SAMPLE. Polynomial functions


 Rodney Quinn
 2 years ago
 Views:
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, TINspire & 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 righthand 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 righthand 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, TINspire & 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 righthand 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 righthand 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 nonzero 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, TINspire & 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 TINspire 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, TINspire & 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, TINspire & 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 TINspire 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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 subfamil 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 nonzero. That is, the gradient of the graph of = f () ispositive for all nonzero 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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 TINspire 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, TINspire & 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, TINspire & 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, TINspire & 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, TINspire & 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 nonlinear 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, TINspire & 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, TINspire & 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 TINspire 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, TINspire & 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, TINspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
37 164 Essential Mathematical Methods 3&4CAS Using the TINspire 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, TINspire & 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, TINspire & 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, TINspire & 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 =. Multiplechoice 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, TINspire & 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 thirddegree 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, TINspire & 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 Shortanswer questions (technologfree) 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, TINspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
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 informationPolynomial 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 informationQuadratic 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 informationChapter 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 informationColegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radioactive substance at time t hours is given by. m = 4e 0.2t.
REPASO. The mass m kg of a radioactive 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 informationNorth Carolina Community College System Diagnostic and Placement Test Sample Questions
North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College
More informationIdentifying 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 informationClick here for answers.
CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the ais and the tangent
More informationPolynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005
Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division
More informationPolynomials. Teachers Teaching with Technology. Scotland T 3. Teachers Teaching with Technology (Scotland)
Teachers Teaching with Technology (Scotland) Teachers Teaching with Technology T Scotland Polynomials Teachers Teaching with Technology (Scotland) POLYNOMIALS Aim To demonstrate how the TI8 can be used
More informationRoots, Linear Factors, and Sign Charts review of background material for Math 163A (Barsamian)
Roots, Linear Factors, and Sign Charts review of background material for Math 16A (Barsamian) Contents 1. Introduction 1. Roots 1. Linear Factors 4. Sign Charts 5 5. Eercises 8 1. Introduction The sign
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationChapter 8. Lines and Planes. By the end of this chapter, you will
Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes
More informationMATH 185 CHAPTER 2 REVIEW
NAME MATH 18 CHAPTER REVIEW Use the slope and intercept to graph the linear function. 1. F() = 4   Objective: (.1) Graph a Linear Function Determine whether the given function is linear or nonlinear..
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationSOLVING POLYNOMIAL EQUATIONS
C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra
More informationHIBBING COMMUNITY COLLEGE COURSE OUTLINE
HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE:  Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,
More informationThe Big Picture. Correlation. Scatter Plots. Data
The Big Picture Correlation Bret Hanlon and Bret Larget Department of Statistics Universit of Wisconsin Madison December 6, We have just completed a length series of lectures on ANOVA where we considered
More informationMATH 102 College Algebra
FACTORING Factoring polnomials ls is simpl the reverse process of the special product formulas. Thus, the reverse process of special product formulas will be used to factor polnomials. To factor polnomials
More information3.6 The Real Zeros of a Polynomial Function
SECTION 3.6 The Real Zeros of a Polynomial Function 219 3.6 The Real Zeros of a Polynomial Function PREPARING FOR THIS SECTION Before getting started, review the following: Classification of Numbers (Appendix,
More informationIf A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question
More informationIndiana University Purdue University Indianapolis. Marvin L. Bittinger. Indiana University Purdue University Indianapolis. Judith A.
STUDENT S SOLUTIONS MANUAL JUDITH A. PENNA Indiana Universit Purdue Universit Indianapolis COLLEGE ALGEBRA: GRAPHS AND MODELS FIFTH EDITION Marvin L. Bittinger Indiana Universit Purdue Universit Indianapolis
More informationMathematics Placement Packet Colorado College Department of Mathematics and Computer Science
Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking
More informationSAMPLE. Computer Algebra System (Classpad 330 using OS 3 or above) Application selector. Icolns that access working zones. Icon panel (Master toolbar)
A P P E N D I X B Computer Algebra System (Classpad 330 using OS 3 or above) B.1 Introduction For reference material on basic operations of the calculator, refer to the free downloadable documentation
More informationFind the Relationship: An Exercise in Graphing Analysis
Find the Relationship: An Eercise in Graphing Analsis Computer 5 In several laborator investigations ou do this ear, a primar purpose will be to find the mathematical relationship between two variables.
