Section 7.1 Polynomial Functions 1. To evaluate polynomial functions.. To identify general shapes of the graphs of polynomial functions. I. Terminology A. Polynomials in one variable B. Examples: Determine if each is a polynomial of one variable. 1. x + xy + y II.. a a + 1 3. 1 + n n Polynomial Functions A. Def B. Examples 1. Find P (3) if P( x) = 3x 8x + x 3. Find P(3) P( 3) if P x x x ( ) = + 6 3. Find P( a ) if P( x) = 3x 8x + x 3. Find P( a) + 3 P( a) if 3 P( x) = x x + x 5. Find P( x + ) if P x x x ( ) = + 3 + 1 Precalculus Chapter 7 Page 1
III. Polynomials and Graphs A. Note: 1. degree polynomials left most and right most points will be either positive or negative (Superman Look). Odd degree polynomials a) Left most points will be of right most points (Swimmer Look) b) Will always cross the at least once. 3. Root a) Where graph crosses. b) Number of real and non-real roots is to the of the polynomial. B. Examples: Determine if each graph is even or odd and state the number of roots. 1. 6-6 - - 6 - - -6. - - - -6-8 3. 6 - - Homework: p. 350, 3, 17-1 odds, 0, (-36)/3, 39- all, 6-50 all, 57-67 all Precalculus Chapter 7 Page
Section 7.A Graphing Polynomial Functions and Approximating Zeros 1. To approximate the real zeros of polynomial functions.. To find relative maxima and minima of polynomial functions. 3. To graph complete graphs of polynomial functions. I. Graphing Complete Graphs A. Def A graph is a graph where all are seen. B. Characteristics 1. a) Note: b) Location Principle for Roots Relative extrema ( peeks and valleys) C. Examples 3 1. Graph: f ( x) = x 5x + 3x + Give appropriate view window 3. Graph: f ( x) = x 9x + 5x x + 6 Give appropriate view window II. Finding Characteristics A. Examples: Find the relative extrema, roots (x-intercepts), and y-intercept 3 1. f ( x) = x 5x + 3x +. f x x x x x 3 ( ) = 9 + 5 + 6 Homework: p. 356 1,, 3, 1-6 evens (state the viewing window, find the roots, find the intercepts, and find the relative extrema) Precalculus Chapter 7 Page 3
Section 7.B Modeling Real World Data with Polynomials 1. To model data whose curve of best fit is a polynomial function. Example: The table at the right gives the consumption of peanuts. Use a graphing calculator to draw a scatter plot for the data. State the viewing windows and scale factors that you used. Let 1970 = 0 Year US Peanut Consumption (millions of pounds) 1970 1118 1980 1087 1985 199 1990 19 1991 1639 199 1581 1993 157 Calculate and graph curves of best fit that show how the year is related to peanut consumption. Give the equations for: Linear Reg (P) Quadratic Reg (P3) Cubic Reg (P) Quartic Reg Exponential Reg Write the equation for the curve you think best fits the data. Explain why you think it fits the best. Equation Explain: Based on the cubic regression, what was the consumption in 199?. Can I use this equation to predict consumption in 010? Explain. Can I use this equation to estimate consumption in 1960? Explain. When can I use this equation? Explain. Based on the quartic regression, when will consumption be 1500 million pounds? Homework: p. 359 1-3 all, and p. 369-5 all Precalculus Chapter 7 Page
Section 7.3 Use quadratic Techniques to Solve Polynomial Equations 1. To solve non-quadratic equations by using quadratic techniques. Examples 1. Solve: 3 x + 3x 18x = 0. Graph to verify.. Solve: x 7x + 1 = 0. Graph to verify. 3. Solve: t 3 16 = 0. Graph to verify.. Solve: x 1 3 3 8x + 15 = 0. Graph to verify. 5. Solve: x 3 3 13x + 36 = 0. Graph to verify. 6. Solve: x 8 x + 7 = 0. Graph to verify. 7. Solve: x 13 x + 36 = 0. Graph to verify. Homework: p. 363 11-16 all, 17-7 odds, 37, 39-5 all Precalculus Chapter 7 Page 5
Section 5.3 Dividing Polynomials 1. To divide polynomials using long division.. To divide polynomials using synthetic division. I. Dividing Polynomials by monomials Examples 3 3a b + 6a b + 18ab 1. 3ab. 1x y + 3 x x II. Dividing Polynomials by Polynomials A. Long Division Examples 1. x + x x 3 5 Note: When dividing make sure the polynomial is in order.. ( x 11x + 8) ( x ) 3 3. ( )( ) 1 x 7 x 3. 6 3 x x x + 5 + 1 x + 3 Precalculus Chapter 7 Page 6
III. Synthetic Division A. Def of where you are only using the. B. Note: SD can only be used when dividing by C. Example 3 1. ( 3x x + x ) ( x + 3) 3 x + 3 3x x + x -3 3-1 - 3. ( ) x x + x 3x + ( x ) x x + x 1 ( x + 1) 3 3. ( ) 6x x + (x 3). ( ) Homework: p. 36, 3, (15-)/3, 60-68 all Precalculus Chapter 7 Page 7
