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K. Stewart MHF4U Unit Outline Chapter 1: Polynomial Functions Throughout this chapter, you will explore how the curves represented by polynomial functions are applied in various design-related fields such as civil engineering, architectural design, computer graphics design, interior design, and landscape architecture. By the end of this chapter, you will be assessed on your ability to: C1.0 D1.0 identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions. demonstrate an understanding of average and instantaneous rate of change, and determine, numerically and graphically, and interpret the average rate of change of a function over a given interval and the instantaneous rate of change of a function at a given point. Section 1.1 Power Functions 1.2 Characteristics of Polynomial Functions Learning goals What are the key features of the graphs of power function? How can I recognize polynomial functions? What is the connection between the graphs and equations of power functions? Vocab: power, exponent, power function, polynomial function, trigonometric function, rational function, exponential function, square root function, coefficient, degree, leading coefficient, constant term, linear, quadratic, cubic, quartic, quintic, polynomial, key features, domain, range, end behaviour, symmetry (line symmetry, point symmetry), axis of symmetry, intercepts Notation: x , x – interval notation – < x 0 set notation (– , 0] graphical notation (on a number line) domain notation e.g. {x| x } range notation e.g. {y| y > 0, y } Key equation: Power function is of the form f(x) = axn where n is a whole number Polynomial function is of the form f(x)= an xn +an-1 xn-1 +an-2 xn-2 +…+a1 x1 +a0 where coefficients are real numbers and n is a whole number What are the key features of the graphs of polynomial functions? What is the relationship between finite differences and the equation of a polynomial function? What is the connection between the graphs, equations and tables of values of polynomial functions? Vocab: local minimum (or maximum), absolute (or global) minimum (or maximum), optimal value, finite difference, finite difference table, factorial Notation: n! is read n factorial and is the product (n)(n – 1) (n – 2)…(2)(1) Homework Pg 11 #1 – 4 optional Pg 12 #6, 7, 8, 13 mandatory Skills: match graph with corresponding power function, name function by degree, identify power functions from their equations, use interval notation, set notation and graphical notation interchangeably, identify key features of power functions (domain and range, end behaviour, symmetry, intercepts, degree, positive or negative leading coefficient), solve problems involving power functions Pg 26 #1 – 4, 12 optional Pg 27 #5 – 8, 11 mandatory Skills: match graph with corresponding polynomial function, name function by degree, identify polynomial functions from their equations, use interval notation, set notation and graphical notation interchangeably, identify key features of polynomial functions (domain and range, end behaviour, symmetry, intercepts, degree, positive or negative leading coefficient), solve problems involving polynomial functions, determine value of finite difference using a difference table and algebraically, use a regression analysis on a graphing calculator to determine the equation of best fit. (Continued next page) K. Stewart 1.3 Equations and Graphs of Polynomial Functions 1.4 Transformations 1.5 Slopes of Secants and Average Rate of Change 1.6 Slopes of Tangents and Instantaneous Rate of Change What is the connection between the factored form of a polynomial function and its graph? How can I sketch graphs of polynomial functions from the equations? How is symmetry represented in the equation of a polynomial function? Vocab: order of an intercept, even function, odd function, constants are coefficients of x0, Key equations: even function f(x) = f(–x) odd function –f(x) = f(–x) (day 1) Pg 39 #1, 2 optional Pg 40 # 9, 11, 12 mandatory What are the roles of a, k, d, and c in polynomial functions of the form y = a[k(x-d)]n+c where n N? How can I describe transformations from an equation and use them to sketch a graph? How can I determine an equation give the graph of a transformed function? Vocab: transformation, reflection, translation, stretch, compression, opening Notation: y = a[k(x-d)] n+c How can I connect average rate of change and slope? How can I calculate and interpret average rates of change from any given representation? Vocab: average rate of change, slope, secant, radius What is the connection between the slopes of secants, the slope of a tangent and the instantaneous rate of change? How can I estimate an instantaneous rate of change from any representation? Vocab: instantaneous rate of change, tangent, velocity Pg 49 #1 – 7 optional Pg 50 #8, 10, 11, 12 mandatory Skills: write an equation of a polynomial function from a description of its key features, identify zeros of an equation, sketch a graph from an equation (day 2) Pg 39 #3 – 5 optional Pg 40 #6, 7, 8, 10, 15 mandatory Skills: write an equation from a graph of a polynomial function, identify symmetry of graph from equation, determine whether polynomial function is an even function, odd function or neither. Skills: applying transformations to sketch a graph, identify and describe transformations from a graph or an equation, determine equation from graph using transformations, solving for parameters given graph or value of function at a point. Pg 62 #1 – 5, 9, 12 optional Pg 62 #6, 7, 8, 10 mandatory Skills: connect rate of change to slope, calculate and interpret average rates of change from a graph, table of values or an equation. Pg 71 # 1, 2, 7, 8 optional Pg 71 #3, 4, 5, 10, 11 mandatory Skills: describe relationship between slope of secants and the slope of a tangent, estimate instantaneous rate of change of a point from a graph, table of values or an equation, differentiate between the rate of change at a point and the value of the function at that point. The Chapter will wrap up with a review, the Chapter 1 Task and a written test. Class discussions, class work, homework and quizzes will help you to determine how well you understand the course material in preparation for the summative assessments. Extra help is available with me upon request. There is free tutoring and instructional videos available on the class moodle.