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More informationCOLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5
More informationPolynomial Operations and Factoring
Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.
More informationSection A3 Polynomials: Factoring APPLICATIONS. A22 Appendix A A BASIC ALGEBRA REVIEW
A Appendi A A BASIC ALGEBRA REVIEW C In Problems 53 56, perform the indicated operations and simplify. 53. ( ) 3 ( ) 3( ) 4 54. ( ) 3 ( ) 3( ) 7 55. 3{[ ( )] ( )( 3)} 56. {( 3)( ) [3 ( )]} 57. Show by
More informationMany common functions are polynomial functions. In this unit we describe polynomial functions and look at some of their properties.
Polynomial functions mctypolynomial20091 Many common functions are polynomial functions. In this unit we describe polynomial functions and look at some of their properties. In order to master the techniques
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationThe degree of a polynomial function is equal to the highest exponent found on the independent variables.
DETAILED SOLUTIONS AND CONCEPTS  POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
More informationAlgebra 1 Course Information
Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through
More informationThe Australian Curriculum Mathematics
The Australian Curriculum Mathematics Mathematics ACARA The Australian Curriculum Number Algebra Number place value Fractions decimals Real numbers Foundation Year Year 1 Year 2 Year 3 Year 4 Year 5 Year
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationSECTION P.5 Factoring Polynomials
BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The
More information9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.
9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n1 x n1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role
More informationVISUAL ALGEBRA FOR COLLEGE STUDENTS. Laurie J. Burton Western Oregon University
VISUAL ALGEBRA FOR COLLEGE STUDENTS Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS TABLE OF CONTENTS Welcome and Introduction 1 Chapter 1: INTEGERS AND INTEGER OPERATIONS
More information63. Graph y 1 2 x and y 2 THE FACTOR THEOREM. The Factor Theorem. Consider the polynomial function. P(x) x 2 2x 15.
9.4 (927) 517 Gear ratio d) For a fixed wheel size and chain ring, does the gear ratio increase or decrease as the number of teeth on the cog increases? decreases 100 80 60 40 20 27in. wheel, 44 teeth
More informationTHIS CHAPTER INTRODUCES the Cartesian coordinate
87533_01_ch1_p001066 1/30/08 9:36 AM Page 1 STRAIGHT LINES AND LINEAR FUNCTIONS 1 THIS CHAPTER INTRODUCES the Cartesian coordinate sstem, a sstem that allows us to represent points in the plane in terms
More informationSECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS
SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31
More informationFactoring bivariate polynomials with integer coefficients via Newton polygons
Filomat 27:2 (2013), 215 226 DOI 10.2298/FIL1302215C Published b Facult of Sciences and Mathematics, Universit of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Factoring bivariate polnomials
More informationDeterminants can be used to solve a linear system of equations using Cramer s Rule.
2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationSECTION A3 Polynomials: Factoring
A3 Polynomials: Factoring A23 thick, write an algebraic epression in terms of that represents the volume of the plastic used to construct the container. Simplify the epression. [Recall: The volume 4
More informationy 1 x dx ln x y a x dx 3. y e x dx e x 15. y sinh x dx cosh x y cos x dx sin x y csc 2 x dx cot x 7. y sec 2 x dx tan x 9. y sec x tan x dx sec x
Strateg for Integration As we have seen, integration is more challenging than differentiation. In finding the derivative of a function it is obvious which differentiation formula we should appl. But it
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationCHAPTER 7: FACTORING POLYNOMIALS
CHAPTER 7: FACTORING POLYNOMIALS FACTOR (noun) An of two or more quantities which form a product when multiplied together. 1 can be rewritten as 3*, where 3 and are FACTORS of 1. FACTOR (verb)  To factor
More informationALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section
ALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 53.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 64.2 Solving Equations by
More informationt hours This is the distance in miles travelled in 2 hours when the speed is 70mph. = 22 yards per second. = 110 yards.
