Notes - Polynomial Graphs PDF

5 ­ Notes ­ Characteristics of Polynomials and Graphing from Equations b4.notebook August 23, 2023 Warm­Up Determine the leading coefficient, degree, end behavior, max number of...

5 ­ Notes ­ Characteristics of Polynomials and Graphing from Equations b4.notebook August 23, 2023 Warm­Up Determine the leading coefficient, degree, end behavior, max number of U­turns, and y­intercept for the following: 1. 2. 3. Use a graphing calculator to find the characteristics of the following graph. (Domain, range, relative and absolute extrema, intervals of increase and decrease, x and y­intercepts, and end behavior) Characteristics of Polynomial Functions MGSE9­12.F.IF.4: Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities. Sketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior 5 ­ Notes ­ Characteristics of Polynomials and Graphing from Equations b4.notebook August 23, 2023 Vocabulary Multiplicity ­ the number of times that a root occurs at a given point of a polynomial equation Relative Minimum ­ the y­value at the lowest point on the graph of a function in its "neighborhood" Relative Maximum ­ the y­value at the highest point on the graph of a function in its "neighborhood" Absolute Minimum/Maximum ­ the y­value at the highest or lowest point on the entire graph. Defines the range. Zero ­ the x­value at which f (x) = 0. Graphically, the zeros are the x­intercepts What am I learning today? How to determine the characteristics of the graphs of polynomial functions How will I show that I learned it? Sketch the graph of a polynomial function given its equation in factored form 5 ­ Notes ­ Characteristics of Polynomials and Graphing from Equations b4.notebook August 23, 2023 The Cubic Parent Function x f(x)=x3 ­2 ­1 ­.5 0.5 1 2 Characteristics End Behavior: as as Domain: Range: Increase: Decrease: never Minimum: none Maximum: none Zeros: (0, 0) Turning Points A turning point is where a graph changes from increasing to decreasing or from decreasing to increasing. This point will always correspond to a relative minimum or maximum. ** A polynomial function of degree n has at most n x­intercepts (zeros). ** A polynomial function of degree n has at most (n ­ 1) turning points. 5 ­ Notes ­ Characteristics of Polynomials and Graphing from Equations b4.notebook August 23, 2023 Relative Minimum and Maximum Values f(a) is a relative minimum, if, for all x in an open interval that contains a, f(x) f(a) f(a) is a relative maximum, if, for all x in an open interval that contains a, f(x) f(a) Multiplicity If a root’s multiplicity is odd, the graph will cross the x­axis at that value. ­ If it is one, the graph will cross the x­axis in a linear fashion. ­ If it is more than one, it will cross in a cubic fashion. If a root’s multiplicity is even, the graph will touch the x­axis at that value. Example: 4x2(x + 2)(x ­ 5)3 5 ­ Notes ­ Characteristics of Polynomials and Graphing from Equations b4.notebook August 23, 2023 For each of the following polynomials, identify: a) the degree of the polynomial b) the sign of the leading coefficient c) the end behavior d) the zeros and their multiplicity e) y ­ intercept Then sketch the graph. 1. a) degree b) sign of leading coefficient c) end behavior d) zeros and multiplicity e) y­intercept 2. a) degree b) sign of leading coefficient c) end behavior d) zeros and multiplicity e) y­intercept 5 ­ Notes ­ Characteristics of Polynomials and Graphing from Equations b4.notebook August 23, 2023 a) degree b) sign of leading coefficient c) end behavior d) zeros and multiplicity e) y­intercept 4. a) degree b) sign of leading coefficient c) end behavior d) zeros and multiplicity e) y­intercept 5 ­ Notes ­ Characteristics of Polynomials and Graphing from Equations b4.notebook August 23, 2023 5. a) degree b) sign of leading coefficient c) end behavior d) zeros and multiplicity e) y­intercept