MS - 12ZZ 745231 The Woodlands SS - 1.1.1 Defining Polynomial Functions.pdf

1.1.1 POLYNOMIAL FUNCTIONS – Defining a Polynomial Function McGraw-Hill Ryerson Advanced Functions 12 Textbook MHR KEY Corrections 1.1 Power Functions pp 4-14 1.2 Characteristics of Polynomial Functions pp 15-29 Nelson Advanced Fu...

1.1.1 POLYNOMIAL FUNCTIONS – Defining a Polynomial Function McGraw-Hill Ryerson Advanced Functions 12 Textbook MHR KEY Corrections 1.1 Power Functions pp 4-14 1.2 Characteristics of Polynomial Functions pp 15-29 Nelson Advanced Functions 12 Textbook Nelson KEY Corrections 1.1 Functions pp 4-13 #1 – 3, 5, 6, 7b-f, 9, 10, 12 1.2 Exploring Absolute Value pp 14-16 1.3 Properties of Graphs of Functions pp 17-25 3.1 Exploring Polynomial Functions pp 124-128 3.2 Characteristics of Polynomial Functions pp 129-138 CEMC Exercises: Relations and Functions CEMC KEY 1 CEMC KEY 2 1. What is a Function? A function is a relation where the independent “input” variable (x) corresponds to only one value of the dependent “output” variable (y) and can be represented numerically, algebraically, or graphically. Therefore a Vertical Line Test can be used to test whether a graph represents a function where drawing a vertical line can only intersect the graph at one point. Numerically: Set of Ordered Pairs listed in Set Notation, Table of Values, Mapping Diagram Let A = X Y { (a,x) , a x (b,x) , b x (c,z) } c z Let B = X Y { (a,x) , a x (a,z) , a z (b,y) } b y X Y Let C = a y { (a,y) , b y (b,y) , (b,z) , b z (c,x) , (c,y) } c x c y  Algebraically or Graphically: 2 2 1 𝑦 =𝑥 𝑦 = 𝑥 𝑦= 𝑥 𝑦= 𝑥 2 2 𝑓(𝑥) = 5 𝑠𝑖𝑛 (2𝑥) + 3 𝑦 = |𝑥| 𝑥 +𝑦 = 9 𝑥=𝑎 2. Polynomial Function expressed as a Polynomial Expression has the Standard Form: n is a whole number {0, 1, 2, 3, 4, 5, etc}. x is the variable represented by an alphabetic letter. xn is the degree of the function because n is the exponent of the greatest power of x. a0 , … , an are real number coefficients. an is the leading coefficient because it is the coefficient of the greatest power of x. a0 is the constant coefficient since n is zero and thus the term is without the variable x. written in descending order of powers of x, but may be missing because some terms may have 2 3 zero as a coefficient, for example, 0𝑥 such as in the polynomial function: 𝑓(𝑥) = 4𝑥 + 2𝑥 − 1 domain of a polynomial function is the set of real numbers where 𝐷 = {𝑥 | 𝑥 ϵ 𝑅} range of a polynomial function may be all numbers with no bounds where 𝑅 = {𝑦 | 𝑦 ϵ 𝑅}, or it may have a lower bound, or it may have an upper bound, but not both graphs of polynomial functions do not have horizontal or vertical asymptotes 4 3 CLASS EXERCISE: What is the degree, leading coefficient, and constant of 𝑦 =− 5𝑥 + 𝑥 + 1? Degree = 4, Leading Coefficient = 5, Constant = 1  Some types of Relations that are Functions but are NOT Polynomial Functions: Trigonometric Function Exponential Function Logarithmic Function 𝑥 𝑓(𝑥) = 𝑡𝑎𝑛 𝑥 𝑔(𝑥) = 𝑎 ℎ(𝑥) = 𝑙𝑜𝑔𝑎 𝑥 CLASS EXERCISE: Which of the following is a polynomial function? 1 −1 y = 10x y = 5𝑥 2 f(x) = 3 x f(x) = x-2 3. Review Terminologies for the Characteristics (Key Features) of Graphs of Polynomials: a) Continuous Functions: any function that does not contain any holes or breaks over its entire domain Constant Linear Quadratic Cubic Absolute Value b) Discontinuous Functions: any function that does contain a hole or break which can occur anywhere on its domain Domain: 𝑥 ϵ (− ∞, 2) ∪ (2, + ∞) Domain: 𝑥 ϵ (− ∞, 3] ∪ (3, + ∞) Domain: 𝑥 ϵ (− ∞, 3) ∪ (3, + ∞)  Range: 𝑦 ϵ (− ∞, 7) ∪ (7, + ∞) Range: 𝑦 ϵ (− ∞, 1] ∪ (4, + ∞) Range: 𝑦 ϵ (− ∞, 0) ∪ (0, + ∞) c) Asymptote: a vertical, horizontal or oblique (also called diagonal or slant) line that a curve approaches more and more closely but does not intersect d) Intercepts: X-intercept Y-intercept the distance from the origin the distance from the origin of the point where a line or of the point where a line or curve crosses the x-axis curve crosses the y-axis occurs when y = 0 as an occurs when x = 0 as an point is (x, 0) point is (0, y) e) Turning Points: a turning point on a graph is where the slope of the function changes from an interval of decrease to interval of increase (minimum point) or from an interval of increase to interval of decrease (maximum point) Interval of Decrease: interval(s) within a function’s domain where the y-values of the function get smaller when moving from left to right on the graph Interval of Increase: interval(s) within a function’s domain where the y-values of the function get larger when moving from left to right on the graph f) Inflection Points: note that inflection points are NOT the same as turning points an inflection point on a graph is where the concavity of the function changes from concave down to concave up or from concave up to concave down Concave Down: a line segment joining any two points on a curve is entirely below the curve Concave Up: a line segment joining any two points on a curve is entirely above the curve g) Extrema Points: Absolute or Global Maxima Absolute or Global Minima the one highest point over the entire the one lowest point over the entire domain of a function domain of a function an unbound polynomial function an unbound polynomial function 𝐷 = {𝑥 ϵ 𝑅 | − ∞ < 𝑥