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# PRECALC ALG & TRIG MAC 1147

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This 26 page Class Notes was uploaded by Marquise Graham on Saturday September 19, 2015. The Class Notes belongs to MAC 1147 at University of Florida taught by Larissa Williamson in Fall. Since its upload, it has received 9 views. For similar materials see /class/207048/mac-1147-university-of-florida in Calculus and Pre Calculus at University of Florida.

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Date Created: 09/19/15

L11 Quadratic Functions and Models Polynomial Functions and Inequalities Quadratic Function The quadratic function is a function of the form fx axz bxc Where a b and c are real numbers and a i 0 Example Use the graph of fx x2 as a reference to sketch the graphs of the following functions hx 7x2 7 2 gxx278xl7xi42l By completing the square the equation of a quadratic function fxax2bxc a720 can be written in the form fX 006402 k 2 where h b andk4ac b 2a or kfh The Graph of fx 0m2 bx c a 72 O is Vertex Axis Parabola opens up if f h k is the minimum value of f Parabola opens down if f h k is the maximum value of f Compared with the graph of f x x2 the graph of fc01c2 bxc is stretched vertically if lal gt1 compressed vertically if 0 lt a lt l 133 Example Find the vertex axis intercepts domain and range Graph the parabola fx 2x2 12x10 Applications of the Quadratic Function Example Suppose that a baseball is tossed straight up and its heights in feet as a function of time t in seconds is given by st l6t2 64t 6 with t 0 corresponding to the instant when the ball is released When does the ball reach the maximum height What is the maximum height of the ball 135 Economics Demand Equations The price p and the quantity x sold of a certain product obey the demand equation p x18 OSxS9O a Express the revenue R as a function of x b What is the revenue if 80 units are sold c What quantity x maximizes revenue What is the maximum revenue d What price should the company charge to maximize the revenue 136 Enclosing the Most Area A builder has 800 feet of fencing left over from a job He wants to fence in a rectangular plot of land except for a 20foot strip to be used as a driveway 2039 Express the area A of the plot as a function of x What is the domain of A For what x is the area A a maximum 137 Polynomial Functions The polynomial function is a function of the form fx anx 01n1c 1 alx do where anan1a1a0 are real numbers and n 2 O is an integer Degree 0 fx a0 a0 5 0 Degree 1 fxa1xa0 a1 0 Degree 2 fx a2x2 alx do 012 72 0 Features of the Polynomial Function l The domain is the set of all real numbers 2 The graph has no holes gaps or jumps a polynomial function is continuous 3 The graph has no sharp corners or cusps a polynomial function is smooth Power Function A power function of degree n is a function of the form f X Xquot where n gt O is an integer 138 Properties of the Power Functions n 1 y x where n is even m a 5 Symmetry A 2 3 2 1 1 2 3 End Behavior is 78 an 2 y xquot where n is add Symmetry End Behavior The greater the value of n the steeper graph of y xquot when x lt 71 or x gt 1 and the atter graph of y xquot when 71lt x lt1 139 Graphing Polynomials A number C is called a zero of a polynomial f x if fC0 x c is a zero of a polynomial f x if and only if x c is a factor of f The number of times the factor x 6 occurs in the polynomial is called the multiplicity of the zero C Example Find all zeros and their multiplicities in parentheses fx 2x2 x 43 x 2c2 1 The Fundamental Theorem of Algebra Every polynomial of degree 1 or more has at least one complex zero Number of Zeros Theorem A polynomial of degree n has at most n distinct zeros Note The real zeros of a function are the xintercepts 140 Multiplicities of Zeros and Graph Sketching Example Find the intervals on which f x gt O and fx lt 0 if fx x32x l Solution Zeros of f x x 3 of multiplicity 2 even x 1 of multiplicity l odd Intervals gt 00 3 x 3 391 x 1 1900 x 32 O X 1 O fxx32x 1 O O Observe 1 If the multiplicity of a zero is even the corresponding to the zero factor does not change sign passing through it If the multiplicity is odd the factor changes sign 2 f x changes sign at a zero if and only if the factor corresponding to this zero changes its sign l4l Therefore 1 does not change sign touches the xaXis when passing through a zero