SAMPLE. Polynomial functions

Size: px
Start display at page:

Download "SAMPLE. Polynomial functions"

Transcription

1 Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through the use of transformations. To divide polnomials. To use the factor theorem to solve cubic equations and quartic equations. To use the remainder theorem. To draw and use sign diagrams. To find equations for given graphs of polnomials. To appl polnomial functions to problem solving. 4.1 Polnomials In an earlier chapter, linear functions were discussed. This famil of functions is a member of a larger famil of polnomial functions. A function with rule P() = a + a 1 + a + +a n n,where a, a 1, a, a 3,...,a n are real constants and n is a positive integer, is called a polnomial in over the real numbers. The numbers a, a 1, a, a 3,...,a n are called the coefficients of the polnomial. Assuming a n =, the term a n n is called the leading term. The integer n of the leading term is the degree of the polnomial. Foreample: f () = a + a 1 is a degree one polnomial (a linear function). f () = a + a 1 + a is a degree two polnomial (a quadratic function). f () = a + a 1 + a + a 3 3 is a degree three polnomial (a cubic function). f () = a + a 1 + a + a a 4 4 is a degree four polnomial (a quartic function). The polnomials above are written with ascending powers of. The can also be written with descending powers of, for eample: f () = 3 + 1, f () = + + 3, f () = Polnomials are often written in factorised form, e.g. (3 + ),4( 1) 3 +. Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard 18

2 Chapter 4 Polnomial functions 19 Although it is fairl simple to epand such polnomials when the degree is small, sa two or three, the binomial theorem discussed in Appendi A facilitates the epansion of polnomials of larger degree. Foreample: ( + 3) 4 = ( 4 ) () 4 (3) + ( 4 1 ) () 3 (3) 1 + ( 4 ) () (3) + ( 4 3 ) () 1 (3) 3 + ( 4 4 ) () (3) 4 Equating coefficients = If P() = a + a 1 + a + +a n n and Q() = b + b 1 + b + +b m m are equal, i.e. if P() = Q() for all, then the degree of P() = degree of Q() and a = b, a 1 = b 1, a = b,...,etc. Eample 1 If = a( + 3) + b for all R, find the values of a and b. Epanding the right-hand side of the equation gives: a( + 3) + b = a( ) + b = a + 6a + 9a + b If = a + 6a + 9a + b for all R, then b equating coefficients: (coefficient of ) (coefficient of ) (coefficient of, the constant term of the polnomial) Hence a = 1 and b = 5. Eample 1 = a 6 = 6a 4 = 9a + b a If = a( + 1) 3 + b for all R, find the values of a and b. b Show that cannot be written in the form a( + c) 3 + b for real numbers a, b and c. a The epansion of the right-hand side of the equation gives: a( + 1) 3 + b = a( ) + b = a 3 + 3a + 3a + a + b Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

3 13 Essential Mathematical Methods 3&4CAS b If = a 3 + 3a + 3a + a + b for all R, equating coefficients gives: (coefficient of 3 ) (coefficient of ) (coefficient of ) (coefficient of, the constant term of the polnomial) Hence a = 1 and b = 7. The epansion of the right-hand side of the equation gives: a( + c) 3 + b = a( 3 + 3c + 3c + c 3 ) + b = a 3 + 3ca + 3c a + c 3 a + b 1 = a 3 = 3a 3 = 3a 8 = a + b If = a 3 + 3ca + 3c a + c 3 a + b for all R: (coefficient of 3 ) 1 = a (1) (coefficient of ) 6 = 3ca () (coefficient of ) 6 = 3c a (3) (coefficient of, the constant term of the polnomial) 8 = c 3 a + b (4) From (1), a = 1 From (), c = For (3), the right-hand side 3c a equals 1, which is a contradiction. Division of polnomials The division of polnomials was introduced in Essential Mathematical Methods 1&CAS. The general result for polnomial division is: For non-zero polnomials, P() and D(), if P() (the dividend) is divided b D() (the divisor), then there are unique polnomials, Q() (the quotient) and R() (the remainder), such that P() = D()Q() + R() Either the degree of R() < D(), or R() = When R() =, then D() iscalled a divisor of P() and P() = D()Q() The following eample illustrates the process of dividing. Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

4 Chapter 4 Polnomial functions 131 Eample 3 Divide b. ) Thus = ( )( ) + 58 or, equivalentl, = In this eample isthe dividend, isthe divisor and the remainder is 58. Using the TI-Nspire Use propfrac (b 371)asshown. Using the Casio ClassPad Enter and highlight ( )/( ) and tap Interactive > Transformation > propfrac. Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

5 13 Essential Mathematical Methods 3&4CAS The remainder theorem and factor theorem The following two results are recalled from Essential Mathematical Methods 1&CAS. The remainder theorem ( ) b If a polnomial P()isdivided b a + b the remainder is P. a This is easil proved b observing if P() = D()(a + b) + R() and = b ( ) ( )( a b b P = D a b ) ( ) b a a a + b + R a and thus ( ) ( ) b b P = R a a The factor theorem An immediate consequence of the remainder theorem is the factor theorem. ( ) b a + b is a factor of the polnomial P() ifandonl if P =. a Eample 4 Find the remainder when P() = isdivided b + 1. B the remainder theorem the remainder is: ( P 1 ) ( = ( + ) 1 ( + ) 1 ) + 1 Eample 5 = = = 5 8 Given that + 1 and are factors of a 6 + b, find the values of a and b. Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

6 Let P() = a 6 + b. B the factor theorem, P( 1) = and P() =. Hence, a b = (1) and a 1 + b = () Rearranging gives: a + b = 13 (1 ) 4a + b = 76 ( ) Subtract (1 ) from ( ): 3a = 63 Therefore a = 1 and, from (1 ), b = 8. Solving polnomial equations The factor theorem ma be used in the solution of equations. Eample 6 Factorise P() = and hence solve the equation = P(1) = = P( 1) = = P() = = isafactor. Dividing b reveals that: P() = ( )( 15) = ( )( 5)( + 3) Therefore = or 5 = or + 3 = = or = 5 or or = 3 Using the TI-Nspire Use factor( ) (b 3) and solve( ) (b 31)asshown. Chapter 4 Polnomial functions 133 Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

7 134 Essential Mathematical Methods 3&4CAS Using the Casio ClassPad Enter and highlight then tap Interactive > Transformation > factor. Cop and paste the answer to the net entr line and tap Interactive > Equation/inequalit > solve and ensure the variable set is. Eercise 4A 1 Find the values of A and B such that A( + 3) + B( + ) = for all real numbers. Find the values of A, B and C in each of the following if: a = A( + B) + C for all R b = A( + B) + C for all R c = A( + B) 3 + C for all R 3 For each of the following, divide the first term b the second: a , 3 b , + 1 c , 4 a Find the remainder when isdivided b +. b Find the value of a for which (1 a) + 5a + (a 1)(a 8) is divisible b ( ) but not b ( 1). 5 Given that f () = : a Find the remainder when f () isdivided b. b Find the remainder when f () isdivided b +. c Factorise f () completel. 6 a Prove that the epression 3 + (k 1) + (k 9) 7isdivisible b + 1 for all values of k. b Find the value of k for which the epression has a remainder of 1 when divided b. 7 The function f () = 3 + a b + 3 has a factor ( + 3). When f () isdivided b ( ), the remainder is 15. a Calculate the values of a and b. b Find the other two factors of f (). Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

