphs. General equation is = a + + c + d + e. Basic shape of quartic graphs: (a) If a > : Chapter Graphs and polnomials 7 = a = a + c, c = a ( )( c) c = a( ) ( c) c = a( )( c) 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.

48 5_6_56_MQVMM - _t Page 8 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 8 Maths Quest Mathematical Methods G Quartic graphs Mathcad Mathcad Quartic graphs factor form Quartic graphs Eample 7 Sketch the graph of each of the following showing all intercepts. Verif the shape of the graph using a graphics calculator. a = ( )( + )( )( + ) = ( )( + )( 5) c = d = + e = + 9 f = + g = h = EXCEL SkillSHEET.8 Eample 8 Solving quartic equations Spreadsheet Sketch the graph of each of the following equations, showing the coordinates of all intercepts. Use a graphics calculator to find the coordinates of the turning points, rounding to decimal places as appropriate. a = ( )( ) = ( + ) ( ) c = ( ) ( + )( + ) d = ( + ) ( ) Quartic graphs factor form multiple choice Consider the function f () = a When factorised, f () is equal to: A ( + )( )( )( + ) B ( )( )( + ) C ( + )( )( )( + ) D ( ) ( + ) E ( ) ( + ) The graph of f () is est represented : A B C D 6 E

49 5_6_56_MQVMM - _t Page 9 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 9 c If the domain of f () is restricted to [, ], then the range is: A [, 6] B [, ] C [, ] D R + E [, ) d If the range of f () is restricted to (, 5) then the maimal domain is: A [, ) B (, ) C (, ) D (, ) E (, ) e If the domain of f () is restricted to (, ), then the range is: A (, 6) B (, ) C (, 9) D (9, 6) E [9, ) f If the domain of f () is restricted to [, ), then the range is: A R B R + C [, ) D [, 6) E [, ) Eample 9 Determine the equation of each of the following graphs. a 6 - c d 8

50 5_6_56_MQVMM - _t Page 5 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 5 Maths Quest Mathematical Methods Eample 5 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 = ( ), R = ( )( )( + ), [, ] c =, [, ) d = , (, ] e = ( ) ( + ), (, ] f = 6 7, [, ) g = ( + ) ( ), [, ) h =, [, ] Verif our answers using a graphics calculator. 6 The function f () = + a has -intercepts (, ) and (, ). Find the values of a and. 7 The function f () = + a has -intercepts (, ) and (, ). Find the values of a and. 8 The functions = (a ) and = + (a + 5) oth have an -intercept of. Find the value of a and. Mathcad Single function grapher Quartics and eond Use a graphing program such as Graphmatica or one of the Maths Quest Mathcad files to assist in answering the following questions. Investigate graphs of functions of the form f() = n for values of n from to 9. What do graphs of functions for which n is even have in common? What does an odd value of n do to the graph? Investigate graphs of functions of the form = ( a) n ( ) m ( c) p for various values of the pronumerals in the equation, for m, n and p. Write a report on our findings.

51 5_6_56_MQVMM - _t Page 5 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomialhs 5 summar Pascal s triangle Binomial theorem (a + ) n = n (a) n o + n (a) n n (a) n + n (a) o n n n Notes: Indices add to n. There are n + terms in the epansion. The (r + )th term is n (a) n r () r. r Polnomials If P() = a n n + a n n a + a + a and n is a non-negative integer then P() is a polnomial of degree n and a n, a n,..., a, a 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:. 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. General equation is a + + c = or = m + c where m = gradient c = -intercept The gradient m = Equation if a point and the gradient is known: = m( ) Parallel lines have the same gradient.

52 5_6_56_MQVMM - _t Page 5 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 5 Maths Quest Mathematical Methods If m and m are the gradients of perpendicular lines, then: m m = or m = Quadratic graphs Quadratic equations are polnomials of degree. General equation is = a + + c ± Quadratic formula is ac = a Discriminant = ac and. if ac >, there are -intercepts (and if ac is a perfect square, the intercepts are rational). if ac =, there is -intercept. if ac <, there are no -intercepts. The turning point form of the quadratic is: = a( h) + k and the turning point is (h, k). The equation of the ais of smmetr of a paraola is a The ais of smmetr is halfwa etween the -intercepts. Cuic graphs Cuic equations are polnomials of degree. General equation is = a + + c + d Basic shapes of cuic graphs: Positive cuic Negative cuic Basic form m = a( ) + c (, c) Factor form Repeated factor = a( )( c)( d) where a > a c d = ( a) ( ) If a <, then the reflections through the -ais of the tpes of graph in the aove figures are otained.

