Tuesday, 3/28 : Ch. 9.8 Cubic Functions ~ Ch. 9 Packet p.67 #(1-6) Thursday, 3/30 : Ch. 9.8 Rational Expressions ~ Ch. 9 Packet p.

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1 Ch. 9.8 Cubic Functions & Ch. 9.8 Rational Expressions Learning Intentions: Explore general patterns & characteristics of cubic functions. Learn formulas that model the areas of squares & the volumes of cubes. Explore the graphs of cubic functions & transformations of these graphs. Write the equation of a cubic function from its graph. Tuesday, 3/28 : Ch. 9.8 Cubic Functions ~ Ch. 9 Packet p.67 #(1-6) Thursday, 3/30 : Ch. 9.8 Rational Expressions ~ Ch. 9 Packet p.68 #(1-3)

2 Parent Functions & Their Relationships Cubic Function: y = x 3 Cube Root Function: y = 3 x

3 cubing function: the function f x = x 3, which gives the cube of a number. cube root: the cube root of a number a is the number b such that a = b 3. 3 the cube root of a is denoted a. 3 ex.) The cube root of 64 equals = 4 3 The cube root of 16 equals perfect cube: a number, a, that is equal to the cube of an integer. Given: a = b 3 = b b b ex.) -125 is a perfect cube because -125 = ( 5) 3 Perfect cubes: a: { 0, ±1, ±8, ±27, ±64, ±125, ±216, ±343, ±512, ±729, ±1000 } Cube roots: b: { 0, ±1, ±2, ±3, ±4, ±5, ±6, ±7, ±8, ±9, ±10 } (h, k) x 3 (-2) 3 ( 1) 3 (0) 3 (1) 3 (2) 3 (3) 3 What should we call the point (h, k) of a cubic function? If it s not a vertex (pt. of parabola s directional ), then what is it?? n 2 n 2 3 n 3

4 Transformation of Cubic Parent Function y = x 3 Replace x x h a y k h = horizontal shift a = horizontal dilation b & y y = x 3 = ( x h y = b( y k b a )3 x h a )3 + k k = vertical shift b = vertical dilation A square is to a rectangle like a cube is to a. (h, k) = the inflection point or, the center of rotational symmetry

5 Transformation of Cubic Parent Function y = x 3 Replace x x h a y k h = horizontal shift a = horizontal dilation b & y y = x 3 = ( x h y = b( y k b a )3 x h a )3 + k k = vertical shift b = vertical dilation A square is to a rectangle like a cube is to a. Answer: PRISM. Prism: a solid geometric figure whose two end faces are similar, equal, & parallel, however, the sides are parallelograms, but NOT equal to the ends. l w h V = lwh l = w = h V = s 3

6 Parent Cubic Function Cubic Function Transformations

7 RECALL: General Quadratic Function Factored Form ax 2 + bx + c a(x r 1 )(x r 2 ) General Cubic Function Factored Form??? ax 3 + bx 2 + cx + d a(x r 1 )(x r 2 )(x r 3 ) local maximum (5, 2,000) End behavior: As x increases, y also increases. local minimum (15, 0) End behavior: As x decreases, y also decreases.

8 General Cubic Function Factored Form ax 3 + bx 2 + cx + d a(x r 1 )(x r 2 )(x r 3 ) 4x 3 120x x + 0 4(x r 1 )(x r 2 )(x r 3 ) factor out GCF & use M:A find roots & use factored form equation 4x(x 2 30x + 225) = 4(x 0)(x 15)(x r 3 ) 4x(x 15)(x 15) = 4(x 0)(x 15) 2 Thus, y = 4x 3 120x x also equals y = 4x(x 15) 2 in factored form. local maximum (5, 2,000) End behavior: As x increases, y also increases. End behavior: As x decreases, y also decreases. local minimum (15, 0) AND a DOUBLE ROOT When a-value is positive

9 Writing the Factored Form of a Cubic Function (given a graph) a.) Identify the zeros or roots of the cubic function. r 1 = r 2 = r 3 = b.) Find the y-intercept. Use this point to solve for the dilation, a. c.) Write the factored form of the function. y = a(x r 1 )(x r 2 )(x r 3 )