of an even multiplicity 2 f x changes sign crosses the xaXis when passing through a zero of an odd multiplicity Turning Points T uming points of the graph are points of local maxima or local minima A polynomial of degree n has at most n l turning points 142 End Behavior Example Describe the end behavior of the graph of fxx3 7x2 79x9 Hint factor out 63 and consider lxl no x For large values of the polynomial fxanxquotan71xquot 1a1xa0 a 70 n behaves as its leading term y anxquot Also as x 00 the polynomial has the sign of its leading coef cient an Example The graph of a polynomial f x is given Tell a if the degree of fx is even or odd b if the leading coef cient is positive or negative c what is the smallest possible degree of f x Example Match each function to the graph a fx 2x4 2x2 b gx x3 xZ 2x c hx l4x5 34x3 x Analyzing the Graph of a Polynomial Function l 3 4 5 6 7 Check for symmetry Find all real zeros xintercepts and their multiplicities Find the yintercept Give the number of turning points Locate the turning points if possible Analyze the end behavior Plot all intercepts Using the end behavior and multiplicities of the zeros determine the signs of the polynomial on each interval with xintercepts as endpoints 8 9 Plot a few additional points if necessary Draw the graph 145 Example Draw the graph of fx 73x4 12x2 Solving Polynomial Inequalities 1 4 Move over all terms of the inequality to the lefthand side with a O on the right side and simplify in order to obtain one of the inequalities fXSO fx20 fxlt0 fxgt0 where f x is a polynomial Find all real zeros of f x and their multiplicities Put the zeros on the number line and label them as o if the inequality is non strict f x S O or f x Z O the zeros will be included in the answer otherwise label them as 0 Determine the signs of f x on each interval with the zeros as endpoints To do that First find the sign of the leading coefficient and set it on the rightmost interval the end behavior when x gt 00 Then starting from the rightmost interval and moving to the left on the number line alternate Sl when passing through a zero of odd multiplicity leave the same Sl when passing through a zero of even multiplicity 5 6 7 Select intervals with the desired sign according to the inequality in Step 1 Give closed intervals in your answer if the zeros are included otherwise give open intervals Use the symbol U to join two or more intervals 147 Example Solve the inequalities 2x2 3x lt 5 2x 51 x2 3 x3 Z O 148 L11 Quadratic Functions and Models Polynomial Functions and Inequalities Quadratic Function The quadratic function is a function of the form x Where a b and c are real numbers and a i 0 Example Use the graph of f x x2 as a reference to sketch the graphs of the following functions hx 7x2 7 2 By completing the square the equation of a quadratic function fx avc2 bxc 11 0 can be written in the form fx 01x7h2 k 2 where h2 and k 401671 211 4a or kfh gxx278xl7xi42l The Graph of fx ax2 bx c a i 0 is Vertex Axis Parabola opens up if fh k is the minimum value of f Parabola opens down ifi fh k is the maximum value of f Compared with the graph of f x x2 the graph of fxavc2 bxc is stretched vertically if M gt 1 compressed vertically if 0 lt a lt1 Example Find the vertex axis intercepts domain Applications of the Quadratic Function and range Graph the parabola f x 2x2 12x 10 Example Suppose that a baseball is tossed straight up and its heights in feet as a function of time I in seconds is given by sl 71612 641 6 with I 0 corresponding to the instant when the ball is released When does the ball reach the maximum height What is the maximum height of the ball Economics Demand Eguations The price p and the quantity x sold of a certain product obey the demand equation px1s 0Sx 90 a Express the revenue R as a function of x b What is the revenue if 80 units are sold c What quantity x maximizes revenue What is the maximum revenue d What price should the company charge to maximize the revenue Enclosing the Most Area A builder has 800 feet of fencing left over from a job He wants to fence in a rectangular plot of land except for a 20foot strip to be used as a driveway Express the areaA of the plot as a function of x What is the domain ofA For What x is the area A a maximum Polynomial Functions The palynamiulfunclian is a function of the form x anxquot aHx39 a1x do where an any a1 a0 are real numbers and n 2 