The area under a graph often gives useful information. Velocittime graphs Constant velocit The sketch shows the velocittime graph for a car that is travelling along a motorwa at a stead 7 mph. 7 The
More informationAlgebraic Concepts Algebraic Concepts Writing
Curriculum Guide: Algebra 2/Trig (AR) 2 nd Quarter 8/7/2013 2 nd Quarter, Grade 912 GRADE 912 Unit of Study: Matrices Resources: Textbook: Algebra 2 (Holt, Rinehart & Winston), Ch. 4 Length of Study:
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationo Graph an expression as a function of the chosen independent variable to determine the existence of a minimum or maximum
Two Parabolas Time required 90 minutes Teaching Goals:. Students interpret the given word problem and complete geometric constructions according to the condition of the problem.. Students choose an independent
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More information4 Constrained Optimization: The Method of Lagrange Multipliers. Chapter 7 Section 4 Constrained Optimization: The Method of Lagrange Multipliers 551
Chapter 7 Section 4 Constrained Optimization: The Method of Lagrange Multipliers 551 LEVEL CURVES 2 7 2 45. f(, ) ln 46. f(, ) 6 2 12 4 16 3 47. f(, ) 2 4 4 2 (11 18) 48. Sometimes ou can classif the critical
More informationMATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab
MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring noncourse based remediation in developmental mathematics. This structure will
More informationMEP Pupil Text 12. A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued.
MEP Pupil Text Number Patterns. Simple Number Patterns A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued. Worked Example Write down the
More informationEXPANDING THE CALCULUS HORIZON. Hurricane Modeling
EXPANDING THE CALCULUS HORIZON Hurricane Modeling Each ear population centers throughout the world are ravaged b hurricanes, and it is the mission of the National Hurricane Center to minimize the damage
More informationThis unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide.
COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the
More informationSOLVING POLYNOMIAL EQUATIONS BY RADICALS
SOLVING POLYNOMIAL EQUATIONS BY RADICALS Lee Si Ying 1 and Zhang DeQi 2 1 Raffles Girls School (Secondary), 20 Anderson Road, Singapore 259978 2 Department of Mathematics, National University of Singapore,
More informationMathematics Review for MS Finance Students
Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
More informationIntroduction to Quadratic Functions
Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2
More informationA.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it
Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply
More informationAdministrative  Master Syllabus COVER SHEET
Administrative  Master Syllabus COVER SHEET Purpose: It is the intention of this to provide a general description of the course, outline the required elements of the course and to lay the foundation for
More informationRoots of Polynomials
Roots of Polynomials (Com S 477/577 Notes) YanBin Jia Sep 24, 2015 A direct corollary of the fundamental theorem of algebra is that p(x) can be factorized over the complex domain into a product a n (x
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More information1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives
TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of
More informationPolynomials and Factoring
Lesson 2 Polynomials and Factoring A polynomial function is a power function or the sum of two or more power functions, each of which has a nonnegative integer power. Because polynomial functions are built
More informationSPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 111 Factors and Factoring 112 Common Monomial Factors 113 The Square of a Monomial 114 Multiplying the Sum and the Difference of Two Terms 115 Factoring the
More informationCOLLEGE ALGEBRA LEARNING COMMUNITY
COLLEGE ALGEBRA LEARNING COMMUNITY Tulsa Community College, West Campus Presenter Lori Mayberry, B.S., M.S. Associate Professor of Mathematics and Physics lmayberr@tulsacc.edu NACEP National Conference
More informationMonday 13 May 2013 Afternoon
Monday 3 May 203 Afternoon AS GCE MATHEMATICS (MEI) 475/0 Introduction to Advanced Mathematics (C) QUESTION PAPER * 4 7 5 6 2 0 6 3 * Candidates answer on the Printed Answer Book. OCR supplied materials:
More informationInteger roots of quadratic and cubic polynomials with integer coefficients
Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street
More informationz 0 and y even had the form
Gaussian Integers The concepts of divisibility, primality and factoring are actually more general than the discussion so far. For the moment, we have been working in the integers, which we denote by Z
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationSimplification Problems to Prepare for Calculus
Simplification Problems to Prepare for Calculus In calculus, you will encounter some long epressions that will require strong factoring skills. This section is designed to help you develop those skills.
