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PRECALCULUS I Quadratic Functions Dr. Claude S. Moore Danville Community College 1 Polynomial Function A polynomial function of degree n is n f ( x) an x an 1 x n 1 a0 where the a’s are real numbers and the n’s are nonnegative integers and an 0. Quadratic Function A polynomial function of degree 2 is called a quadratic function. It is of the form 2 f ( x) ax bx c a, b, and c are real numbers and a 0. Axis of Symmetry For a quadratic function of the form 2 f ( x) ax bx c b x 2a gives the axis of symmetry. Standard Form A quadratic function of the form 2 f ( x) a( x h) k , a 0 is in standard form. axis of symmetry: x = h vertex: (h, k) Characteristics of Parabola a>0 axis of symmetry vertex: maximum vertex: minimum axis of symmetry a<0 PRECALCULUS I Higher Degree Polynomial Functions Dr. Claude S. Moore Danville Community College 7 Characteristics The graph of a polynomial function… 1. Is continuous. 2. Has smooth, rounded turns. 3. For n even, both sides go same way. 4. For n odd, sides go opposite way. 5. For a > 0, right side goes up. 6. For a < 0, right side goes down. Leading Coefficient Test: n odd n f ( x) an x an 1 x n 1 a0 graphs of a polynomial function for n odd: . an > 0 an < 0 f ( x) f ( x) as x as x f ( x) f ( x) as x as x Leading Coefficient Test: n even n f ( x) an x an 1 x n 1 a0 graphs of a polynomial function for n even: . an < 0 an > 0 f ( x) f ( x) as x as x f ( x) f ( x) as x as x Roots, Zeros, Solutions The following statements are equivalent for real number a and polynomial function f : 1. x = a is root or zero of f. 2. x = a is solution of f (x) = 0. 3. (x - a) is factor of f (x). 4. (a, 0) is x-intercept of graph of f (x). Repeated Roots (Zeros) 1. If a polynomial function contains k a factor (x - a) , then x = a is a repeated root of multiplicity k. 2. If k is even, the graph touches (not crosses) the x-axis at x = a. 3. If k is odd, the graph crosses the x-axis at x = a. Intermediate Value Theorem If a < b are two real numbers and f (x)is a polynomial function with f (a) f (b), then f (x) takes on every real number value between f (a) and f (b) for a x b. NOTE to Intermediate Value Let f (x) be a polynomial function and a < b be two real numbers. If f (a) and f (b) have opposite signs (one positive and one negative), then f (x) = 0 for a < x < b. PRECALCULUS I Polynomial and Synthetic Division Dr. Claude S. Moore Danville Community College 15 Full Division Algorithm If f (x) and d(x) are polynomials with d(x) 0 and the degree of d(x) is less than or equal to the degree of f(x), then q(x) and r (x) are unique polynomials such that f (x) = d(x) ·q(x) + r (x) where r (x) = 0 or has a degree less than d(x). Short Division Algorithm f (x) = d(x) ·q(x) + r (x) dividend quotient divisor remainder where r (x) = 0 or has a degree less than d(x). Synthetic Division 3 ax + 2 bx k a + cx + d divided by x - k b c d ka a coefficients of quotient r remainder 1. Copy leading coefficient. 2. Multiply diagonally. 3. Add vertically. Remainder Theorem If a polynomial f (x) is divided by x - k, the remainder is r = f (k). Factor Theorem A polynomial f (x) has a factor (x - k) if and only if f (k) = 0. PRECALCULUS I Real Zeros of Polynomial Functions Dr. Claude S. Moore Danville Community College 21 Descartes’s Rule of Signs n f ( x) an x an 1 x n 1 a0 a’s are real numbers, an 0, and a0 0. 1. Number of positive real zeros of f equals number of variations in sign of f(x), or less than that number by an even integer. 2. Number of negative real zeros of f equals number of variations in sign of f(-x), or less than that number by an even integer. Example 1: Descartes’s Rule of Signs 3 2 f ( x) 4 x 5 x 6 a’s are real numbers, an 0, and a0 0. 1. f(x) has two change-of-signs; thus, f(x) has two or zero positive real roots. 2. f(-x) = -4x3 5x2 + 6 has one change-ofsigns; thus, f(x) has one negative real root. Example 2: Descartes’s Rule of Signs 3 2 f ( x) 4 x 5 x 6 x Factor out x; f(x) = x(4x2 5x + 6) = xg(x) 1. g(x) has two change-of-signs; thus, g(x) has two or zero positive real roots. 2. g(-x) = 4x2 + 5x + 6 has zero change-ofsigns; thus, g(x) has no negative real root. Rational Zero Test n f ( x) an x an 1 x n 1 a0 If a’s are integers, every rational zero of f has the form rational zero = p/q, in reduced form, and p and q are factors of a0 and an, respectively. Example 3: Rational Zero Test 3 2 f(x) = 4x 5x + 6 p {1, 2, 3, 6} q {1, 2, 4} p/q {1, 2, 3, 6, 1/2, 1/4, 3/2, 3/4} represents all possible rational roots of f(x) = 4x3 5x2 + 6 . Upper and Lower Bound f(x) is a polynomial with real coefficients and an > 0 with f(x) (x - c), using synthetic division: 1. If c > 0 and each # in last row is either positive or zero, c is an upper bound. 2. If c < 0 and the #’s in the last row alternate positive and negative, c is an lower bound. Example 4: Upper and Lower Bound 3 2x -3 12x + 8 divided by x + 3 2 -3 -12 8 2 3x -6 27 -45 2 -9 15 -37 c = -3 < 0 and #’s in last row alternate positive/negative. Thus, x = -3 is a lower bound to real roots.