8 Chapter 4 Polnomial functions The epression a 5 + b leaves remainders of 8 and 1 when divided b ( 3) and ( 3) respectivel. Calculate the values of a and b. 9 Find the remainder when ( + 1) 4 is divided b 1 Let P() = a Show that neither ( 1) nor ( + 1) is a factor of P() b Given that P() can be written in the form ( 1)Q() + a + b where Q() isa polnomial and a and b are constants, hence or otherwise, find the remainder when P()isdivided b 1 11 a Show that both ( 3) and ( + 3) are factors of b Hence write down one quadratic factor of , and find a second quadratic factor. 1 Solve each of the following equations for : a ( )( + 4)( )( 3) = b 3 ( ) = c ( 1) 3 ( ) = d ( + ) 3 ( ) = e 4 4 = f 4 9 = g = h = i = 13 Find the -ais intercepts and the -ais intercepts of the graphs of each of the following: a = 3 b = c = d = e = 3 + f = g = h = i = The epressions p q and 4 3 p q 8have acommon factor. Find the values of p and q. 15 Find the remainder when f () = is divided b Factorise each of the following polnomials using a calculator to help find at least one linear factor: a b c d Factorise each of the following: a b Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

9 136 Essential Mathematical Methods 3&4CAS 18 Factorise each of the following polnomials, using a calculator to help find at least one linear factor: a b c d Quadratic functions The following is a summar of material assumed to have been covered in Essential Mathematical Methods1&CAS. The general epression of a quadratic function is = a + b + c, R. To sketch the graph of a quadratic function (called a parabola) use the following: If a >, the function has a minimum value. If a <, the function has a maimum value. The value of c gives the -ais intercept. b The equation of the ais of smmetr is = a The -ais intercepts are determined b solving the equation a + b + c = A quadratic equation ma be solved b: factorising e.g., = ( 3)( + 4) = = 3 or 4 completing the square e.g., + 4 = ( ) b Add and subtract to complete the square = ( + 1) 5 = ( + 1) = =± 5 = 1 ± 5 b ± b 4ac using the general quadratic formula = a e.g., = = ( 1) ± ( 1) 4( 3)( 7) ( 3) = 6 ± 15 3 Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

10 Chapter 4 Polnomial functions 137 Using the discriminant of the quadratic function f () = a + b + c If b 4ac >, the graph of the function has two -ais intercepts. If b 4ac =, the graph of the function touches the -ais. If b 4ac <, the graph of the function does not intersect the -ais. Eample 7 Sketch the graph of f () = 3 1 7busing the quadratic formula to calculate the -ais intercepts. Since c = 7 the -ais intercept is (, 7). Find the turning point coordinates. Ais of smmetr, = b a = ( 1) 3 = and f ( ) = 3( ) 1( ) 7 = 5 turning point coordinates are (, 5). Calculate the -ais intercepts = 6 = b ± b 4ac a 4 = ( 1) ± ( 1) 4( 3)( 7) ( 3) = 1 ± = 1 ± = = 6 ± ± 3.87 (to nd decimal place) 8 3 = 3.9 or.71 Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

11 138 Essential Mathematical Methods 3&4CAS Use of completing the square to sketch quadratics For a >, the graph of the function f () = a is obtained from the graph of f () = badilation of factor a from the -ais. The graphs on the right are those of =, = and = 1, i.e. a = 1, and 1. For h >, the graph of f () = ( + h) is obtained from the graph of f () = batranslation of h units in the negative direction of the -ais. For h <, the graph of f () = ( + h) is obtained from the graph of f () = batranslation of h units in the positive direction of the -ais. The graphs of = ( + ) and = ( ) are shown: For k >, the graph of f () = + k is obtained from the graph of f () = b atranslation of k units in the positive direction of the -ais. For k <, the translation is in the negative direction of the -ais. Foreample, the graph of the function f () = ( 3) + is obtained b translating the graph of the function f () = b 3units in the positive direction of the -ais and units in the positive direction of the -ais. Note that the verte is at (3, ). 1 = = 1 = (1, ) 1 (, 4) (1, 1) (1,.5) = ( + ) = ( ) = + = (, 11) = ( 3) + The graph of the function f () = ( ) + 3 is obtained from the graph of f () = b the following transformations: dilation of factor from the -ais translation of units in the positive direction of the -ais translation of 3 units in the positive direction of the -ais. Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard (, 11) (, 3) (3, )

12 Chapter 4 Polnomial functions 139 B completing the square, an quadratic function can be written in the form: = a( + h) + k Eample 8 Solve 4 5 = bepressing 4 5inthe form = a( + h) + k Use this to help ou sketch the graph of f () = 4 5 f () = ( 4 5 = 5 ) ( = ) ( ) (adding b and subtracting to complete [ = ( + 1) 7 ] the square ) [ = ( 1) 7 ] = ( 1) 7 Therefore, the graph of f () = 4 5ma be obtained from the graph of = b the following transformations: dilation of factor from the -ais translation of 1 unit in the positive direction of the -ais translation of 7 units in the negative direction of the -ais. The -ais intercepts can also be determined b this process. To solve = 4 5 = ( 1) 7 ( 1) = =± 7 = 1 ± 7 7 = 1 + or = 1.87 or.87 This information can now be used to sketch = 4 5 the graph. Since c = 5 the -ais intercept is (, 5) Turning point coordinates are (1, 7). 4 -ais intercepts are (.87, ) and 5 (.87, ) to two decimal places. 6 7 Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

13 14 Essential Mathematical Methods 3&4CAS Eercise 4B 1 Without sketching the graphs of the following functions, determine whether the cross, touch or do not intersect the -ais: a f () = 5 + b f () = c f () = d f () = 8 3 e f () = f f () = 1 Sketch the graphs of the following functions: a f () = ( 1) b f () = ( 1) c f () = ( 1) ( d f () = 4 ( + 1) e f () = f f () = ( + 1) ) 1 g f () = 3( ) 4 h f () = ( + 1) 1 i f () = 5 1 j f () = ( + 1) 4 3 Epress each of the following functions in the form = a( + h) + k and hence find the maimum or minimum value and the range in each case: a f () = + 3 b f () = c f () = d f () = e f () = 5 f f () = 7 3 g f () = Sketch the graphs of the following functions, clearl labelling the intercepts and turning points: a = + b = c = d = 5 6 e f () = + 3 f f () = g f () = h f () = i = j = a Which of the graphs shown could represent the graph of the equation = ( 4) 3? b Which graph could represent = 3 ( 4)? A B 13 C D Which of the curves shown could be defined b each of the following? a = 1 3 ( + 4)(8 ) b = + 1 c = 1 + ( 1) d = 1 (9 ) Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

14 Chapter 4 Polnomial functions 141 A B C D = Forwhich values of m does the equation m m + 3 = have: a two solutions for? b one solution for? 8 Show that the equation (k + 1) k = has a solution for all values of k. 9 Forwhich values of k does the equation k k = 5have: a two solutions for? b one solution for? 1 Show that the equation a (a + b) + b = has a solution for all values of a and b. 4.3 Determining the rule for a parabola Given sufficient information about a curve, a rule for the function of the graph ma be determined. Foreample, if the coordinates of three points on a parabola of the form = a + b + c are known, the rule for the parabola ma be found, i.e. the values of a, b and c ma be found. Sometimes a more specific rule is known. For eample, the curve ma be a dilation of =. It is then known to be of the = a famil, and the coordinates of one point (with the eception of the origin) will be enough to determine the value for a. In the following it is assumed that each of the graphs is that of a parabola, and each rule is that of a quadratic function in. 1 This is of the form = a (since the graph has its verte at the origin). When =, = 5 5 = a() a = 5 4 The rule is = 5 4 (, 5) Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