More information5. Factoring by the QF method
5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the
More information2010 Solutions. a + b. a + b 1. (a + b)2 + (b a) 2. (b2 + a 2 ) 2 (a 2 b 2 ) 2
00 Problem If a and b are nonzero real numbers such that a b, compute the value of the expression ( ) ( b a + a a + b b b a + b a ) ( + ) a b b a + b a +. b a a b Answer: 8. Solution: Let s simplify the
More informationCourse Outlines. 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit)
Course Outlines 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit) This course will cover Algebra I concepts such as algebra as a language,
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More informationthe points are called control points approximating curve
Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.
More informationMTH124: Honors Algebra I
MTH124: Honors Algebra I This course prepares students for more advanced courses while they develop algebraic fluency, learn the skills needed to solve equations, and perform manipulations with numbers,
More informationIntroduction to the Graphing Calculator
Unit 0 Introduction to the Graphing Calculator Intermediate Algebra Update 2/06/06 Unit 0 Activity 1: Introduction to Computation on a Graphing Calculator Why: As technology becomes integrated into all
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationAP Physics 1 and 2 Lab Investigations
AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks
More informationIn the Herb Business, Part III Factoring and Quadratic Equations
74 In the Herb Business, Part III Factoring and Quadratic Equations In the herbal medicine business, you and your partner sold 120 bottles of your best herbal medicine each week when you sold at your original
More informationFIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is
More informationMath Course Descriptions & Student Learning Outcomes
Math Course Descriptions & Student Learning Outcomes Table of Contents MAC 100: Business Math... 1 MAC 101: Technical Math... 3 MA 090: Basic Math... 4 MA 095: Introductory Algebra... 5 MA 098: Intermediate
More informationFor example, estimate the population of the United States as 3 times 10⁸ and the
CCSS: Mathematics The Number System CCSS: Grade 8 8.NS.A. Know that there are numbers that are not rational, and approximate them by rational numbers. 8.NS.A.1. Understand informally that every number
More informationQuestions. Strategies August/September Number Theory. What is meant by a number being evenly divisible by another number?
Content Skills Essential August/September Number Theory Identify factors List multiples of whole numbers Classify prime and composite numbers Analyze the rules of divisibility What is meant by a number
More informationLines and Planes 1. x(t) = at + b y(t) = ct + d
1 Lines in the Plane Lines and Planes 1 Ever line of points L in R 2 can be epressed as the solution set for an equation of the form A + B = C. The equation is not unique for if we multipl both sides b
More informationPrerequisites: TSI Math Complete and high school Algebra II and geometry or MATH 0303.
Course Syllabus Math 1314 College Algebra Revision Date: 82115 Catalog Description: Indepth study and applications of polynomial, rational, radical, exponential and logarithmic functions, and systems
More information2After completing this chapter you should be able to
After completing this chapter you should be able to solve problems involving motion in a straight line with constant acceleration model an object moving vertically under gravity understand distance time
More informationConnections Across Strands Provides a sampling of connections that can be made across strands, using the theme (integers) as an organizer.
Overview Context Connections Positions integers in a larger context and shows connections to everyday situations, careers, and tasks. Identifies relevant manipulatives, technology, and webbased resources
More informationSome facts about polynomials modulo m (Full proof of the Fingerprinting Theorem)
Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem) In order to understand the details of the Fingerprinting Theorem on fingerprints of different texts from Chapter 19 of the
More informationHigh School Functions Interpreting Functions Understand the concept of a function and use function notation.
Performance Assessment Task Printing Tickets Grade 9 The task challenges a student to demonstrate understanding of the concepts representing and analyzing mathematical situations and structures using algebra.
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More information