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T05_Polynomials_and_Rationals

# T05_Polynomials_and_Rationals - Mathematics Department(MD...

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PMC 0045 Pre-Calculus for Engineering Foundation in Engineering ONLINE NOTES Topic 5 Polynomial and Rational Functions FOSEE , MULTIMEDIA UNIVERSITY (436821-T) MELAKA CAMPUS, JALAN AYER KEROH LAMA, 75450 MELAKA, MALAYSIA. Tel 606 252 3594 Fax 606 231 8799 URL: http://fosee.mmu.edu.my/~asd/ Mathematics Department (MD) Centre for Foundation Studies and Extension Education (FOSEE)

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PMC0045 Pre-Calculus for Engineering Topic 5 TOPIC 5: POLYNOMIAL AND RATIONAL FUNCTIONS References : 1. Sullivan, M. (2002), Algebra & Trigonometry , 6 th Ed., Prentice Hall. 2. Blitzer, R., Goh Wei Wei, et.al. (2005), Algebra & Trigonometry , Prentice Hall. Objectives: 1. Identify the properties of polynomial functions. 2. Graph the polynomial functions using the graphing techniques. 3. Know how to graph rational functions by using method of shifting and reflections. 4. Know and understand the Remainder and Factor Theorems also the Synthetic Division. Contents: 5.1 Quadratic Functions 5.2 Polynomial Functions and Their Graphs 5.3 Dividing Polynomials; Remainder and Factor Theorems 5.4 Rational Functions and Their Graphs 5.1 QUADRATIC FUNCTIONS A quadratic function is a function of the form: f ( x ) = ax 2 + bx + c , where a , b and c are real numbers and 0 a Graphing a Quadratic Function . Characteristics of the graph of a Quadratic Function c bx ax x f + + = 2 ) ( . 1. Vertex =  - - a b f a b 2 , 2 2. Axis of symmetry: The line a b x 2 - = 3. Parabola opens up if a > 0, f min =  - a b f 2 4. Parabola opens down if a < 0, f max =  - a b f 2 The x -intercepts of a Quadratic Function. b 2 - 4 ac No.of distinct x -intercepts > 0 2 = 0 1 (at its vertex) < 0 0 (did not cross / touch x -axis) ____________________________________________________________________________________ Mathematics Dept 2/10
PMC0045 Pre-Calculus for Engineering Topic 5 The y -intercept of a Quadratic Functions - the value of f ( x ) when x = 0 Example: Graph f ( x ) = -3 x 2 + 6 x +1 Solution: We have a = -3, b = 6, c = 1. Since a = -3 < 0, the parabola opens down. For vertex, x -coordinate = a b 2 - = 6 6 - - = 1 ; y -coordinate = f (1) = 4 So, the vertex is (1,4) and f has a maximum value of 4. Axis of symmetry: x = 1 x -intercept: b 2 - 4 ac = 6 2 - 4(-3)(1) = 36 + 12 = 48 >0 ( 2 x -intercepts ) x = a ac b b 2 4 2 - ± - x = -0.15 and x = 2.15 y -intercept : f (0) = 1 ie. The intercepts are (-0.15,0), (2.15,0), (0,1) Note: 1. By using the axis of symmetry and y -intercept, we can locate point (2,1). 2. We must plot at least 3 points on the graph. ----------------------------------------------------------------------------------------------------------- - 5.2 POLYNOMIAL FUNCTIONS AND THEIR GRAPHS Definition: A polynomial function is a function of the form: 0 1 1 1 ...... ) ( a x a x a x a x f n n n n + + + + = - - where a n , a n-1 ,….., a 1 , a 0 are real numbers and n is nonnegative integers. Note: 1) D f = ) , ( -∞ ____________________________________________________________________________________ Mathematics Dept 3/10

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PMC0045 Pre-Calculus for Engineering Topic 5 2) Degree of a polynomial function = degree of the polynomial in 1 variable.
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