ch3_ref_notes

# ch3_ref_notes - x = c Depending on the degree of the...

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The end behavior of n th degree polynomial function p ( x ) depends on whether n is even or odd and whether a n is positive or negative. n a n > 0 a n < 0 x → ∞ , p ( x ) → ∞ x → ∞ , p ( x ) → -∞ EVEN x → -∞ , p ( x ) → ∞ x → -∞ , p ( x ) → -∞ x → ∞ , p ( x ) → ∞ x → ∞ , p ( x ) → -∞ ODD x → -∞ , p ( x ) → -∞ x → -∞ , p ( x ) → ∞ Let a 6 = 0 and the c i are real numbers and the n i be positive integers. Then the graph of the polynomial p ( x ) = a ( x - c 1 ) n 1 ( x - c 2 ) n 2 ··· ( x - c k ) n k , (1) crosses the x-axis at x = c i if n i is odd . (2) tangent (touches) to the x-axis at x = c i if n i is even . (3) has convexity change at x = c i if n i 3 and n i is odd If a polynomial p ( x ) has a factor of the form ( x - c ) k , where k > 1, then x = c is a repeated root of p ( x ) of multiplicity k . (1) If k is even, the graph is tangent to the x-axis (or touches the x-axis) at x = c (2) If k is odd, the graph has a convexity change (the graph ﬂattens out and crosses the x-axis) at
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Unformatted text preview: x = c . Depending on the degree of the numerator and the denominator, it is possible to have horizontal or slant/oblique asymptotes . If f ( x ) = p ( x ) q ( x ) = a n x n + a n-1 x n-1 + ··· + a 1 x + a b m x m + b m-1 x m-1 + ··· + b 1 x + b , then (1) If n < m then the x-axis ( y = 0) is a horizontal asymptote. (2) If n = m then y = a n b n is horizontal asymptote. (3) If n = m + 1 then ∃ a slant (oblique) asymptote ( which is found by polynomial division). (4) If n > m + 1, there is no horizontal asymptote. 1...
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