15 14 Essential Mathematical Methods 3&4CAS This is of the form = a + c (since the graph is smmetric about the -ais). For (, 3) 3 = a() + c implies c = For ( 3, 1) 1 = a( 3) = 9a + 3 a = 9 the rule is = This is of the form = a( 3) As the point ( 1, 8) is on the parabola: 8 = a( 1 3) And hence 4a = 8 Therefore a = The rule is = ( 3) This is of the form = a + b + c As the point with coordinates ( 1, ) is on the parabola: = a b + c (1) The -ais intercept is and therefore c = () As the point with coordinates (1, ) is on the parabola: = a + b + c (3) Substitute c = in(1) and (3) = a b + From (1) = a b (1a) From (3) = a + b (3a) Subtract (3a) from (1a) = b b = 1 Substitute b = 1 and c = in(1). ( 1, 8) 1 ( 3, 1) (, 3) 3 1 = a 1 + = a + 1 a = 1 the quadratic rule is = + + Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

16 Chapter 4 Polnomial functions 143 Eercise 4C 1 Determine the equation of each of the following parabolas: a c e g ( 1, ) (3, ) (1, ) (1, 3) b d f h ( 1, 3) (, ) 4 Find quadratic epressions representing the two curves shown in the diagram, given that the coefficient of in each case is 1. A is (4, 3), B is (4, 1), C is (, 5) and D is (, 1). 4 A 3 1 D B 4 5 C 5 3 The graph of the quadratic function f () = A( + b) + B has a verte at (, 4) and passes through the point (, 8). Find the values of A, b and B. Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

17 144 Essential Mathematical Methods 3&4CAS 4.4 Functions of the form f: R R, f() = a( + h) n + k, where n is a natural number In the previous section it was shown that ever quadratic polnomial can be written in the form a( + h) + k. This is not true for polnomials of higher degree. This was shown in Eample of this chapter. However, there are man polnomials that can be written in this form. In Chapter 3 the famil of power functions was introduced. In this section the sub-famil of power functions with rules of the form f () = n where n is a natural number are considered. f() = n where n is an odd positive integer The diagrams below shows the graphs of = 3 and = 5 ( 3, 7) = 3 (3, 7) ( 3, 43) = 5 (3, 43) The diagram on the right shows both functions graphed on the one set of aes for a smaller domain. The following properties can be observed for a = 3 (1, 1) function f () = n where n is an odd integer: f () = = 5 f (1) = 1 and f ( 1) = 1 ( 1, 1) f () = f ( ), i.e. f is an odd function. f ( ) = ( ) n = ( 1) n () n = f () asn is odd. As, f () As, f () From Essential Mathematical Methods 1&CASou will recall that the gradient function of f () = n has rule n n 1. Hence the gradient is when =. As n 1isevenforn odd, n n 1 > for all non-zero. That is, the gradient of the graph of = f () ispositive for all non-zero and zero when =. Recall that the stationar point at =, for functions of this form, is called a stationar point of infleion. Comparing the graphs of = n and = m where n and m are odd and n > m n < m when < < 1 ( n m = m ( n m 1) < ) n = m when = n > m when 1 < < ( n m = m ( n m 1) > asn m is even, m is odd and is negative and greater than 1) Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

18 n = m for = 1 and n = m for = 1 n > m for > 1 n < m for < 1 It should be noted that the appearance of graphs is dependent on the scales on the - and -aes. Odd power functions are often depicted as shown. Chapter 4 Polnomial functions 145 Transformations of graphs of functions with rule f() = n where n is an odd positive integer Transformations of these graphs result in graphs with rules of the form = a( + h) n + k where a, h and k are real constants. Eample 9 Sketch the graph of f () = ( ) This can be done b noting that a translation (, ) ( +, + 1) maps the graph of = 3 to = ( ) Note: The point with coordinates (, 1) is a point of zero gradient. For the aes intercepts, consider this: When =, = ( ) = 7 When =, = ( ) = ( ) 3 1 = and hence = 1 The reflection in the -ais described b the transformation rule (, ) (, ) and applied to = 3 results in the graph of = 3 (, 1) (1, ) (, 7) = f() Eample 1 = 3 Sketch the graph of = ( 1) 3 + Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

19 146 Essential Mathematical Methods 3&4CAS A reflection in the -ais followed b a translation with rule (, ) ( + 1, + ) applied to the graph of = 3 results in the graph = ( 1) 3 + Note: (1, ) is a point of zero gradient. For the aes intercepts consider this: When =, = ( 1) 3 + = 3 When =, = ( 1) 3 (, 3) + (1, ) ( 1) 3 = 1 = 1 (.6, ) 3 and hence = Eample 11 Sketch the graph of = ( + 1) 3 + A dilation of factor from the -ais followed b the translation with rule (, ) ( 1, + ) maps the graph of = 3 to the graph of = ( + 1) 3 + Note: ( 1, ) is a point of zero gradient. For the aes intercepts consider this: When =, = + = 4 When =, = ( + 1) = ( + 1) 3 1 = + 1 and hence = ( 1, ) (, ) Eample 1 The graph of = a( + h) 3 + k has a point of zero gradient at (1, 1) and passes through the point (, 4). Find the values of a, h and k. h = 1 and k = 1 Therefore as the graph passes through (, 4): 4 = a + 1 a = 3 (, 4) Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

20 Chapter 4 Polnomial functions 147 a b Eample 13 Find the rule for the image of the graph of = 5 under the following sequence of transformations: reflection in the -ais dilation of factor from the -ais translation of units in the positive direction of the -ais and 3 units in the positive direction of the -ais Find a sequence of transformations which takes the graph of = 5 to the graph of = 6 ( + 5) 5 a (, ) (, ) (, ) ( +, + 3) Let (, )bethe image of (, ) under this transformation. Hence = + and = + 3 Thus =, which implies = and = 3 ( ) Therefore, the graph of = 5 maps to the graph of 5 3 =, i.e., to the graph of = 1 3 ( )5 + 3 b Rearrange = 6 ( + 5) 5 to 6 = ( + 5) 5 Therefore, = 6 and = + 5 and = + 6 and = 5 The sequence of transformations is: reflection in the -ais dilation of factor from the -ais translation of 5 units in the negative direction of the -ais and 6 units in the positive direction of the -ais. f() = n where n is an even positive integer The graphs of = and = 4 are shown on the one set of aes. ( 1, 1) = 4 (1, 1) = The following properties can be observed for a function f () = n where n is an even integer: Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

21 148 Essential Mathematical Methods 3&4CAS f () = f (1) = 1 and f ( 1) = 1 f () = f ( ), i.e. f is an even function, f ( ) = ( ) n = ( 1) n () n = f () asn is even As, f () As, f () From Essential Mathematical Methods 1&CASou will recall that the gradient function of f () = n has rule n n 1. Hence the gradient is when =. As n 1isodd for n even, n n 1 > for > and n n 1 < for <. That is, the gradient of the graph of = f () is positive for positive, negative for negative and zero when =. Comparing the graphs of = n and = m where n and m are even and n > m n < m when < < 1 ( n m = m ( n m 1) < ) n = m when = n < m when 1 < < ( n m = m ( n m 1) < asn m is even, m is even and is negative and greater than 1) n = m for = 1 and n = m for = 1 n > m for > 1 n > m for < 1 It should be noted that the appearance of graphs is dependent on the scales on the - and -aes. Power functions of even degree are often depicted as shown. Eample 14 The graph of = a( + h) 4 + k has a turning point at (, ) and passes through the point (, 4). Find the values of a and h and k. h = and k = Therefore as the graph passes through (, 4): 4 = 16a + a = 1 8 Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

