ch01 - Chapter 1 Polynomial Functions Review of...

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Review of Prerequisite Skills 2. g. h. i. 3. c. e. f. 4. y 3 + y 2 – 5 y – 5 = y 2 ( y + 1) – 5 ( y + 1) = ( y + 1)( y 2 – 5) 60 y 2 + 10 y – 120 = 10 (6 y 2 y + 12) = 10 (3 y + 4) (2 y – 3) 5. a. 36 2 25 2 62 5 2 5 2 5 2 12 651 0 1 2651 0 12 4 5 12 16 22 xy u y xyu y xyu y xyu x y –– –– – () = []() [] = + =+ + + 5 u xyz x z xx z z y xz y xzyxzy 222 2 2 2 2 2 + = = + 44 1 21 2 yy z z yz ++ hhh hh h 32 2 2 1 11 +++ + x x 4 2 16 422 + 49 77 2 2 ux y uxy uxy + xn + =++ + 2 9 33 c. e. Section 1.1 Investigation 1: Cubic Functions 2. There can be 1, or 3 real roots of a cubic equation. 3. a. Find the x -intercepts, i.e., the zeros of the function x = 2, x = 2 3, x = 4, y = 24, and the y -intercepts. Since the cubic term has a positive coefficient, start at the lower left, i.e., the third quadrant, crossing the x -axis at –3, then again at 2 and at 4, ending in the upper right of the first quadrant. b. 4. x y y x y 24 x –3 2 4 pp z z py z z z 2 2 1 + = + + 92 1 42 363484363484 14 7 2 z z z z z z xyzxyz z x + + + ( [ + + + y z + 7 yyyyy y y 54 32 1 1 = + + = () ++ Chapter 1: Polynomial Functions 1 Chapter 1 • Polynomial Functions
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5. When the coefficient of x 3 is negative, the graph moves from the second quadrant to the fourth. Investigation 2: Quartic Functions 2. There can be 0, 2, or 4 real roots for a quartic equation. 3. a. Find the x -intercepts at the function, i.e., x – 3, 2, – 1, – 4. Find the y -intercept, i.e., y = –24. Begin in the second quadrant crossing the x -axis at – 4, – 3, – 1, and 2 and end in the first quadrant; draw a smooth curve through intercepts. b. 4. 5. If the coefficient of x 4 is negative, the quartic function is a reflection of quartic with a positive coefficient of x 4 , i.e. the graph moves from the second to the fourth quadrant, passing through the x -axis a maximum of four times. Investigation 3 1. y x –2 1 x y y x y x –4 –3 –2 –1 12 –24 2 Chapter 1: Polynomial Functions yx x =+ () 11 3 yx x 21 2 2. Exercise 1.1 3. a. Also includes the reflections of all these graphs in the x -axis. 4. a. y y x x y x y y yy x x xx y x x y y x y x –1 1
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b. y y x x Section 1.2 Investigation 1: Cubic Functions 2. xf ( x ) f ( x ) 2 f ( x ) 2 f ( x ) 11 28 32 7 m For quadratic functions, the second finite differences are constant. For cubic functions, the third finite differences are constant. It appears that for a polynomial function, a constant finite difference occurs at that difference that is the same as the degree of the polynomial. 62 46 18 6 mm + () + = 1 1 7 31 5 19 6 18 22 m ++ =+ m m + + =++ 3 1 5 1 9 33 2 m + 2 3 m + 2 61 86 12 6 + + = 5 1 9 39 7 6 1 2 m m m mmm m + + + 21 3 9 7 2 m + 1 3 m + 1 266 6 + + = 7 1 6 6 m m m + () =+ + 13 3 1 3 m 3 666 6 + ( ) = 1 1 6 m + = –– 3 3 2 m –1 3 m 66 6 6 = 1 7 6 6 m + + = m m ––– 12 3 9 7 2 ( ) m –2 3 m 30 24 6 = 61 37 24 = 64 27 37 = 24 18 6 = 37 19 18 = 27 8 19 = 18 12 6 = 1 971 2 = 817 = Chapter 1: Polynomial Functions 3 Exercise 1.2 1.
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ch01 - Chapter 1 Polynomial Functions Review of...

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