Unit 4 POLYNOMIAL FUNCTIONS

Transcription

1 Unit 4 POLYNOMIAL FUNCTIONS

2 Not polynomials: 3 x 8 5y 2 m 0.75 m 2b 3 6b 1 2 x What is the degree? 8xy 3 a 2 bc 3 a 6 bc 2 2x 3 + 4x 2 + 3x - 1 3x + 4x 5 + 3x 7-1

3 Addition and Subtraction (2x x) + (5x x + x 3 ) (3 2x 2 ) (x x)

4 POLYNOMIAL RACE Number Off Carry a calculator, pencil and paper with you Line up where I direct you You will work your problem and deliver to the next person Check the work from the person delivering to you before working yours 4 th person have the 1 st person check work Bring final answer to me

5 Multiplication 4y 2 (y 2 + 3) fg(f 4 + 2f 3 g 3f 2 g 2 + fg 3 ) (a 3)(2 5a + a 2 ) (y 2 7y + 5)(y 2 y 3)

6 A standard Burly Box is p ft by 3p ft by 4p ft. A large Burly Box has 1.5 ft added to each dimension. Write a polynomial V(p) in standard form that can be used to find the volume of a large Burly Box. The volume of a large Burly Box can be modeled by V(p) = 12p p p

7 Mr. Silva manages a manufacturing plant. From 1990 through 2005 the number of units produced (in thousands) can be modeled by N(x) = 0.02x x + 3. The average cost per unit (in dollars) can be modeled by C(x) = 0.004x 2 0.1x + 3. Write a polynomial T(x) that can be used to model the total costs. Mr. Silva s total manufacturing costs, in thousands of dollars, can be modeled by T(x) = x x x x + 9

8 (a + 2b)3 (x + 4)4

9 Pascal's triangle is named after the French mathematician and philosopher Blaise Pascal ( ), who wrote a Treatise on the Arithmetical Triangle describing it. But Pascal was not the first to draw out this triangle or to notice its amazing properties! Long before Pascal, 10th century Indian mathematicians described this array of numbers as useful for representing the number of combinations of short and long sounds in poetic meters. The triangle also appears in the writings of Omar Khayyam, the great eleventh-century astronomer, poet, philosopher, and mathematician, who lived in what is modern-day Iran.

10 The Chinese mathematician Chu Shih Chieh depicted the triangle and indicated its use in providing coefficients for the binomial expansion of (a+b) n in his 1303 treatise The Precious Mirror of the Four Elements. On the next slide is a reproduction of the triangle from Chu Shih Chieh, in Chinese numerals and another representation in Arabic.

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12 Pascal's work on the triangle stemmed from the popularity of gambling. A French nobleman had approached him with a question about gambling with dice. Pascal shared the question with another famous mathematician, Fermat, and Pascal's Arithmetical Triangle was the result. Using Pascal's triangle, one can in fact find the number of ways of choosing k items from a set of n items simply by looking at the kth entry on the nth row of the triangle. So, to see how many different trios you could form using the 45 members of your jazz band, you would look at the 3nd entry on the 45th row. (The "1" at the top of the triangle is considered the "0"th row, and the first entry on each row is labeled the "0"th entry on the row.) Since Pascal's time, mathematicians have found numerous patterns in Pascal's triangle. Some of the most interesting patterns are obtained by coloring in multiples of various numbers in Pascal's triangle; the results form endlessly repeating patterns called fractals.

13 Fill in the sides with a 1 Fill in the rest of the cells with the sum of the 2 numbers above the cell Number the rows

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15 Notice the coefficients of the variables in the final product of (a + b) 3. these coefficients are the numbers from the third row of Pascal's triangle.

