Unit 2.1 – Evaluate and graph polynomial functions Math 3 – Coach Bianco

Georgia performance standards MM3A1b - Understand the effects of the following on the graph of a polynomial functions: degree, lead coefficient, and multiplicity of real zeros. MM3A1c – Determine whether a polynomial function has symmetry and whether it is even, odd, or neither. MM3A1d – Investigate and explain characteristics of polynomial functions, including domain and range, intercepts, zeros, relative and absolute extrema, intervals of increase and decrease, and end behavior.

vocabulary A polynomial is a monomial or a sum of monomials. The degree of a polynomial function is the exponent in the term where the variable is raised to the greatest power. The leading coefficient is the coefficient in the term of a polynomial function that has the greatest exponent. A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.

vocabulary Synthetic substitution is another way to evaluate a polynomial function, involving fewer operations than direct substitution. The end behavior of a polynomial function’s graph is the behavior of the graph as x approaches infinity ( + ∞) or negative infinity ( - ∞). End Behavior Rules: positive coefficient, up on right end. negative coefficient, down on right end. even power, left end same as right end. odd power, left end opposite from right end.

Review: Domain & range Domain: Possible “x” values of a function Range: Possible “y” values of a function Hint: Think about the alphabet (D before R and X before Y)

Leading coefficient and degree Put in standard form and identify the degree: 1.) -6x + 4x2 – 8x5 + x7 2.) 3x4-2x+7x4-3x8

Identify polynomial functions Decide whether the function is a polynomial function. If so, write it in standard form and state its degree and leading coefficient. 1.) g(x) = -5x + 6x-2 2.) t(x) = − 𝟓𝒙 + 2

Answer 1.) The function g(x) is not a polynomial function because the term 6x-2 has an exponent that is not a whole number. 2.) the function t(x) is a polynomial function written in standard form. It has a degree 1 and a leading coefficient of − 𝟓𝒙

Direct substitution g(x) = -4x3 + 3x2 – 7 when x = -2

Synthetic substitution g(x) = -4x3 + 3x2 – 7 when x = -2

Describing end behavior

Graph and analyze a polynomial function Graph and analyze the function f(x) = -2x4 + 4x2 -2. Graph the function Find the domain and range of the function Describe the degree and leading coefficient of the function Decide whether the function is even, odd, or neither.