7 1 Polynomial Functions Objectives 1 Evaluate polynomial

7. 1 Polynomial Functions Objectives: 1. Evaluate polynomial functions. 2. Identify general shapes of graphs of polynomial function.

Polynomials • Polynomials – a monomial or sum of monomials • 3 x 3 – 2 x 2 + 6 x – 7 is a polynomial in one variable, since it only contains the variable x. • Polynomial in One Variable – A polynomial of degree n in one variable is an expression of the form • a 0 xn + a 1 xn-1+…+an-1 x + an – where the coefficients a 0, a 1, a 2, …an, represent real numbers, a 0 is not zero, and n represents a nonnegative integer.

Polynomials • 4 x 5 – 4 x 4 + 3 x 3 – 2 x 2 + 4 – a 0 = 4, a 1 = - 4, a 2 = 3, a 3 = -2, a 4 = 0, a 5 = 4 • The degree of a polynomial in one variable is the greatest exponent of its variable. • The leading coefficient of a polynomial is the coefficient of the term with the highest degree. Find the degree and leading coefficient – -4 x 4 + 2 x – 7 – Degree = 4, Leading Coefficient = -4 – ½ t 7 – ¼ t 6 + t 5 – t 4 + 3 t 3 – 2 t 2 + t – 6. – Degree = 7, Leading Coefficient = ½

Determine Attributes of Polynomials • x 2 + 3 x – ½ Degree = 2, Leading Coefficient = 1 • 2 y – 4 + 6 x 3 Not a polynomial in one variable. • -4 h 3 + 6 h – 7 h 6 + 2 Rewrite as -7 h 6 – 4 h 3 + 6 h + 2 Degree = 6, Leading Coefficient = - 7 • z 3 – 3/z + 7 z 2 – 2 Not a polynomial, 3/z is not a monomial

Polynomial Functions • A polynomial function of degree n can be described by an equation of the form P(x) = a 0 xn + a 1 xn-1 +…+an-1 x + an where the coefficients a 0, a 1, a 2, …, an represent real numbers, a 0 is not zero, and n represents a nonnegative integer. • Examples f(x) = 4 x 4 – ½ x 3 + x 2 – x + 4 n = 4, a 0 = 4, a 1 = - ½ , a 2 = 1, a 3 = -1 , a 4 = 4

Evaluating Polynomials • Given p(x) = 3 x 4 – 2 x 2 + 7, find p(-3) = 3(-3)4 – 2(-3)2 + 7 p(-3) = 3(81) – 2 (9) + 7 p(-3) = 243 – 18 + 7 p(-3) = 232

Find functional values of variables • Given f(x) = -3 x 4 + ½ x 3 – 4 x 2 + x, find f(a) = -3(a)4 + ½ (a)3 – 4(a)2 + a f(a) = -3 a 4 + ½ a 3 – 4 a 2 + a • Given p(y) = y 3 – 2 y, find p(t + 1) = (t + 1)3 – 2(t + 1) p(t + 1) = (t + 1)(t + 1) – 2 t – 2 p(t + 1) = (t 2 + 2 t + 1)(t + 1) – 2 t – 2 p(t + 1) = t 3 + 2 t 2 + 2 t + 1 – 2 t – 2 p(t + 1) = t 3 + 3 t 2 + t – 1

Graphs of polynomial functions Common Graphs constant function degree = 0 cubic function, degree = 3 linear function degree = 1 Quartic Function degree = 4 quadratic function degree = 2 Quintic Function, degree = 5

End Behavior of Graphs • This is the behavior of the graph as x approaches + ∞ or - ∞. (positive and negative infinity) • If the function is EVEN, degree = 2, 4, etc. , the ends of the graph point the same way either up if leading coefficient is > 0, or down if leading coefficient is < 0. • If the function is ODD, degree = 3, 5, etc. , the ends of the graph point in opposite directions either down to the left/up to the right if leading coefficient is > 0 or up on the left/down on the right if leading coefficient is < 0. • You can also tell the degree of the graph by counting how many times the line changes direction.

End Behavior Examples End Behavior? Even Leading Coefficient? Negative Degree? 2 End Behavior? Odd Leading Coefficient? Positive Degree? 5 End Behavior? Odd Leading Coefficient? Negative Degree? 3

Homework p. 350, 16 -44 even