16 Expand the following: (x + y) 7 (a + b) 10 (c + d) 12

17 DIVISION SYNTHETIC LONG linear only!!! (2x 2 +7x+7) (x+1) (2x 2 +7x+7) (x+1)

18 YOU TRY, both methods (x 2 x 6) (x 3) (3x 3 + 9x 2 14) (x + 3)

19 YOU TRY, both methods ( y 2 + 2y ) (y 3) (15x 2 + 8x 12) (3x + 1) = 2y 2 + 5y y 3 = 5x x + 1

20 This means that in the previous problem where the remainder was 13 that f(-1/3) = -13. Check it! This is called synthetic substitution. (15x 2 + 8x 12) (3x + 1)

21 Use synthetic substitution to evaluate the polynomial for the given value and check it. P(x) = 2x 3 + 5x 2 x + 7 for x = 2 P(x) = 6x 4 25x 3 3x + 5 for x = 1 3

22 If a polynomial is divisible by an expression, that expression is a factor of the polynomial. The remainder is 0.

23 Determine whether the given binomial is a factor of the polynomial P(x). (x + 1); (x 2 3x + 1) (x + 2); (3x 4 + 6x 3 5x 10) f(-1) =? f(-2) =?

24 Determine whether the given binomial is a factor of the polynomial P(x). (x + 2); (4x 2 2x + 5) (3x 6); (3x 4 6x 3 + 6x 2 + 3x 30) f(-2) =? f(2) =?

25 Kuta Dividing Polynomials Do odd numbers with BOTH methods

26 Determine if (x+3) is a factor of 2x x 2 + 6x - 18 by synthetic substitution, long division and synthetic division. DO ALL 3!!!!!

27 SIDEBAR PROPERTIES OF RADICALS x y = xy x y = x y 24 6 = 24 6 = 4= = 27 3 = 9= 3 n x m = x m n x = x x 2 = x 2 3

28 Matho Radical Functions

29 FACTORS OF POLYNOMIALS

30 WHY DO WE FACTOR? How do we solve for x?? TO SOLVE FOR X 5 options! Look at the graph Factor Taking Square Roots Quadratic Formula Completing the Square ** REMEMBER - x intercepts are also called roots, solutions or zeros.

31 FACTORING REVIEW What we have done so far 1. Standard Form? 2. GCF? 3. How many terms? 2 difference of squares sum/difference of cubes 3 guess and check, AC method, perfect square trinomial 4 - grouping

32 Find the Solutions by Factoring x 2 64 = 0 x 2 13x + 40 = 0 5x 2 19x + 12 = 0 7x 2 + 9x = 0 28x x 3 80x 2 = 0 7x 2 32x - 60 = 0 IF you struggle with factoring, Ch 7 in the Alg 1 book for practice

33 Sketch the parabola x 2 + 5x + 6 Factored form? (x + 2)(x + 3) Vertex form? (x ) Factoring Dominos

34 What if you can t factor? QUADRATIC FORMULA FRACTIONS!!! No decimals 9x 2 5x - 10 = 0 9x 2 + 7x - 56 = 0 2x 2 7x - 13 = -10 9x 2 = 4 + 7x

35 What is the discriminant? b 2 4ac is called the discriminant If b 2 4ac > 0 (positive), then there are 2 real roots If b 2 4ac = 0, then there is 1 real root If b 2 4ac < 0 (negative), then there are no real roots, and 2 complex roots FOLDABLE

36 What if there is a negative under the radical? Imaginary Numbers! FOLDABLE i = i 2 = ( 1) 2 = -1 i 3 = i * i 2 = -i i 4 = -i * i = -i 2 = 1 6i 2 9i 2-3i 2-103i 2 6i 3 9i 3-3i 3-103i 3 i 33 i 72 i 106 i 19

37 Operations Complex Numbers a + bi (3 + 4i) (6+5i) (3 + 4i)(6+5i) CARD MATCH work together in table groups

38 What about division? (3 + 4i) (6+5i) Practice Kuta Rationalizing Imaginary Denominators

39 MATHO Complex Numbers

40 So back to our Quadratic Formula! Solve for x 8x 2 + 6x = -5 10x = x 9x 2-3x 8 = -10-9x 2 = -8x + 8

41 5th option Look at the graph Factor Taking Square Roots Quadratic Formula Completing the Square ** REMEMBER - x intercepts are also called roots, solutions or zeros.

