Advanced Algebra Notes Section 5.2: Evaluate and Graph Polynomial Functions A __________________ is a number, a variable, or the product of numbers and variables. A _________________________ is a monomial or the sum of monomials. A _________________________________________ is a function of the following form: where, the exponents must all be whole numbers, and the coefficients are all real numbers. For this function, _______ is the leading coefficient, _______ is the degree, and _______ is the constant term. A polynomial is said to be in ____________________________ if its terms are written in descending order of exponents from left to right. term polynomial Polynomial function standard form

Common Polynomial Functions Degree Type Standard Form Example 0 Constant f(x) = f(x) = -8 1 Linear f(x) = f(x) = 3x Quadratic f(x) = f(x) = 5x 2 – x Cubic f(x) = f(x) = 2x 3 – x 2 + 4x Quartic f(x) = F(x) = x 4 -6x 2 + 2x - 7 Examples: Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient This is not a function The exponents must be whole numbers This is a function

3. f(x) = 4. Example: Use direct substitution to evaluate the function. 5. f(x) = -3x 3 + x 2 -12x -5, when x = 2 This is a function This is NOT a function

Another way to evaluate a polynomial function is to use ___________________________. This method involves fewer operations and you don’t have to work with the variables. Example: Use synthetic substitution to evaluate the function. 6. f(x) = -2x 4 –x 3 + 4x -5 ; x = -1 The _______________________________ of a function’s graph is the behavior of the graph as x approaches positive infinity (+) or negative infinity ( - ). For the graph of a polynomial function, the end behavior is determined by the function’s ________________ and the ______________ of the leading coefficient. synthetic substitution end behavior degree sign -2 (-1) f(-1)=-10 You will get the same answer if you use direct or synthetic substitution.

End Behavior of Polynomial Functions Degree Leading Coefficient Left Side of Graph Right Side of Graph Odd Positive as Odd Negative as Even Positive as Even Negative as Example: Describe the degree and leading coefficient of the polynomial function whose graph is shown. Degree: ___________ 7. Leading Coefficient: ________________ odd negative

To graph a polynomial function, first plot points to determine the shape of the graph’s middle portion. Then use what you know about end behavior to sketch the ends of the graph. Example: Graph polynomial functions. 8. f(x) = -x 4 + 4x 3 – x f(x) = x 3 – 3x 2 + x Find all Zeros 2. Find all Turning Points 3. Use the table to find as many points as possible Turning PointsZeros