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pre-calc section 2.3 & 2.4

pre-calc section 2.3 & 2.4 - Section 2.3 Polynomial...

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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 0 n a . The function defined by 0 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 : 0 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. For n (degree) even: 1.) If the leading coefficient is positive ( n a > 0), the graph rises to the left and right.
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