42 When we can t factor we have a second option COMPLETING THE SQUARE We create a perfect square trinomial and solve by taking square roots 9x 2 24x + 16 x 2 + x x 2 36x + x 2 = 16 (x +2) 2 = 16 (x +2) 2 = -16

43 Steps: x x 38 = 0 x x = 38 x x + = 38 + x x + 49 = Move constant to right Add a blank to both sides (b/2) 2 in blanks (x +7) 2 = 87 Factor the left, simplify the right (x +7) = + 87 Take the square root of both sides x = Solve for x RADICALS TRAVEL IN PAIRS!!!

44 Give it a try with your table partners: x 2 + 2x = -20 x 2-4x + 1 = -5 x x 51 = 0

45 What if a > 1? Add 2 steps!!! 2x 2 + 3x 7 = 0 5x x 15 = 0 2x 2 + 3x = 7 2(x x) = 7 x x = 7 2 Pull out a Divide both sides by a Follow remaining steps

46 Let s Practice Number off 1 10 Get with your partner Draw a problem from the bag 1 person use quadratic, other use completing the square When your answers are the same, draw again

47 So let s factor some polynomials of degree higher than 2 Factor: x 3 x 2 25x (x 1)(x 5)(x + 5) Put this in your calculator. In the table where do you see y = 0? How many times? In the graph, how many times does the function cross the x axis? Where does it cross?

48 You try. What are the roots/zeros/solutions of these polynomials? x 3 2x 2 9x (x 2)(x 3)(x + 3) x=2, x=3, x=-3 4x 6 + 4x 5 24x 4 = 0 4x 4 (x + 3)(x - 2) x=0, x=-3, x=2 x = 26x 2 (x 5)(x + 5)(x + 1) (x - 1) x=5, x=-5, x=-1, x=1

49 Two More 2x 6 10x 5 12x 4 = 0 x 3 2x 2 25x = 50 The roots are 0, 6, and 1. The roots are 5, 2, and 5.

50 The volume of a plastic storage box is modeled by the function V(x) = x 3 + 6x 2 + 3x 10. Identify the values of x for which V(x) = 0, then use the graph to factor V(x). Can we factor this polynomial? Let s look at the graph or table to find where y = 0 V(x) has three real zeros at x = 5, x = 2, and x = 1. Use synthetic division with one of the roots to get down to a quadratic, then factor OR use SD again If the model is accurate, the box will have no volume if x = 5, x = 2, or x = 1.

51 The volume of a rectangular prism is modeled by the function V(x) = x 3 8x x 12, which is graphed below. Identify the values of x for which V(x) = 0, then use the graph to factor V(x). V(x) has three real zeros at x = 1, x = 3, and x = 4. If the model is accurate, the box will have no volume if x = 1, x = 3, or x = 4.

52 What about this one? How many roots does it have? 3x x x 3 = 0 Sometimes a polynomial equation has a factor that appears more than once. This creates a multiple root. In 3x x x 3 = 0 has two multiple roots, 0 and 3. For example, the root 0 is a factor three times because 3x 3 = 0. The root -3 is a factor twice. The multiplicity of root r is the number of times that x r is a factor of P(x). When a real root has even multiplicity, the graph of y = P(x) touches the x-axis but does not cross it. When a real root has odd multiplicity greater than 1, the graph bends as it crosses the x-axis.

53 You cannot always determine the multiplicity of a root from a graph. It is easiest to determine multiplicity when the polynomial is in factored form.

54 Identify the roots of each equation. State the multiplicity of each root. Check it in your calculator. x 3 + 6x x + 8 = 0 x 4 + 8x x 2 27 = 0 x + 2 is a factor three times. The root 2 has a multiplicity of 3. x 1 is a factor once, and x + 3 is a factor three times. The root 1 has a multiplicity of 1. The root 3 has a multiplicity of 3.

55 x 4 8x x 2 32x + 16 = 0 2x 6 22x x x 3 = 0 x 2 is a factor four times. The root 2 has a multiplicity of 4. x is a factor three times, x + 1 is a factor once, and x 6 is a factor two times.

