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 3x 3 – 2x 2 + 6x – 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 x n + a 1 x n-1 +…+a n-1 x + a n – where the coefficients a 0, a 1, a 2,…a n, represent real numbers, a 0 is not zero, and n represents a nonnegative integer.

Polynomials 4x 5 – 4x 4 + 3x 3 – 2x – 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 – -4x 4 + 2x – 7 – Degree = 4, Leading Coefficient = -4 – ½ t 7 – ¼ t 6 + t 5 – t 4 + 3t 3 – 2t 2 + t – 6. – Degree = 7, Leading Coefficient = ½

Determine Attributes of Polynomials x 2 + 3x – ½ Degree = 2, Leading Coefficient = 1 2y – 4 + 6x 3 Not a polynomial in one variable. -4h 3 + 6h – 7h Rewrite as -7h 6 – 4h 3 + 6h + 2 Degree = 6, Leading Coefficient = - 7 z 3 – 3 / z + 7z 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 x n + a 1 x n-1 +…+a n-1 x + a n where the coefficients a 0, a 1, a 2,…, a n represent real numbers, a 0 is not zero, and n represents a nonnegative integer. Examples f(x) = 4x 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) = 3x 4 – 2x 2 + 7, find p(-3) p(-3) = 3(-3) 4 – 2(-3) p(-3) = 3(81) – 2 (9) + 7 p(-3) = 243 – p(-3) = 232

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

Graphs of polynomial functions Common Graphs constant function degree = 0 linear function degree = 1 quadratic function degree = 2 cubic function, degree = 3 Quartic Function degree = 4 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 Even Leading Coefficient? Negative Degree? 2 End Behavior? Odd Leading Coefficient? Positive Degree? 5 End Behavior?Odd Leading Coefficient? Negative Degree?3 End Behavior?

Homework p. 350, even