22 Chapter 4 Polnomial functions 149 Eercise 4D 1 Sketch the graphs of each of the following. State the coordinates of the point of zero gradient and the aes intercepts. a f () = 3 b g() = 3 c h() = d f () = 3 4 e f () = ( + 1) 3 8 f f () = ( 1) 3 g g() = ( 1) 3 + h h() = 3( ) 3 4 i f () = ( 1) 3 + j h() = ( 1) 3 4 k f () = ( + 1) 5 3 l f () = ( 1) 5 The graph of = a( + h) 3 + k has a point of zero gradient at (, 4) and passes through the point (1, 1). Find the values of a, h and k. 3 The graph of = a( + h) 4 + k has a turning point at (1, 7) and passes through the point (, 3). Find the values of a, h and k. 4 Find the equation of the image of = 3 under each of the following transformations: a a dilation of factor 3 from the -ais b a translation with rule (, ) ( 1, + 1) c a reflection in the -ais followed b a translation with rule d e (, ) ( +, 3) a dilation of factor from the -ais followed b a translation with rule (, ) ( 1, ) a dilation of factor 3 from the -ais. 5 B appling suitable transformations to = 4,sketch the graph of each of the following: a = 3( 1) 4 b = ( + ) 4 c = ( ) 4 6 d = ( 3) 4 1 e = 1 ( + 4) 4 f = 3( ) The general cubic function Not all cubic functions are of the form f () = a( + h) 3 + k. In this section the general cubic function is considered. The form of a general cubic function is: f () = a 3 + b + c + d It is impossible to full investigate cubic functions without the use of calculus. Cubic functions will be revisited in Chapter 9. The shapes of cubic graphs var. Below is a galler of cubic graphs demonstrating the variet of shapes that are possible. Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

23 15 Essential Mathematical Methods 3&4CAS 1 f () = 3 + Note: (, ) is not a point of zero gradient. There is one root,. f () = 3 Note: The turning points do not occur smmetricall between consecutive -ais intercepts as the do for quadratics. Differential calculus must be used to determine them. There are three roots: 1, and 1. 3 f () = 3 3 Note: There are two roots: 1 and f () = The graphs of f () = 3 and f () = are shown. The are the reflection in the -ais of the graphs of 1 and 4 respectivel. = Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

24 Chapter 4 Polnomial functions 151 Sign diagrams A sign diagram is a number line diagram that shows when an epression is positive or negative. For a cubic function with rule f () = ( )( )( ) when > >, the sign diagram is as shown: Eample 15 + α β γ Draw a sign diagram for the epression From Eample 6, f () = ( )( 5)( + 3) We note: f () > for > 5 f () < for < < 5 f () > for 3 < < f () < for < 3 Also, f () = f (5) = f ( 3) = Hence the sign diagram ma be drawn. Eample For the cubic function with rule f () = : a Sketch the graph of = f () using a calculator to find the values of the coordinates of the turning points, correct to two decimal places. b Sketch the graph of = 1 f ( 1) a f () = = (3 )( )( + 5) = ( + 5)( )( 3) We note: f () < for > 3 f () > for < < 3 f () < for 5 < < f () > for < 5 Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

25 15 Essential Mathematical Methods 3&4CAS b Also, f ( 5) = f () = f (3) = Hence the sign diagram ma be drawn The -ais intercepts are at = 5, = and = 3 and the -ais intercept is at = 3. The window has to be adjusted carefull to see the -ais intercepts and the turning points. The local minimum turning point ma be found b selecting 3:Minimum from the F5 menu and then entering the lower bound 5 and the upper bound and pressing ENTER. The local minimum is found to occur at the point with coordinates (.5, 61.88) with values given correct to two decimal places. The local maimum turning point ma be found b selecting 4:Maimum from the F5 menu and then entering the lower bound and the upper bound 3 and pressing ENTER. The local maimum is found to occur at the point with coordinates (.5, 1.88) with values given correct to two decimal places. The rule for the transformation is (, ) ( + 1, 1 ) B the transformation ( 5, ) ( 4, ), (, ) (3, ), (3, ) (4, ), (, 3) (1, 15), (.5, 1.88) (3.5,.94) and (.5, 61.88) ( 1.5, 3.94) 5 3 (.5, 61.88) (.5, 1.88) (3.51,.94) ( 1.5, 3.94) 3 Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

26 Chapter 4 Polnomial functions 153 Eercise 4E 1 Draw sign diagrams for each of the following epressions: a (3 )( 1)( 6) b (3 + )( 1)( + 6) c ( 5)( + 1)( 6) d (4 )(5 )(1 ) a Use a calculator to plot the graph of = f () where f () = b On the same screen plot the graphs of: i = f ( ) ii = f ( + ) iii = 3 f () 4.6 Polnomials of higher degree The techniques that have been developed for cubic functions ma now be applied to quartic functions and in general to functions of higher degree. It is clear that a polnomial P() of degree n has at most n solutions for the equation P() =. It is possible for the graphs of polnomials of even degree to have no -ais intercepts, for eample P() = + 1, but graphs of polnomials of odd degree have at least one -ais intercept. The general form for a quartic function is: f () = a 4 + b 3 + c + d + e Agaller of quartic functions produced with a graphing package is shown below. 1 f () = 4 f () = f () = f () = ( 1) ( + ) Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

27 154 Essential Mathematical Methods 3&4CAS 5 f () = ( 1) 3 ( + ) Eample 17 Draw a sign diagram for the quartic epressions: a ( )( + )( 3)( 5) b 4 + a b Let P() = 4 + P(1) = = 1isafactor. ) P() = ( 1)( ) = ( 1)[ ( + 1) + ( + 1)] = ( 1)( + 1)( + ) Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

28 Chapter 4 Polnomial functions 155 Eample 18 Find the coordinates of the points where the graph of = p(), p() = 4 + 1, crosses the - and -aes, and hence sketch the graph. 1isafactor. p(1) = = p() = ( 1)( 3 + 1) = ( 1)[ ( + 1) ( + 1)] = ( 1)( + 1)( 1) = ( 1) ( + 1) Alternativel, note that: p() = ( 1) = [( 1)( + 1)] = ( 1) ( + 1) Therefore, the -ais intercepts are (1, ) and ( 1, ). When =, = 1. Eercise 4F (, 1) ( 1, ) (1, ) 1 a Use a calculator to plot the graph of = f () where f () = b On the same screen plot the graphs of: ( ) i = f ( ) ii = f () iii = f The graph of = 9 4 is as shown. Sketch the graph of each of the following b appling suitable transformations. a = 9( 1) ( 1) 4 b = 18 4 c = 18( + 1) ( + 1) 4 d = e = (Do not find the -ais intercepts for e.) 9 81, , 4 3 Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

29 156 Essential Mathematical Methods 3&4CAS 4.7 Determining rules for the graphs of polnomials It is first worth noting that the graph of a polnomial function of degree n is completel determined b an n + 1 points on the curve. Foreample, for a cubic function with rule = f (), if it is known that f (a 1 ) = b 1, f (a ) = b, f (a 3 ) = b 3, f (a 4 ) = b 4, then the rule can be determined. Finding the rule for a parabola has been discussed in Section 4.3 of this chapter. Eample 19 For the cubic function with rule f () = a 3 + b + c + d, itisknown that the points with coordinates ( 1, 18), (, 5), (1, 4) and (, 9) lie on the graph of the cubic. Find the values of a, b, c and d. The following equations can be formed. a + b c + d = 18 (1) d = 5 () a + b + c + d = 4 (3) 8a + 4b + c + d = 9 (4) Adding (1) and (3) gives b + d = and as d = 5, b = 6. There are now onl two unknowns. (3) becomes a + c = 7 (3 ) and (4) becomes 8a + c = (4 ) Multipl (3 )band subtract from (4 ). 6a = 6which gives a = 1 and c = 6 Using the TI-Nspire Enter solve( a + b c + d = 18 and d = 5 and a + b + c + d = 4 and 8a + 4b + c + d = 9, {a, b, c, d}). Alternativel, Define f () = a 3 + b + c + d and then solve( f ( 1) = 18 and f () = 5 and f (1) = 4 and f () = 9, {a, b, c, d}). Both methods are shown on the screen to the right. Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