53 5_6_56_MQVMM - _t Page 5 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 5 Quartic graphs Quartic equations are polnomials of degree. General equation is = a + + c + d + e Basic shapes of quartic graphs:. If a > : = a c = a( ) ( c) c = a + c, c = a( )( c) c = a ( )( c) c d e = 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 < }

54 5_6_56_MQVMM - _t Page 5 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 5 Maths Quest Mathematical Methods CHAPTER review Multiple choice A A A B B B C C C D When epanded, ( ) 5 is equal to: A B C D E The coefficient of 5 in the epansion of is: A 96 B 7 C 96 D 6 8 E 6 8 Assuming descending powers of, the fifth term of the epansion of + -- is: A 5 9 B 8 C D 79 E Which of the following epressions is not a polnomial? A + B C + D E The value of P( ) in the polnomial, P() = 5 + +, is: A B 9 C 9 D 6 E 6 The degree of the polnomial ( )( + ) when epanded is: A B 8 C D 6 E 7 The remainder when is divided ( + ) is: A 7 B 5 C 7 D E 8 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 ( + ) C ( + ) D ( ) E ( ) The rule for the graph shown is: A + + = B = C = D + = E + =

55 5_6_56_MQVMM - _t Page 55 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 55 Questions and refer to the graph at right, which has a gradient of. (, ) The value of must e: A 5 B C D E The -intercept is: A (, ) B (, ) C (, -- ) D (, ) E ( --, ) If + 5 =, then the value of the discriminant is: A 76 B C D -- E 76 Questions and 5 refer to the function with the rule: = + 8 where ( 6, ). Which one of the following graphs could represent this function? A B C ( 6, ) (, ) ( 6, ) (, ) (, 5) D (, ) (6, ) D E E D ( 6, ) E (, ) ( 6, 6) (,.6) 5 The range of this function is: A ( 8, ) B (, ) C [ 8, ) D [ 8, ] E (, ) 6 The graph of = could e: A B C D E E F 7 Which of the following intercepts does the graph of f () = have? A ( --, ), (, ), (, ) and (, 6) B (, ), (, ), (, ) and (, 6) C ( --, ), (, ), (, ) and (, 6) D (, ), (, ), (, ) and (, 6) E ( --, ), (, ), (, ) and (, 6) F

56 5_6_56_MQVMM - _t Page 56 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 56 Maths Quest Mathematical Methods F 8 The rule for the graph shown at right could e: A f () = ( ) ( + ) B f () = ( + )( ) C f () = ( + ) ( ) D f () = ( )( + ) E f () = ( )( + ) f() G 9 The rule for the graph shown at right could e: A f () = ( + ) B f () = ( ) C f () = ( ) D f () = ( ) E f () = ( ) f() G The graph of = ( + ) ( )( ) is est represented : A B C D E A B C D E Short answer Epand each of the following: a ( ) If ( ) is a factor of P() = 7 + a , then find the values of a and. 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 (, ). The line which is perpendicular to the line with equation + = and passes through the point (, ). 5 Sketch the graph of = 8, laelling the turning point and all intercepts. State its domain and range.

57 5_6_56_MQVMM - _t Page 57 Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Chapter Graphs and polnomials 57 6 Sketch the graph of = + 8, [, ). State the range of this function. 7 a If ( + ) is a factor of f () = + + a 8 and g() = a + 75, then find the values of a and. Sketch the graph of f () laelling all intercepts. 8 Sketch the graph of = , [, ). State the range of this function. 9 Solve the equation = Sketch the graph of f () = Analsis 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, 8 cm aove the tale. a The stem of the glass is 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. c d State the domain and range of the function. If there is fruit juice in the ottom of the glass to a depth of cm, find the coordinates of the point where the ants first touch the juice. Round answers to the nearest whole numer. Using function notation, write the rule for the surface of the cross-section of the juice in the glass. E F F G G A rogue satellite has its distance from Earth, d thousand kilometres, modelled a cuic function of time, t das after launch. After da it reaches a maimum distance from Earth of kilometres, then after das it is kilometres awa. It effectivel returns to Earth after 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. The moon is approimatel kilometres from Earth. d e f Which is closer to Earth after 8 das, the satellite or the moon? B how far? The satellite is programmed to selfdestruct. This happens when it is 9 kilometres from Earth. What is the life span of the satellite? State the domain and range of d(t).