56 What happens if we can t factor?? x 3 + 3x 2 4x 12 = => 12, -12, 1, -1, -2, 2, 6, -6, 3, -3, 4, -4 These are the POSSIBLE rational roots Factors of 12: -12 and 1, 12 and -1, -2 and 6, 2 and -6, -3 and 4, 3 and -4 Factors of 1: 1 and 1, -1 and -1

57 NOT IN PRINTED SLIDES 2 other strategies to help you find your roots using bounds Other option is using synthetic division, suggestion is start with 1, 2, 3, and then -1, -2, -3. IF your result is ALL positive then the root is an upper bound (does not have to be a factor). If you get alternating positive and negative then it is the lower bound. x 4 3x 2 + x 12 = 0

58 Use the rational root theorem to find the possible rational roots AND the upper and lower bounds of the following: 2x 2 + 3x 5 = 0 6x 4 11x 3 + 8x 2 33x 30 = 0 ± 5 2, ±5, ±1 2, ± = ±5, ±30, ±15, ±10 1 and -3 1 and 3 How do we determine if any of them are ACTUAL roots?

59 2x 2 7x - 3 The same goes for complex roots: 8x 2 + 6x + 5 These pairs are called conjugates

60 Kuta Rational Root Theorem and Irrational and Imaginary Root Theorem

61 The design of a box specifies that its length is 4 inches greater than its width. The height is 1 inch less than the width. The volume of the box is 12 cubic inches. What is the width of the box? The width must be positive, so the width should be 2 inches.

62 A shipping crate must hold 12 cubic feet. The width should be 1 foot less than the length, and the height should be 4 feet greater than the length. What should the length of the crate be? The length must be positive, so the length should be 2 feet.

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64 So, what if I tell you the roots, or you see them in your calculator? Can you find the simplest possible polynomial? 2 and -1

65 Write the simplest polynomial with roots 1, 2 3, and 4 2, 2, 4 0, 2 3, 3 P(x) = x 3 11 x 2 2x P(x) = x 3 4x 2 4x + 16 P(x) = x 3 x 2 + 2x 11 3

66 3 + 2 What about these? 2i, 2 x 2 6x + 7 Check it with the quadratic formula x 3 2x 2 + 4x - 8 Check it..how??

67 Fundamental Theorem of Algebra: The degree tells you how many zeros you have Remember that radicals and complex numbers travel in pairs. The number of positive roots is the same as the number of sign changes. The number of negative roots is the same as the number of sign changes when you plug in (-x)

68 Solve x 4 3x 3 + 5x 2 27x 36 = 0 by finding all roots. Start with calculator, do you see any roots in your table or graph? If not, go to rational root theorem (x+1)(x-4)(x+3i)(x-3i) Multiply this out to check

69 Solve x 4 + 4x 3 x 2 +16x 20 = 0 by finding all roots. (x + 5) (x 1)(x + 2i)(x 2i) = 0 The solutions are 5, 1, 2i, +2i

70 Write the simplest function with zeros 2 + i, 3, and 1. **Remember what comes in pairs?? P(x) = x 5 5x 4 + 6x x 2 27x + 15

71 Write the simplest function with zeros 2i, 1+ 2, and 3. P(x) = x 5 5x 4 + 9x 3 17x x + 12

72 A silo is in the shape of a cylinder with a cone-shaped top. The cylinder is 20 feet tall. The height of the cone is 1.5 times the radius. The volume of the silo is 828 cubic feet. Find the radius of the silo. V cyl = πr 2 h 1 2 x x 2 = 828 V cone = 1 3 πr2 h The graph indicates a positive root of 6. Use synthetic division to verify that 6 is a root, and write the equation as (x 6)( 1 2 x2 + 23x + 138) = 0. The radius must be a positive number, so the radius of the silo is 6 feet.

73 Find the roots 27x 9 + 8x 6 27x 3 8 = 0

74 Kuta Fundamental Theorem of Algebra

75 What are some hints or clues we can use to determine what our function looks like? Each graph, based on the degree, has a distinctive shape and characteristics. Count the turning points

76 Degree and the sign on the leading coefficient determine end behavior

77 Identify the leading coefficient, degree, and end behavior. A. Q(x) = x 4 + 6x 3 x + 9 As x, P(x), and as x +, P(x). B. P(x) = 2x 5 + 6x 4 x + 4 As x, P(x), and as x +, P(x) +.