30 Chapter 4 Polnomial functions 157 Using the Casio ClassPad Press and select menu (if necessar tap ). Tap three times to produce the template to enter four simultaneous equations. Enter the equations and set the variables in the variable bo to a,b,c,d then tap. An alternative method is to define f () = a 3 + b + c + d then enter f ( 1) = 18, f () = 5, f (1) = 4 and f () = 9asthe simultaneous equations to be solved with variables a,b,c,d as above. The following eamples provide further procedures for finding the rules of cubic functions. It should be noted that a similar facilit is available for quartics. Eample The graph shown is that of a cubic function. Find the rule for this cubic function. From the graph the function can be seen to be of the form = a( 4) ( 1) ( + 3). 3 (, 4) It remains to find the value of a. The point with coordinates (, 4) is on the graph. Hence: 4 = a( 4)( 1)3 a = 1 3 and = 1 ( 4)( 1)( + 3) Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

31 158 Essential Mathematical Methods 3&4CAS Eample 1 The graph shown is that of a cubic function. Find the rule for this cubic function. From the graph it can be seen to be of the form = k( 1)( + 3).Itremains to find the value of k.asthe value (, 9) is on the graph: 9 = k( 1)(9) k = 1 and = ( 1)( + 3) (, 9) 3 1 Note: Because the graph reveals that there are two factors, ( + 3) and ( 1), there must be a third linear factor (a + b), but this implies that there is an intercept b a. Thus a + b = k 1 ( + 3) or k ( 1). A consideration of the signs reveals that ( + 3) is a repeated factor. Eample The graph of a cubic function passes through the point with coordinates (, 1), (1, 4), (, 17) and ( 1, ). Find the rule for this cubic function. The cubic function will have a rule of the form: = a 3 + b + c + d The values of a, b, c and d have to be determined. As the point (, 1) is on the graph, d = 1. B using the points (1, 4), (, 17) and ( 1, ), three simultaneous equations are produced: 4 = a + b + c = 8a + 4b + c + 1 = a + b c + 1 These become: 3 = a + b + c (1) 16 = 8a + 4b + c () 1 = a + b c (3) Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

32 Chapter 4 Polnomial functions 159 Add (1) and (3) to find: b = 4 i.e., b = Substitute in (1) and (): 1 = a + c (4) 8 = 8a + c (5) Multipl (4) b and subtract from (5). Hence 6 = 6a and a = 1 From (4) c = Thus = Eercise 4G 1 Determine the rule for the cubic function with graph as shown: Determine the rule for the cubic function with graph as shown: 5 (, 11) (, 5) ( 1, ) 3 Find the rule for the cubic function that passes through the following points: a (, 1), (1, 3), ( 1, 1) and (, 11) b (, 1), (1, 1), ( 1, 1) and (, 7) c (, ), (1, ), ( 1, 6) and (, 1) 6 3 Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

33 16 Essential Mathematical Methods 3&4CAS 4 Find epressions which define the following cubic curves: a b d e 1 (, 3) (, 3) (1,.75) 5 Find the equation of the cubic function for which the graph passes through the points with coordinates: a (, 135), (1, 156), (, 115), (3, ) b (, 3), (, 13), (1, 5), (, 11) 6 Find the equation of the quartic function for which the graph passes through the points with coordinates: a ( 1, 43), (, 4), (, 7), (6, 1618), (1, 67) b ( 3, 119), (, 3), ( 1, 9), (, 8), (1, 11) c ( 3, 6), ( 1, ), (1, ), (3, 66), (6, 17) 4.8 of literal equations and sstems of equations Literal equations Solving literal linear equations and simultaneous equations was undertaken in Section.. In this section other non-linear epressions are considered. The certainl can be solved with a CAS calculator but full setting out is shown here. Eample 3 Solve each of the following literal equations for : a + k + k = b 3 3a + a = c ( a) = 3 c 1 1 Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

34 Chapter 4 Polnomial functions 161 a Completing the square gives + k + k 4 + k k ( 4 = + k ) = k 4 k + k =± k 4k 4 = k ± k 4k A real solution eists onl for k 4k ; that is for k 4ork b 3 3a + a = ( 3a + a ) = ( a)( a) = = or = a or = a c ( a) = implies = or = a or = a In the following, the propert that for suitable values of a, b and an odd natural number n, b n = a is equivalent to b = a 1 n and also b = n a. Also b p q q = a is equivalent to b = a p suitable values of a and b and integers q and p. Care must be taken with the application of these. For eample = isequivalent to =± and 3 = 4isequivalent to = 8or = 8. Eample 4 Solve each of the following equations for : a a 3 b = c b 3 5 = a c 1 n = a where n is a natural number and is a positive real number d a ( + b) 3 = c e a 1 5 = b f 5 c = d a a 3 b = c a 3 = b + c 3 = b + c a b + c = 3 a ( ) 1 b + c 3 or = a b 3 5 = a is equivalent to = a 5 3 c 1 n = a is equivalent to = a n d a ( + b) 3 = c for ( + b) 3 = c ( a c ) b = a ( c ) 1 3 = b a Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

35 16 Essential Mathematical Methods 3&4CAS e a 1 5 = b f 1 5 = b ( a b = a ) 5 5 c = d 5 = c + d = (c + d) 1 5 Simultaneous equations In this section, methods for finding the coordinates of the points of intersection of different graphs are discussed. Eample 5 Find the coordinates of the points of intersection of the parabola with equation = with the straight line with equation = + 4 Consider + 4 = Then = 3 6 = 3 ± = 3 ± 33 The points of intersection have coordinates ( 3 33 A, 11 ) ( and B, 11 + ) 33 Using the TI-Nspire Use solve( ) from the Algebra menu (b 31)asshown. The word and can be tped directl or found in the catalog (b 31). Use the up arrow ( )tomoveuptothe answer and use the right arrow ( )to displa the remaining part of the answer. 4 A 4 = + 4 B = Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

36 Chapter 4 Polnomial functions 163 Using the Casio ClassPad In the simultaneous equation entr screen, enter the equations = and = + 4 and set the variables as,. Note the at the end of the answer line indicates that ou must scroll to the right to see all the solutions. Eample 6 Find the coordinates of the points of intersection of the circle with equation ( 4) + = 16 and the line with equation = Rearrange = tomake the subject. Substitute = into the equation of the circle. i.e., ( 4) + = = 16 i.e., 8 = ( 4) = = or = 4 The points of intersection are (, ) and (4, 4). Eample 7 Find the point of contact of the line with equation = and the curve with equation 3 = 1 Rewrite the equations as = and = 1 Consider = = 9 and + 6 = 9 Therefore = and ( 3) = i.e. = 3 The point of intersection is ( 3, 1 3 ). = (4, ) = (4, 4) 1 Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

37 164 Essential Mathematical Methods 3&4CAS Using the TI-Nspire Use solve( ) from the Algebra menu (b 3 1)asshown. Note that the multiplication sign is required between and. Using the Casio ClassPad In the simultaneous equation entr screen, enter the equations 9 + = 3 and = 1 and set the variables as,. Note that for this eample, since the fraction entr is on the same screen as the simultaneous equation in the keboard menu, the fractions can be entered using the fraction entr ke. Eercise 4H 1 Solve each of the following literal equations for : a k + + k = b 3 7a + 1a = c ( 3 a) = d k + k = e 3 a = f 4 a 4 = g ( a) 5 ( b) = h (a ) 4 (a 3 )( a) = Solve each of the following equations for : a a 3 b = c b 3 7 = a c 1 n + c = a where n is a natural number and is a positive real number d a ( + b) 3 = c e a 1 3 = b f 3 c = d 3 Find the coordinates of the points of intersection for each of the following: a = = b = = c = = + 1 Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