10 SOLUTIONS: Zeros or Roots: a.) r 1 = -5, r 2 = 3 & r 3 = 7 -> y = a(x 5)(x 3)(x 7) y-intercept: b.) (0, d) = (0, 105) Factored form cubic function: c.) y = 1(x + 5)(x 3)(x 7) Let (x,y) = (0, 105) ; solve for a-value. y = a(x + 5)(x 3)(x 7) 105 = a(0 + 5)(0 3)(0 7) 1 = a Thus, The factored form cubic function is: y = 1(x + 5)(x 3)(x 7)

11 SOLUTIONS: Zeros or Roots: a.) r 1 = 1, r 2 = -3 & r 3 = -3 -> y = a(x 1)(x 3)(x 3) y-intercept: b.) (0, d) = (0, -18) Factored form cubic function: c.) y = 2(x 1)(x + 3) 2 Let (x,y) = (0, -18) ; solve for a-value. y = a(x 1)(x + 3)(x + 3) -18 = a(0 1)(0 + 3) 2-18 = -9a 2 = a Thus, The factored form cubic function is: y = 2(x 1)(x + 3) 2

12 Ch. 9.8 Rational Expressions Part II p.544 #13.) Simplify each rational expression completely. State any restrictions on the variable. a.) x + 4 x2 + 4x + 4 x + 2 x 2 16 b.) x 2 + 2x x 2 4 x 2 x 2 6x + 8 c.) x + 1 x 2 + 6x + 9 x + 3 d.) x 1 x x +1

13 SOLUTIONS: p.544 #13.) Simplify each rational expression completely. State any restrictions on the variable. a.) x + 4 x2 + 4x + 4 x + 2 x 2 16 x + 4 (x+2)(x+2) x + 2 (x+4)(x 4) (x+2) (x 4) b.) D:{all Real x s s.t. x -2, -4 or 4} x 2 + 2x x 2 4 x 2 x 2 6x + 8 x(x+ 2) (x 4)(x 2) (x+4)(x 4) x 2 (x+ 2)(x 2) x(x+4) or x 2 4 x 2 + 4x D:{all Real x s s.t. x 0, or ±4} c.) x + 1 x 2 + 6x + 9 x + 3 x + 1 (x+3)(x+3) x + 3 x + 1 (x+3)(x+3) x + 3 (x+3) x+3 x+1(x+3) (x+3) 2 x+x+3 (x+3) 2 2x+3 (x+3) 2 D:{all Real x s s.t. x -3} d.) x 1 4 x 2 1 x +1 x 1 4 (x 1)(x+1) x +1 x 1 4 (x 1)(x+1) x +1 (x 1) x 1 x 1 4(x 1) (x 1)(x+1) x (x 1) 4(x 1) (x 1)(x+1) 3 x + 1 D:{all Real x s s.t. x -1 or 1}

14 Lesson 9.8 Rational Expressions State any restrictions on the variable. Reduce each rational expression to lowest terms. p.68 #1e.) #2i.) x 20x x 2 5x 6 x2 4x 12 x 2 + 4x + 3 x 2 + 5x + 6

15 SOLUTIONS: Lesson 9.8 Rational Expressions State any restrictions on the variable. Reduce each rational expression to lowest terms. p.68 #1e.) #2i.) x 20x 4(1 + 5x) 4(5x) 4(1+5x) 4(5x) 1 + 5x 5x Factor out GCF from numerator. Factor that same value from the denominator. Reduce. Warning: DO NOT EVER divide only part of the numerator by the denominator. i.e X x 2 5x 6 x2 4x 12 x 2 + 4x + 3 x 2 + 5x + 6 x 2 5x 6 x2 + 5x + 6 x 2 + 4x + 3 x 2 4x 12 (x+1)(x 6) (x+3)(x+2) (x+1)(x+3) (x 6)(x+2) (x + 1)(x + 3) (x 6)(x + 2) (x + 1)(x + 3) (x 6)(x + 2) 1 Multiply by the Reciprocal. Factor. Reduce. (1+5) 5 = 6 5 = 1. 2