78 Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. P(x) is of odd degree with a negative leading coefficient.

79 A turning point is where a graph changes from increasing to decreasing or from decreasing to increasing. A turning point corresponds to a local maximum or minimum. A polynomial function of degree n has at most n 1 turning points and at most n x-intercepts. If the function has n distinct roots, then it has exactly n 1 turning points and exactly n x- intercepts. You can use a graphing calculator to graph and estimate maximum and minimum values.

80 Get a piece of construction paper per table Cut a small square out of each corner.same size in each corner!

81 An artist plans to construct an open box from a 15 in. by 20 in. sheet of metal by cutting squares from the corners and folding up the sides. Find the maximum volume of the box and the corresponding dimensions. Find a formula to represent the volume and graph it The graph has a local maximum of about when x So the largest open box will have dimensions of 2.83 in. by 9.34 in. by in. and a volume of in 3.

82 A welder plans to construct an open box from a 16 ft. by 20 ft. sheet of metal by cutting squares from the corners and folding up the sides. Find the maximum volume of the box and the corresponding dimensions. The graph has a local maximum of about when x So the largest open box will have dimensions of 2.94 ft by ft by ft and a volume of ft 3.

83 3-7 Challenge

84 Intermediate Value Theorem

85 Warm Up: Find a polynomial with roots: 3, 2, , 5, -4+2i Write factors FIRST!!!!!

86 Before we move on to rational functions:

87 Unit 5 RATIONAL FUNCTIONS A function with a variable in the denominator Parent function 1 x Graph is a hyperbola

88 A direct variation is a relationship between two variables x and y that can be written in the form y = kx, where k 0. In this relationship, k is the constant of variation. We say y varies directly as x. y = kx is a linear function with a y intercept of 0. k is the slope. A joint variation is a relationship among three variables that can be written in the form y = kxz, where k is the constant of variation. For the equation y = kxz, y varies jointly as x and z.

89 The perimeter P of a regular dodecagon varies directly as the side length s, and P = 18 in. when s = 1.5 in. Find s when P = 75 in s The volume V of a cone varies jointly as the area of the base B and the height h, and V = 12 ft 3 when B = 9 ft 3 and h = 4 ft. Find b when V = 24 ft 3 and h = 9 ft. The base is 8 ft 2.

90 A third type of variation describes a situation in which one quantity increases and the other decreases. For example, the table shows that the time needed to drive 600 miles decreases as speed increases. This type of variation is an inverse variation. An inverse variation is a relationship between two variables x and y that can be written in the form y = k x, where k 0. For the equation y = k x, y varies inversely as x.

91 The time t needed to complete a certain race varies inversely as the runner s average speed s. If a runner with an average speed of 8.82 mi/h completes the race in 2.97 h, what is the average speed of a runner who completes the race in 3.5 h? s 7.48

92 You can use algebra to rewrite variation functions in terms of k. Notice that in direct variation, the ratio of the two quantities is constant. In inverse variation, the product of the two quantities is constant.

93 Determine whether each data set represents a direct variation, an inverse variation, or neither. A. B. x y x y In each case xy = 52. The product is constant, so this represents an inverse variation. In each case y = 6. The x ratio is constant, so this represents a direct variation.

94 A combined variation is a relationship that contains both direct and inverse variation. Quantities that vary directly appear in the numerator, and quantities that vary inversely appear in the denominator. The change in temperature of an aluminum wire varies inversely as its mass m and directly as the amount of heat energy E transferred. The temperature of an aluminum wire with a mass of 0.1 kg rises 5 C when 450 joules (J) of heat energy are transferred to it. How much heat energy must be transferred to an aluminum wire with a mass of 0.2 kg raise its temperature 20 C? The amount of heat energy that must be transferred is 3600 joules (J).

95 5-1 Challenge

96 Operations with Rational Functions Adding Rational Functions Puzzle

97 AVERAGE SPEED Kuta Distance Rate and Time

98 Rational functions may have asymptotes. The function f(x) = 1 x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.

99 The rational function f(x) = 1 can be transformed by using x methods similar to those used to transform other types of functions. x=h is vertical asymptote Domain is all real numbers, x h y=k is horizontal asymptote Range is all real numbers, y k

100 Using the graph of f(x) = 1 x as a guide, describe the transformation and graph each function. g(x) = 1 x+2 Because h = 2, translate f 2 units left. g(x) = 1 x - 3 Because k = 3, translate f 3 units down.

