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Unformatted text preview: Section 2.3 Polynomial Functions and Their Graphs Definition of a Polynomial Function: Let n be a nonnegative integer (no fractions, no variables in the denominator) and let be real numbers with n a . The function defined by 1 2 2 1 1 ) ( a x a x a x a x a x f n n n n + + + + + =-- is called a polynomial function of degree n . -- The number , the coefficient of the variable to the highest power, is called the leading coefficient . Ex. 5 2 3 4 4 2- +- x x x is a polynomial of degree 4 and has a leading coefficient of -3 Examples of expressions that are not polynomials: 2 4 2-- x , x x 7 5 3 1 + , 3 1 + x Smooth and Continuous Curves smooth the graph contains only rounded curves with no sharp corners continuous the graph has no breaks and can be drawn without lifting your pencil End Behavior- What happens at the left and right ends of the graph? Goes up? Or down? Use the Leading Coefficient Test : 1 2 2 1 1 ) ( a x a x a x a x a x f n n n n + + + + + =-- As x increases (to the right) or decreases (to the left) without bound, the graph of a polynomial function eventually rises or falls. For n (degree) odd: 1.) If the leading coefficient is positive ( n a > 0), the graph falls to the left and rises to the right. 2.) If the leading coefficient is negative ( n a < 0), the graph rises to the left and falls to the right....
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