Polynomial Behavior patterns in the graphs

Warm Up List and name the transformations in this diagram

Standard Form of Polynomial

DegreeName using degree Polynomial example Number of terms Name using # terms X x x 3 - 3x 2 + x3 42x 4 + 5x x 5 +4x 2 +2x +14

Lesson 4.4Graphing Polynomial Functions Key Terms: Relative Max / Min Real Zeros (x-intercepts) Example 1Graphing Polynomials A)B) Zeros: 1, -1.4, -3.6 R. Min: (-2.7, -4.5) R. Max: (0, 5)

Left & Right End Behavior Degree 0Degree 2Degree 3Degree 4Degree 1 Degree 2Degree 3Degree 4Degree 1 Negative Polynomials Positive Polynomials

Studying Graphs of Polynomials Odd Degree 1 Positive LC Even Degree 2 Positive LC Odd Degree 3 Positive LC Even Degree 4 Positive LC Odd Degree 5 Positive LC Odd Degree 5 Negative LC Even Degree 4 Negative LC Key Notes: [1] Odd when rt & lf ends go opposite directions [2] Even when rt & lf ends go in same direction [3] Positive right end goes up [4] Negative right end goes down [5] Degree is one more than the number of turns

Example 3Graphs of Polynomial Functions Determine if the polynomial is even/odd, Positive / Negative, find the Degree, and state the number of Real Zeros / Imaginary. a) Even Negative Degree 2 2 Real Zeros b) Odd Positive Degree 5 1 Real / 4 Imaginary Zeros

Determine if the polynomial is even/odd, positive/negative, and give the degree. c) d) Even Negative Degree 4 Odd Positive Degree 5

Power Functions are any function of the form f(x) = ax n where a and n are nonzero constant real numbers n is the exponent; 1.If If n = positive integer (linear, quadratic, cubic, etc) 2.If n = negative integer (rational function) 3.If n = fraction (square root function)

Power Functions are any function of the form f(x) = ax n If If n = positive integer (linear, quadratic, cubic, etc) f(x) = ax 3 f(x) = ax 4

Power Functions are any function of the form f(x) = ax n If n = negative integer (rational function) f(x) = ax -1 = 1/x

Power Functions are any function of the form f(x) = ax n 1.If n = fraction (square root function) f(x) = ax 1/2

Homework a)b) Zeros: c) Rel. Max: Rel. Min: – 4, 0 (0, 0) (-2.7, -9.5)

Warm-Up [1] State the degree and LC: [2], find Degree 4 LC = 5

Example 2Polynomial Applications A)Suppose you have a 12 x 14 sheet of cardboard. You plan to cut a uniform corner from each corner and fold the sheet into an open box. - What is the maximum volume? - What are the dimensions of the maximized box? B)Use a 10 x 12 sheet of cardboard 10 – 2x 12 – 2xx Max V: units 3 Dimensions:1.8 x 6.4 x units x 7.7 x 9.7