Section 7. The Remainder and Factor Theorems 1. To find factors of polynomials using the factor theorem and synthetic division. I. Work Together A. Find the remainder to each problem. 3 x 6x + 8x + 5x + 13 1. x. 5 3 3x 3x + 57 x + B. Find the value for each function 3 1. Find f () if f ( x) = x 6x + 8x + 5x + 13.. Find f ( ) if 5 3 f ( x) = 3x 3x + 57. C. Can you make a conclusion? D. Wrap Up II. Think and Discuss A. Remainder Theorem: If f ( x) ( x a), then f ( a ) the of f ( x) ( x a). B. Factor Theorem Theorem: a factor of f ( x ) iff f ( a ) = 0 1. Examples: 3 a) Is x a factor of x x + x? 8 6 3 b) Is x a factor of x + 7x + x 0? If so find all the factors. - - - c) Find the polynomial (in factored form) for the given graph. -6-8 Homework: p. 368 3, (15-30)/3, 33, 35, 37, 38, 7-5 all, 57-59 all Precalculus Chapter 7 Page 8
Section 7.5 and 7.6 Finding all the Roots (Real and Non-real) Goal: 1. To find all the real and non-real roots of a polynomial.. To find all the factors of a polynomial. 3. To write the polynomial of least degree given zeros. Examples: 1. Find all the factors of 3 f ( x) = x + x x given ( x + ) is a factor.. Find all the roots of 3 f ( x) = x 5x 7x + 51 given that i is a root. 3. Write the polynomial function of least degree with integral coefficients whose zeros include the following. 1 and 1+ i - and + 3 Homework: worksheet Precalculus Chapter 7 Page 9
Section 7.7 Operations on Functions 1. Find the sum, difference, product, and quotient of functions. To find the composition of functions. I. Arithmetic Operations Codes: 1. sum: ( f + g )( x) = f ( x) + g( x). difference: ( f g )( x) = f ( x) g( x) 3. Product: ( f g )( x) = f ( x) g( x) f f ( x). quotient: ( x) =, g ( x ) 0 g g( x) II. Composition of Functions A. Definition When the result of one function is plugged into another function. B. Examples of use 1. Converting 59 F to Kelvin a) Convert 59 F to Celsius b) Convert 15 C to Kelvin. An $5 item on sale for 30% off and from 8am to 10am take an addition 50% off. a) Take 30% off $5 b) Take 50% off $31.50 C. Notation: f g or f ( g( x)) start with the x value in g and take its result and plug it into f. D. Examples: 1. If f = {(1, ),(10,5),(6, 3) } and {(5,1),(,6),( 3,10) } g = find f g and g f.. If f = {(1,3),(,7),(3, ) } and {(7,11),(3, 6),(, 3) } they exist. g = find f g and g f, if 3. If f x = + and g( x) = x + 7 find [ f g]( x) or f ( g( x )) 3 ( ) x 3. If f ( x) = x + 7 and g x = find [ f g]() and g( f ()) ( ) x 5. Iteration (repeating a process): Find f () if f (0) = 7 and f ( n) = f ( n 1) + n Homework: p. 386 3, (18-5)/3, 9-51 all, 55, 58, 59, 63-81 all Precalculus Chapter 7 Page 10
Section 7.8 Inverse Functions and Relations 1. To determine the inverse of a function or relation.. To graph a function and its inverse. A. Definition of Inverse Functions 1. Algebraically: Two relations are inverses iff and.. Geometrically: Two relations are inverses iff their are over the line. 3. Example: Show both algebraically and geometrically that f ( x) = x + 3 and 3 x g( x) =. 3 B. Properties of Inverses 1 1. If f and f are inverses, then f ( a) = b and [i.e., f = ( x, y) and. Examples: f 1 = ( y, x) ] f 1 ( b) = a. a) Find the inverse relation if f = {(, 3),(,1),(5,9) } 1 - -3 - -1 1 3 5-1 - -3 - b) Find the inverse of f ( x) = 3x 5 c) Find the inverse of d) Graph the inverse of Is it a function? f ( x) = x x +. Is it a function? 3 1 - -3 - -1 1 3 5-1 - -3 - Homework: p. 393 1, (15-36)/3,, 3, 6-59 all, 61 Precalculus Chapter 7 Page 11
Section 7.9 Square Root Functions and Inequalities 1. To graph and analyze square root functions.. To graph square root inequalities. I. Analyzing Square Root Functions and Graphing A. Analyzing 1. Note: We are only graphing numbers.. Determine the (what I can plug in) and the (the results produced from the ) 3. Determine the. Example a) State the domain and range of f ( x) = 3x. 6 b) State the x- and y-intercepts of f ( x) = 3x. 6 B. Graphing Example: Graph f ( x) = 3x II. Graphing Square Root Inequalities A. Examples 1. Graph: f ( x) < x + 6 6. Graph: f ( x) x 6 6 - - -6 Homework: p. 397, 15-3 all, 7-31 odds, 3-3 all, 38-9 all Precalculus Chapter 7 Page 1