0 is an integer Degree 0 fx a0 a0 7 0 Degree 1 fxa1xa0 5110 Degree 2 fx azxz a1x a0 all i 0 Features of the Polynomial Function 1 The domain is the set of all real numbers 2 The graph has no holes gaps or jumps 7 a polynomial function is continuous 3 The graph has no sharp corners or cusps 7 a polynomial function is smooth Power Function A pawerfunclian of degree n is a function of the orm f x xquot where n gt 0 is an integer Properties of the Power Functions 1 y xquot where n is even in Symmetry End Behavior 2 y xquot where n is add Symmetry End Behavior 2 The greater the value of n the steeper graph of y xquot when x lt 71 or x gt1 and the atter graph of y xquot when 71lt x lt1 139 Graphing Polynomials A number 0 is called a zero ofa polynomial fx if fc 0 x c is a zero ofa polynomial fx if and only if x c is a factor off The number of times the factor x 0 occurs in the polynomial is called the multiplicit of the zero 0 Example Find all zeros and their multiplicities in parentheses fx 2x2 x 43 x 2x2 1 The F 39 39 Theorem of Algebra Every polynomial of degree 1 or more has at least one complex zero Number of Zeros Theorem A polynomial of degree n has at most n distinct zeros Note The real zeros of a function are the xintercepts 140 Multiplicities of Zeros and Graph Sketching Example Find the intervals on Which f x gt 0 and fx lt 0 iffx x32x l Solution Zeros of fx x 3 of multiplicity 2 7 even x 1 of multiplicity l 7 odd Intervals 0073 x 3 391 x 1 194 x 32 0 x 1 o fxx32x1 0 0 Observe 1 If the multiplicity of a zero is even the corresponding to the zero factor does not change sign passing through it If the multiplicity is E the factor changes sign 2 f x changes sign at a zero if and only if the factor corresponding to this zero changes its sign 141 Therefore 1 does not change sign touches the xaXis when passing through a zero of an even multiplicity 2 f x changes sign crosses the xaXis when passing through a zero of an odd multiplicity Turning Points Turning points of the graph are points of local maxima or local minima A polynomial of degree n has at most n l turning points 142 End Behavior Example Describe the end behavior of the graph of fxx3 x2 9x9 Hint factor out x3 and consider lxl gt so For large values of lx the polynomial fx anxquot aHx H a1x a0 a i O n behaves as its leading term y anxquot Also as x gt 00 the polynomial f x has the sign of its leading coef cient atquot 143 Example The graph of a polynomial f x is given Tell a if the degree of fx is even or odd b if the leading coefficient is positive or negative c What is the smallest possible degree of fx Example Match each function to the graph a fx 2x4 2x2 b gx x3 x2 2x c 11x l4x5 34c3 x 74 73 72 Analyzing the Graph of a Polynomial Function p A 5 6 7 Check for symmetry Find all real zeros xintercepts and their multiplicities Find the yintercept Give the number of turning points Locate the turning points if possible Analyze the end behavior Plot all intercepts Using the end behavior and multiplicities of the zeros determine the signs of the polynomial on each interval With xintercepts as endpoints 8 9 Plot a few additional points if necessary Draw the graph Example Draw the graph of fx 73x4 12Z Solving Polynomial Inequalities 5 e Move over all terms of the inequality to the le hand side with a 0 on the right side and simplify in order to obtain one of the inequalities fXS0 fx20 fxlt0 fxgt0 where f x is a polynomial Find all real zeros of f x and their multiplicities Put the zeros on the number line and label them as ifthe inequality is non strict fx g 0 or fx 2 0 the zeros will be included in the answer otherwise label them as 0 Determine the signs of f x on each interval with the zeros as endpoints To do at First nd the sign of the leading eoe ieient and set it on the rightmost interval the end behavior when x gt 00 Then starting from the rightmost interval and moving to the le on the number line alternate sign when passing through a zero of odd multipliciy39 leave the same sign when passing through a zero of even multiplicig Fquot gt1 Select intervals with the desired sign according to the inequality in Step 1 Give closed intervals in your answer if the zeros are included otherwise give open intervals Use the symbol U to join two or more intervals 147 Example Solve the inequalities 2x2 3x lt 5 2x 51 x23 x3 2 0

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