38 Chapter 4 Polnomial functions Find the coordinates of the points of intersection for each of the following: a + = = 16 d + = 97 + = 13 b + = 15 + = 15 e + = 16 = 4 c + = 185 = 3 5 Find the coordinates of the points of intersection for each of the following: a + = 8 = 187 b + = 51 = 518 c = 5 = 16 6 Find the coordinates of the points of intersection of the straight line with equation = and the circle with equation ( 5) + = 5 7 Find the coordinates of the points of intersection of the curves with equation = and = 8 Find the coordinates of the points of intersection of the line with equation 4 5 = 1 and the circle with equation = 1 9 Find the coordinates of the points of intersection of the curve with equation = 1 3 and the line with equation = + 1 Find the coordinates of the point where the line 4 = touches the parabola with equation = 9 11 Find the coordinates of the point where the line with equation = touches the circle + = 9 1 Find the coordinates of the point where the straight line with equation = touches the curve with equation = 1 13 Find the coordinates of the points of intersection of the curve with equation = and the line = 1 14 Solve the simultaneous equations: a 5 4 = 7 and = 6 b + 3 = 37 and = 45 c 5 3 = 18 and = 4 Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

39 166 Essential Mathematical Methods 3&4CAS 15 What is the condition for + a + b to be divisible b + c? 16 Solve the simultaneous equations = + and = a Solve the simultaneous equations = m and = for in terms of m. b Find the value of m for which the graphs of = m and = touch and give the coordinates of this point. c Forwhich values of m do the graphs not meet? Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

40 Chapter 4 Polnomial functions 167 Chapter summar A function P() = a + a 1 + +a n n,where a, a 1,...,a n are constants, is called a polnomial in. The numbers a, a 1,...a n are the coefficients. Assuming a n =,the term a n n is the leading term. The integer n is the degree of the polnomial. Degree one polnomials are called linear functions. Degree two polnomials are called quadratic functions. Degree three polnomials are called cubic functions. Degree four polnomials are called quartic functions. The ( remainder theorem:ifthe polnomial P() isdivided b a + b, the remainder is P b ) a ( The factor theorem:(a + b)isafactor of P() ifandonl if P b ) = a To sketch the graph of a quadratic function f () = a + b + c (called a parabola), use the following: If a >, the function has a minimum value. If a <, the function has a maimum value. The value of c gives the -ais intercept. b The equation of the ais of smmetr is = a The -ais intercepts are determined b solving the equation a + b + c = A quadratic equation ma be solved b: factorising completing the square b ± b 4ac using the general quadratic formula = a Using the discriminant of the quadratic function f () = a + b + c: If b 4ac >, the graph of the function has two -ais intercepts. If b 4ac =, the graph of the function touches the -ais. If b 4ac <, the graph of the function does not intercept the -ais. B completing the square, an quadratic function can be written in the form = A( + b) + B From this it can be seen that the graph of an quadratic function ma be obtained b a composition of transformations applied to the graph of =. Multiple-choice questions 1 The equation 5 1 inturning point form a( + h) + k b completing the square is: A (5 + 1) + 5 B (5 1) 5 C 5( 1) 5 D 5( + 1) E 5( 1) 7 Review Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

41 168 Essential Mathematical Methods 3&4CAS Review The value(s) of m that will give the equation m = two real roots is/are: A m = 3 B m = 3 C m = D m > 3 E m < factorised over R is equal to: A ( + 3) 3 B ( 3) 3 C ( + 3)( 6 + 9) D ( 3)( ) E ( + 3)( 3 + 9) 4 The equation of the graph shown is: A = ( )( + 4) B = ( + )( 4) C = ( + ) ( 4) D = ( + )( 4) E = ( + ) ( 4) 5 1isafactor of a + 1. The value of a is: A B 5 C 5 D 5 E is equal to: A (3 + )( 4) B (3 )(6 + 4) C (6 4)( + ) D (3 )( + 4) E (6 + )( 8) 7 A part of the graph of the third-degree polnomial function f (), near the point (1, ), is shown below. 1 Which of the following could be the rule for f ()? A f () = ( 1) B f () = ( 1) 3 C f () = ( 1) D f () = ( 1) E f () = ( + 1) 8 The coordinates of the turning point of the graph of the function p() = 3(( ) + 4) are: A (, 1) B (, 4) C (, 1) D (, 4) E (, 1) 9 The diagram below shows part of the graph of a polnomial function. c b Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

42 Chapter 4 Polnomial functions 169 A possible equation for the rule of the function is: A = ( + c)( b) B = ( b)( c) C = ( c)(b ) D = ( c)(b ) E = ( + b) ( c) 1 The number of roots to the equation ( + a)( b)( + c) =, where a, b and c R +, is: A B 1 C D 3 E 4 Short-answer questions (technolog-free) 1 Sketch the graphs of each of the following quadratic functions. Clearl indicate coordinates of the verte and aes intercepts. a h() = 3( 1) + b h() = ( 1) 9 c f () = + 6 d f () = 6 e f () = + 5 f h() = 1 The points with coordinates (1, 1) and (, 5) lie on a parabola with equation of the form = a + b.find the values of a and b. 3 Solve the equation 3 1 = busing the quadratic formula. 4 Sketch the graphs of each of the following. State the coordinates of the point of zero gradient and the aes intercepts: a f () = ( 1) 3 16 b g() = ( + 1) c h() = ( + ) 3 1 d f () = ( + 3) 3 1 e f () = 1 ( 1) 3 5 Draw a sign diagram for each of the following: a = ( + )( )( + 1) b = ( 3)( + 1)( 1) c = d = Without actuall dividing, find the remainder when the first polnomial is divided b the second: a , + 1 b , c , + 7 Determine the rule for the cubic function shown in the graphs: 3 (, 4) 7 Review Cambridge Universit Press Uncorrected Sample Pages 8 Evans, Lipson, Jones, Aver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

Higher. Polynomials and Quadratics 64

Higher. Polynomials and Quadratics 64 hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important

More information

Core Maths C2. Revision Notes

Core Maths C2. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

More information

The Quadratic Function

The Quadratic Function 0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral

More information

Graphing Quadratic Functions

Graphing Quadratic Functions A. THE STANDARD PARABOLA Graphing Quadratic Functions The graph of a quadratic function is called a parabola. The most basic graph is of the function =, as shown in Figure, and it is to this graph which

More information

13 Graphs, Equations and Inequalities

13 Graphs, Equations and Inequalities 13 Graphs, Equations and Inequalities 13.1 Linear Inequalities In this section we look at how to solve linear inequalities and illustrate their solutions using a number line. When using a number line,

More information

Quadratic Functions. MathsStart. Topic 3

Quadratic Functions. MathsStart. Topic 3 MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 3 Quadratic Functions 8 = 3 2 6 8 ( 2)( 4) ( 3) 2 1 2 4 0 (3, 1) MATHS LEARNING CENTRE

More information

Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will

More information

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1 Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

More information

C3: Functions. Learning objectives

C3: Functions. Learning objectives CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the

More information

3.1 Quadratic Functions

3.1 Quadratic Functions Section 3.1 Quadratic Functions 1 3.1 Quadratic Functions Functions Let s quickl review again the definition of a function. Definition 1 A relation is a function if and onl if each object in its domain

More information

8.7 Systems of Non-Linear Equations and Inequalities

8.7 Systems of Non-Linear Equations and Inequalities 8.7 Sstems of Non-Linear Equations and Inequalities 67 8.7 Sstems of Non-Linear Equations and Inequalities In this section, we stud sstems of non-linear equations and inequalities. Unlike the sstems of

More information

4 Non-Linear relationships

4 Non-Linear relationships NUMBER AND ALGEBRA Non-Linear relationships A Solving quadratic equations B Plotting quadratic relationships C Parabolas and transformations D Sketching parabolas using transformations E Sketching parabolas

More information

Functions and Graphs

Functions and Graphs PSf Functions and Graphs Paper 1 Section B 1. The points A and B have coordinates (a, a 2 ) and (2b, 4b 2 ) respectivel. Determine the gradient of AB in its simplest form. 2 2. hsn.uk.net Page 1 Questions

More information

If (a)(b) 5 0, then a 5 0 or b 5 0.

