Section 5.1 Polynomial Functions and Models Objectives: Identify polynomial functions and their degree; graph using transformations; identify zeros and their multiplicity; analyze the graph of a polynomial function. A polynomial function is a sum of terms Ax n where A are real numbers and n is a nonnegative integer. The domain is the set of all real numbers. The degree is the largest power of x. The zero polynomial function f x 0 is not assigned a degree. The constant polynomial function f x b has degree 0 since bx 0 b 1 b . Work #1 – 4 A power function of degree n is a monomial of the form f x ax n where n > 0 and n is an integer. For example, given the polynomial f x 6 x 4 3 x 3 7 , the power function is f x 6x 4 . 5.1 - 2 Graphing Polynomials We will use the end behavior of the power function to determine what happens to a polynomial graph as x becomes very large or very small. If the degree of the polynomial is odd, the ends will point in opposite directions. o Positive leading coefficient: down on left / up on right o Negative leading coefficient: up on left / down on right If the degree of the polynomial is even, the ends will point in the same direction. o Positive leading coefficient: up on left / up on right o Negative leading coefficient: down left / down right The graph of every polynomial function is both smooth (no sharp corners) and continuous (no gaps or holes). If f is a function and r is a real number for which f r 0 , then r is called a real zero of f. Therefore, the following are equivalent: r is a real zero of a polynomial function f (real roots) r is an x-intercept of the graph of f x r is a factor of f Work #5 – 7 5.1 - 3 Zeros and Their Multiplicities If f is completely factored with factors x r m then r is a multiple root that occurs m times. For example, in the polynomial 0 x 54 x 63 x 3 , 5 is a zero with multiplicity 4, 6 is a zero with multiplicity 3, and 3 is a zero with multiplicity 1. If m is odd the graph of f crosses the x-axis at r. If m is even the graph of f touches the x-axis at r. The larger the value of m, the flatter (more compressed) the graph becomes near r. Look for the degree of the polynomial—that is the most real zeros you will have. Work #8 – 11 Behavior Near a Zero To determine behavior near a zero, keep the factor that gave rise to the zero, and let x r in the remaining factors. Turning Points The points where a graph changes direction are called turning points – these are the local maximums and local minimums. If f is a polynomial function of degree n, then f has at most n – 1 turning points. Work #12 - 15