1. 8-1 A TTRIBUTES of Polynomial F UNCTIONS Objectives: To classify polynomials, graph polynomial functions, and describe end behavior. Classifying Polynomials Monomial – a real number, a variable, or a product of a real number and one or more variables with whole-number exponents Polynomial – a monomial or a sum of monomials Degree of a polynomial – for a polynomial in one variable, the greatest degree of the monomial terms Degree of a monomial – the sum of the exponents of the variables

2. 8-1 A TTRIBUTES of Polynomial F UNCTIONS Objectives: To classify polynomials, graph polynomial functions, and describe end behavior. Polynomial function – A polynomial in the variable x defines a polynomial function Polynomial function – A polynomial in the variable x defines a polynomial function

3. 8-1 A TTRIBUTES of Polynomial F UNCTIONS Objectives: To classify polynomials, graph polynomial functions, and describe end behavior. Write each polynomial in standard form. What is the classification of each by degree? By number of terms? quartic trinomial quintic trinomial

4. 8-1 A TTRIBUTES of Polynomial F UNCTIONS Objectives: To classify polynomials, graph polynomial functions, and describe end behavior. End behavior – the direction of the graph to the far left and to the far right Turning point – a point where the graph changes direction from upwards to downwards or from downwards to upwards

5. 8-1 A TTRIBUTES of Polynomial F UNCTIONS Objectives: To classify polynomials, graph polynomial functions, and describe end behavior. Polynomial Functions End Behavior: Up and Up Turning Points: (-1.07, -1.04), (-0.27, 0.17), and (0.22, -0.15) The function is decreasing when x < -1.07 and -0.27 < x < 0.22. The function increases when -1.07 0.22. End Behavior: Down and Down Turning Point: (1, 1) The function is increasing when x < 1 and is decreasing when x > 1.

6. 8-1 A TTRIBUTES of Polynomial F UNCTIONS Objectives: To classify polynomials, graph polynomial functions, and describe end behavior. Determining End Behavior the graph of a polynomial function of odd degree has an even number of turning points. the graph of a polynomial function of even degree has an odd number of turning points.

7. 8-1 A TTRIBUTES of Polynomial F UNCTIONS Objectives: To classify polynomials, graph polynomial functions, and describe end behavior. Consider the leading term of y = -4x 3 + 2x 2 + 7. What is the end behavior of the graph? Recall: up and down

8. 8-1 A TTRIBUTES of Polynomial F UNCTIONS Objectives: To classify polynomials, graph polynomial functions, and describe end behavior. What is the graph of each cubic function? Describe the graph. end behavior: up and down end behavior: up and down end behavior: down and up end behavior: down and up no turning points

9. 8-1 A TTRIBUTES of Polynomial F UNCTIONS Objectives: To classify polynomials, graph polynomial functions, and describe end behavior. For 2nd differences, subtract consecutive 1st differences. For 3rd differences, subtract consecutive 2nd differences. If the pattern suggested by the 3rd differences continues, what is the 8th number in the first column? Justify your reasoning. The first column shows a sequence of numbers. For 1st differences, subtract consecutive numbers in the sequence: -6 - (-4) = -2, 4 - (-6) = 10, and so on. 106 190 36 60 84 24 108 298 644 24 132 430 1074 Since the 3 rd difference is constant at 24, work backwards to find the 2 nd and first differences, and eventually to the 7 th and 8 th numbers in the first column.

10. 8-1 A TTRIBUTES of Polynomial F UNCTIONS Objectives: To classify polynomials, graph polynomial functions, and describe end behavior. a. What is the degree of the polynomial function that generates the data shown below? b. What is an example of a polynomial function whose fifth differences are constant but whose fourth differences are not constant? 4 4

11. 8-1 A TTRIBUTES of Polynomial F UNCTIONS Objectives: To classify polynomials, graph polynomial functions, and describe end behavior. Do you know HOW? cubic binomial 2. Write the equation for the volume of a box with a length that is 4 less than the width and a height that is twice the width. Graph the equation. Let y = volume of box x = width of box Let y = volume of box x = width of box Volume = length x width x height

12. 8-1 A TTRIBUTES of Polynomial F UNCTIONS Objectives: To classify polynomials, graph polynomial functions, and describe end behavior. 3. A polynomial function has three turning points. What are the possible degrees of the polynomial? Do you know HOW? Recall : 1) A polynomial function of degree n will have at most n -1 turning points. 2) The graph of a polynomial function of even degree has an odd # of turning points. End behavior is up and down.