If (a)(b) 5 0, then a 5 0 or b 5 0. chapter Algebra Ke words substitution discriminant completing the square real and distinct imaginar rational verte parabola maimum minimum surd irrational rationalising the denominator Section. Quadratic

More information

Polynomials Past Papers Unit 2 Outcome 1

Polynomials Past Papers Unit 2 Outcome 1 PSf Polnomials Past Papers Unit 2 utcome 1 Multiple Choice Questions Each correct answer in this section is worth two marks. 1. Given p() = 2 + 6, which of the following are true? I. ( + 3) is a factor

More information

y intercept Gradient Facts Lines that have the same gradient are PARALLEL

y intercept Gradient Facts Lines that have the same gradient are PARALLEL CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or

More information

When I was 3.1 POLYNOMIAL FUNCTIONS

When I was 3.1 POLYNOMIAL FUNCTIONS 146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we

More information

11.7 MATHEMATICAL MODELING WITH QUADRATIC FUNCTIONS. Objectives. Maximum Minimum Problems

11.7 MATHEMATICAL MODELING WITH QUADRATIC FUNCTIONS. Objectives. Maximum Minimum Problems a b Objectives Solve maimum minimum problems involving quadratic functions. Fit a quadratic function to a set of data to form a mathematical model, and solve related applied problems. 11.7 MATHEMATICAL

More information

2 Analysis of Graphs of

2 Analysis of Graphs of ch.pgs1-16 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,

More information

2.3 Quadratic Functions

2.3 Quadratic Functions . Quadratic Functions 9. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the

More information

Graphing Quadratic Equations

Graphing Quadratic Equations .4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

More information

Quadratic Functions and Parabolas

Quadratic Functions and Parabolas MATH 11 Quadratic Functions and Parabolas A quadratic function has the form Dr. Neal, Fall 2008 f () = a 2 + b + c where a 0. The graph of the function is a parabola that opens upward if a > 0, and opens

More information

Polynomial Degree and Finite Differences

Polynomial Degree and Finite Differences CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

More information

Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.

Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4. _.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic

More information

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its

More information

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

More information

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

More information

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES 6 LESSON CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES Learning Outcome : Functions and Algebra Assessment Standard 1..7 (a) In this section: The limit concept and solving for limits

More information

Graphing and transforming functions

Graphing and transforming functions Chapter 5 Graphing and transforming functions Contents: A B C D Families of functions Transformations of graphs Simple rational functions Further graphical transformations Review set 5A Review set 5B 6

More information

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Chapter 5 Polnomial and Rational Functions Section 5.1 Polnomial Functions Section summaries The general form of a polnomial function is f() = a n n + a n 1 n 1 + +a 1 + a 0. The degree of f() is the largest

More information

Coordinate Geometry. Positive gradients: Negative gradients:

Coordinate Geometry. Positive gradients: Negative gradients: 8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight

More information

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the

More information

An Insight into Division Algorithm, Remainder and Factor Theorem

An Insight into Division Algorithm, Remainder and Factor Theorem An Insight into Division Algorithm, Remainder and Factor Theorem Division Algorithm Recall division of a positive integer by another positive integer For eample, 78 7, we get and remainder We confine the

More information

HI-RES STILL TO BE SUPPLIED

HI-RES STILL TO BE SUPPLIED 1 MRE GRAPHS AND EQUATINS HI-RES STILL T BE SUPPLIED Different-shaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object

More information

C1: Coordinate geometry of straight lines

C1: Coordinate geometry of straight lines B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

More information

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,

More information

STRAND F: ALGEBRA. UNIT F4 Solving Quadratic Equations: Text * * * Contents. Section. F4.1 Factorisation. F4.2 Using the Formula

STRAND F: ALGEBRA. UNIT F4 Solving Quadratic Equations: Text * * * Contents. Section. F4.1 Factorisation. F4.2 Using the Formula UNIT F4 Solving Quadratic Equations: Tet STRAND F: ALGEBRA Unit F4 Solving Quadratic Equations Tet Contents * * * Section F4. Factorisation F4. Using the Formula F4. Completing the Square UNIT F4 Solving

More information

I think that starting

I think that starting . Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries

More information

The Not-Formula Book for C1

The Not-Formula Book for C1 Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes

More information

Identify a pattern and find the next three numbers in the pattern. 5. 5(2s 2 1) 2 3(s 1 2); s 5 4

Identify a pattern and find the next three numbers in the pattern. 5. 5(2s 2 1) 2 3(s 1 2); s 5 4 Chapter 1 Test Do ou know HOW? Identif a pattern and find the net three numbers in the pattern. 1. 5, 1, 3, 7, c. 6, 3, 16, 8, c Each term is more than the previous Each term is half of the previous term;

More information

STRAND: ALGEBRA Unit 3 Solving Equations

STRAND: ALGEBRA Unit 3 Solving Equations CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic

More information

Polynomials Classwork

Polynomials Classwork Polynomials Classwork What Is a Polynomial Function? Numerical, Analytical and Graphical Approaches Anatomy of an n th -degree polynomial function Def.: A polynomial function of degree n in the vaiable

More information

INTEGRATION FINDING AREAS

INTEGRATION FINDING AREAS INTEGRTIN FINDING RES Created b T. Madas Question 1 (**) = 4 + 10 3 The figure above shows the curve with equation = 4 + 10, R. Find the area of the region, bounded b the curve the coordinate aes and the

More information

5.2 Inverse Functions

5.2 Inverse Functions 78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,

More information

SECTION 2-5 Combining Functions

SECTION 2-5 Combining Functions 2- Combining Functions 16 91. Phsics. A stunt driver is planning to jump a motorccle from one ramp to another as illustrated in the figure. The ramps are 10 feet high, and the distance between the ramps

More information

1.6 Graphs of Functions

1.6 Graphs of Functions .6 Graphs of Functions 9.6 Graphs of Functions In Section. we defined a function as a special tpe of relation; one in which each -coordinate was matched with onl one -coordinate. We spent most of our time

More information

Practice for Final Disclaimer: The actual exam is not a mirror of this. These questions are merely an aid to help you practice 1 2)

Practice for Final Disclaimer: The actual exam is not a mirror of this. These questions are merely an aid to help you practice 1 2) Practice for Final Disclaimer: The actual eam is not a mirror of this. These questions are merel an aid to help ou practice Solve the problem. 1) If m varies directl as p, and m = 7 when p = 9, find m

More information

Families of Quadratics

Families of Quadratics Families of Quadratics Objectives To understand the effects of a, b, and c on the graphs of parabolas of the form a 2 b c To use quadratic equations and graphs to analze the motion of projectiles To distinguish

More information

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

More information

Quadratic Equations and Functions

Quadratic Equations and Functions Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In

More information

GRAPHS OF RATIONAL FUNCTIONS

GRAPHS OF RATIONAL FUNCTIONS 0 (0-) Chapter 0 Polnomial and Rational Functions. f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0) 0. GRAPHS OF RATIONAL FUNCTIONS In this section Domain Horizontal and Vertical Asmptotes Oblique

More information

Lesson 9.1 Solving Quadratic Equations

Lesson 9.1 Solving Quadratic Equations Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte

More information

THE POWER RULES. Raising an Exponential Expression to a Power

THE POWER RULES. Raising an Exponential Expression to a Power 8 (5-) Chapter 5 Eponents and Polnomials 5. THE POWER RULES In this section Raising an Eponential Epression to a Power Raising a Product to a Power Raising a Quotient to a Power Variable Eponents Summar

More information

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

More information

The Graph of a Linear Equation

The Graph of a Linear Equation 4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

