Graphing Calculator, Notebook U7D1 Have Out: Pencil, Highlighter, Graphing Calculator, Notebook Bellwork: Answer the following questions in your packet. CF – 1 Solve each of the following for x. a) (x – 1) = 0 b) (x – 1)(x + 1) = 0 c) (x – 1)2 = 0 CF – 2 Sketch a graph of each of the following functions and indicate where each intersects the x–axis. a) y = (x – 1) b) y = (x – 1)(x + 1) c) y = (x – 1)2

Solve each of the following for x. CF – 1 Solve each of the following for x. a) (x – 1) = 0 b) (x – 1)(x + 1) = 0 c) (x – 1)2 = 0 x = 1 x – 1 = 0 x + 1 = 0 x – 1 = 0 x = 1 x = –1 x = 1 CF – 2 Sketch a graph for each of the following functions and indicate where each intersects the x–axis. a) y = (x – 1) b) y = (x – 1)(x + 1) c) y = (x – 1)2 Parabola LINE Parabola 5 –5 x y 5 –5 x y 5 –5 x y (1, 0) (–1, 0) (1, 0) (1, 0)

Definitions: Roots vs. Zeros CF – 3 Definitions: Roots vs. Zeros Roots: solutions to an equation For example, the roots of _________ are ______ and _____. x2 – 9 = 0 x = –3 x = 3 Zeros: the value of x when function f(x) = 0, that is, the locations where f(x) crosses the x–axis For ex., the zeros of ______________ are ______ and _____. f(x) = x2 – 3x – 4 x = –1 x = 4 CF – 4 Find the zeros for each of the following functions. a) y = x2 – 6x + 8 b) y = x2 – 6x + 9 c) y = x3 – 4x 0 = x2 – 6x + 8 0 = x2 – 6x + 9 0 = x3 – 4x 0 = (x – 2)(x – 4) 0 = (x – 3)(x – 3) 0 = x(x2 – 4) 0 = x(x – 2)(x + 2) x – 2 = 0 or x – 4 = 0 x – 3 = 0 x = 0 or x – 2 = 0 or x + 2 = 0 x = 3 x = 2, 4 x = 0, ±2

CF – 5 Using your graphing calculator, sketch the graph for the following functions. a) y = (x – 1)2(x + 1) b) y = (x – 1)2(x + 1)2 c) y = x3 – 4x (–1, 0) (1, 0) (–1, 0) (1, 0) (–2, 0) (2, 0) (0, 0) Parts (a) and (b) have examples of double zeros because the graphs “bounce” off the x–axis at these zeros.

Let’s review the basics: CF – 6 Let’s review the basics: 3x2, y10, –2 Monomials: Expression that is a ___, ______, or any ___________. # Example: _________ variable combination Example: _________ 9x2 + 14 Binomial: _____ terms added or subtracted. two Trinomial: _____ terms added or subtracted. three Example: _________ x2 + 6x + 9 Polynomial: _____ terms added or subtracted. many Example: ____________ x3 – 5x2 + 3x – 1

Polynomials can be written in the form: CF – 6 Polynomials can be written in the form: There are several characteristics of polynomials: coefficients  The ____________ are _______ numbers. real  The _______ _______, a0, is not _______. first number zero  The _______ or ________ are __________ _________. powers exponents positive integers

The highest power of x in any of these terms is called the _________. CF – 6 Essentially, a polynomial can be written as the sum of the terms in the form: positive whole number ___________________ ________________  x any number Ex: x3 – 5x2 + 3x – 1 The highest power of x in any of these terms is called the _________. degree Ex: x3 – 5x2 + 3x – 1 The degree is 3. leading coefficient The _______ __________ is the coefficient of the term with the highest degree. Ex: x3 – 5x2 + 3x – 1 The leading coefficient is 1. descending We always write polynomials in ____________ order. Ex: –7x5 – 2x4 + 19x3 – x2 + 8x – 10 In the above example, the degree is ____ and the leading coefficient is ___. 5 –7

Examples: a) degree = 5 leading coefficient = 8 b) degree = 6 The following are examples of polynomial functions. Identify the degree of the polynomial and the leading coefficient. Examples: a) degree = 5 leading coefficient = 8 b) degree = 6 leading coefficient = c) degree = 3 leading coefficient = 7 Short cut: just count the number of factors with an “x”.

Examples: The following are NOT polynomial functions. Explain why not. The exponent is a variable, but it’s supposed to be a whole number. can be written as , which is a fractional exponent. b) can be written as c) Polynomials cannot have negative exponents.

Using a graphing calculator, work on the worksheet titled: “Graphs of Polynomial Functions”

Graphs of Polynomial Functions Using your graphing calculator, sketch the graph of the following functions. a) b) c) Degree: Zeros: Degree: Zeros: 1 Degree: Zeros: 2 none x = ± 2

Graphs of Polynomial Functions Using your graphing calculator, sketch the graph of the following functions. d) e) f) Degree: Zeros: 2 Degree: Zeros: 3 Degree: Zeros: 4 x = –3 x = –4,–1,0,3 x = ± 2, 3

Graphs of Polynomial Functions Using your graphing calculator, sketch the graph of the following functions. g) h) i) Degree: Zeros: 3 Degree: Zeros: 4 Degree: Zeros: 3 x = –3,–1, 3 x = 0, 2 x = –4, –1, 2

a) How does the degree compare to the maximum number of real zeros? The maximum number of zeros equals the degree. b) What do you notice about the shapes of the graphs for even–degree polynomial functions and odd–degree polynomial functions? Anything even degree is similar to a parabola’s ends. Anything odd degree is similar to cubic’s ends. c) The _______ and ___________________ determine the graph’s ______________. degree leading coefficients end behavior

Even degree + leading coefficient Even degree – leading coefficient Odd degree + leading coefficient Odd degree – leading coefficient

Do not move until everyone’s name is called. It’s time to Change Seats! Do not move until everyone’s name is called.