More information

Summer Review For Students Entering Algebra 2

Summer Review For Students Entering Algebra 2 Summer Review For Students Entering Algebra Board of Education of Howard Count Frank Aquino Chairman Ellen Flnn Giles Vice Chairman Larr Cohen Allen Der Sandra H. French Patricia S. Gordon Janet Siddiqui

More information

ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )

ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 ) SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as

More information

INTRODUCTION. Approach

INTRODUCTION. Approach INTRODUCTION In its paper for discussion and development Advanced Calculators and Mathematics Education the Scottish Consultative Council on the Curriculum states that, in its view, calculators have great

More information

Quadratic Equations in One Unknown

Quadratic Equations in One Unknown 1 Quadratic Equations in One Unknown 1A 1. Solving Quadratic Equations Using the Factor Method Name : Date : Mark : Ke Concepts and Formulae 1. An equation in the form a + b + c, where a, b and c are real

More information

Functions and Equations

Functions and Equations Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

More information

Graphing Nonlinear Systems

Graphing Nonlinear Systems 10.4 Graphing Nonlinear Sstems 10.4 OBJECTIVES 1. Graph a sstem of nonlinear equations 2. Find ordered pairs associated with the solution set of a nonlinear sstem 3. Graph a sstem of nonlinear inequalities

More information

8 Graphs of Quadratic Expressions: The Parabola

8 Graphs of Quadratic Expressions: The Parabola 8 Graphs of Quadratic Epressions: The Parabola In Topic 6 we saw that the graph of a linear function such as = 2 + 1 was a straight line. The graph of a function which is not linear therefore cannot be

More information

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

More information

Methods to Solve Quadratic Equations

Methods to Solve Quadratic Equations Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a second-degree

More information

2.4 Inequalities with Absolute Value and Quadratic Functions

2.4 Inequalities with Absolute Value and Quadratic Functions 08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we

More information

Rational Functions. 7.1 A Rational Existence. 7.2 A Rational Shift in Behavior. 7.3 A Rational Approach. 7.4 There s a Hole In My Function, Dear Liza

Rational Functions. 7.1 A Rational Existence. 7.2 A Rational Shift in Behavior. 7.3 A Rational Approach. 7.4 There s a Hole In My Function, Dear Liza Rational Functions 7 The ozone laer protects Earth from harmful ultraviolet radiation. Each ear, this laer thins dramaticall over the poles, creating ozone holes which have stretched as far as Australia

More information

More Equations and Inequalities

More Equations and Inequalities Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities

More information

x 2 would be a solution to d y

x 2 would be a solution to d y CATHOLIC JUNIOR COLLEGE H MATHEMATICS JC PRELIMINARY EXAMINATION PAPER I 0 System of Linear Equations Assessment Objectives Solution Feedback To use a system of linear c equations to model and solve y

More information

Chapter 6 Quadratic Functions

Chapter 6 Quadratic Functions Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Chapter 3 Eponential and Logarithmic Functions Section 3.1 Eponential Functions and Their Graphs Objective: In this lesson ou learned how to recognize, evaluate, and graph eponential functions. Course

More information

Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.

Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t. REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()

More information

Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.

Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system. _.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial

More information

2.1 Equations of Lines

2.1 Equations of Lines Section 2.1 Equations of Lines 1 2.1 Equations of Lines The Slope-Intercept Form Recall the formula for the slope of a line. Let s assume that the dependent variable is and the independent variable is

More information

Section C Non Linear Graphs

Section C Non Linear Graphs 1 of 8 Section C Non Linear Graphs Graphic Calculators will be useful for this topic of 8 Cop into our notes Some words to learn Plot a graph: Draw graph b plotting points Sketch/Draw a graph: Do not plot,

More information

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real

More information

LESSON EIII.E EXPONENTS AND LOGARITHMS

LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential

More information

STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Axis Combining Transformations of Graphs

STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Axis Combining Transformations of Graphs 6 CHAPTER Analsis of Graphs of Functions. STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Ais Combining Transformations of Graphs In the previous

More information

The Distance Formula and the Circle

The Distance Formula and the Circle 10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,

More information

Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

More information

2.5 Library of Functions; Piecewise-defined Functions

2.5 Library of Functions; Piecewise-defined Functions SECTION.5 Librar of Functions; Piecewise-defined Functions 07.5 Librar of Functions; Piecewise-defined Functions PREPARING FOR THIS SECTION Before getting started, review the following: Intercepts (Section.,

More information

Transformations of Function Graphs

Transformations of Function Graphs - - - 0 - - - - - - - Locker LESSON.3 Transformations of Function Graphs Teas Math Standards The student is epected to: A..C Analze the effect on the graphs of f () = when f () is replaced b af (), f (b),

More information

Filling in Coordinate Grid Planes

Filling in Coordinate Grid Planes Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the

More information

1.2 GRAPHS OF EQUATIONS

1.2 GRAPHS OF EQUATIONS 000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the - and -intercepts of graphs of equations. Write the standard forms of equations of

More information

Chapter 3A - Rectangular Coordinate System

Chapter 3A - Rectangular Coordinate System - Chapter A Chapter A - Rectangular Coordinate Sstem Introduction: Rectangular Coordinate Sstem Although the use of rectangular coordinates in such geometric applications as surveing and planning has been

More information

PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS

PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS A ver important set of curves which has received considerabl attention in recent ears in connection with the factoring of large numbers

More information

Year 9 set 1 Mathematics notes, to accompany the 9H book.

Year 9 set 1 Mathematics notes, to accompany the 9H book. Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H

More information

7.7 Solving Rational Equations

7.7 Solving Rational Equations Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate

More information

Objectives. By the time the student is finished with this section of the workbook, he/she should be able

Objectives. By the time the student is finished with this section of the workbook, he/she should be able QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a

More information

Identifying second degree equations

Identifying second degree equations Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +

More information

Section V: Quadratic Equations and Functions. Module 1: Solving Quadratic Equations Using Factoring, Square Roots, Graphs, and Completing-the-Square

Section V: Quadratic Equations and Functions. Module 1: Solving Quadratic Equations Using Factoring, Square Roots, Graphs, and Completing-the-Square Haberman / Kling MTH 95 Section V: Quadratic Equations and Functions Module : Solving Quadratic Equations Using Factoring, Square Roots, Graphs, and Completing-the-Square DEFINITION: A quadratic equation

More information

17.1 Connecting Intercepts and Zeros

17.1 Connecting Intercepts and Zeros Locker LESSON 7. Connecting Intercepts and Zeros Teas Math Standards The student is epected to: A.7.A Graph quadratic functions on the coordinate plane and use the graph to identif ke attributes, if possible,

More information

Years t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304

Years t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304 Section The Circle 65 Dollars Purchase price P Book value = f(t) Salvage value S Useful life L Years t FIGURE 3 Straight-line depreciation. The Circle Definition Anone who has drawn a circle using a compass

More information

HOW TO COMPLETE THE SQUARE

HOW TO COMPLETE THE SQUARE HOW TO COMPLETE THE SQUARE LANCE D. DRAGER. Introduction To complete the square means to take a quadratic function f() = a 2 + b + c and do the algebra to write it in the form f() = a( h) 2 + k, for some

More information

CHAPTER 9. Polynomials

CHAPTER 9. Polynomials CHAPTER 9 In this chapter ou epand our knowledge of families of functions to include polnomial functions. As ou investigate the equation! graph connection for polnomials, ou will learn how to search for

More information

FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) -

More information

Graphing Linear Equations

Graphing Linear Equations 6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are

More information

Mathematics Paper 1 (Non-Calculator)

Mathematics Paper 1 (Non-Calculator) H National Qualifications CFE Higher Mathematics - Specimen Paper A Duration hour and 0 minutes Mathematics Paper (Non-Calculator) Total marks 60 Attempt ALL questions. You ma NOT use a calculator. Full

More information