101 Identify the asymptotes, domain, and range of the function g(x) = Vertical asymptote: x = 3 Domain: {x x 3} Horizontal asymptote: y = 2 Range: {y y 2} 1 x+3 2. h = 3, k = 2. Check Graph the function on a graphing calculator. The graph suggests that the function has asymptotes at x = 3 and y = 2.

102 Identify the asymptotes, domain, and range of the function g(x) = Vertical asymptote: x = -3 Domain: {x x -3} Horizontal asymptote: y = 5 Range: {y y 5} 1 x+3 5 h = 3, k = 5. Check Graph the function on a graphing calculator. The graph suggests that the function has asymptotes at x = 3 and y = 5.

103 Rational Inequalities You can also solve rational inequalities algebraically. You start by multiplying each term by the least common denominator (LCD) of all the expressions in the inequality. However, you must consider two cases: the LCD is positive or the LCD is negative. Look at the graph of 1, it has 2 branches, where x is negative x and where x is positive

104 Solve 6 x 8 3 algebraically. Case 1 LCD is positive. Step 1 Solve for x. 6 x 8 6 3x 24 (x 8) 3(x 8) Step 2 Consider the sign of the LCD. x 8 > 0 x > x 10 x x 10 For Case 1, the solution must satisfy x 10 and x > 8, which simplifies to x 10.

105 Case 2 LCD is negative. Step 1 Solve for x. 6 x 8 (x 8) 3(x 8) 6 3x x 10 x x 10 Step 2 Consider the sign of the LCD. x 8 < 0 x < 8 For Case 2, the solution must satisfy x 10 and x < 8, which simplifies to x < 8. The solution set of the original inequality is the union of the solutions to both Case 1 and Case 2. The solution to the inequality is x < 8 or x 10, or {x x < 8 x 10}.

106 Let s try 5-5 Practice A inequality problems

107 A discontinuous function is a function whose graph has one or more gaps or breaks. A hyperbola and many other rational functions are discontinuous functions. A continuous function is a function whose graph has no gaps or breaks. The functions you have studied before this, including linear, quadratic, polynomial, exponential, and logarithmic functions, are continuous functions.

108 The graphs of some rational functions are not hyperbolas. Consider the rational function f(x) = (x 3)(x+2) and its graph. (x+1) The numerator of this function is 0 when x = 3 or x = 2. Therefore, the function has x-intercepts at 2 and 3. The denominator of this function is 0 when x = 1. So there is a vertical asymptote at the line x = 1.

109 My words (not in your slides): Zeros (x intercepts) - Set the top =0 and solve Vertical asymptote Set the bottom =0 and solve

110 Identify the zeros and vertical asymptotes of f(x) = x2 +3x 4 x+3. Factor the numerator. Zeros: 4 and 1 (x + 4)(x 1) x + 3 Vertical asymptote: x = 3 Graph the function. Plot the zeros and draw the asymptote. Then make a table of values to fill in missing points. Vertical asymptote: x = 3 x y

111 Identify the zeros and vertical asymptotes of f(x) = x2 +7x+6 x+3. Zeros: 6 and 1 Vertical asymptote: x = 3

112 My words: If degree top>bottom no HA If degree top<bottom y=0 If degree top = bottom LC top LC bottom

113 NOT IN YOUR PRINTED COPY!!!!! When the degree on top > degree on the bottom there is not a horizontal asymptote BUT There is a SLANT or Oblique asymptote if the degree on top is EXACTLY one more than the degree on bottom. You find this asymptote by using long or synthetic division on the rational function. The remainder is ignored. Let s do example 5 from pg 202 of the book: x 2 x 2 x 1 1. Use long or synthetic division to divide the denominator into the numerator. Let s hope we get y = x

114 Identify the zeros and asymptotes of the function. Then graph. f(x) = x2 3x 4 x Zeros: 4 and 1 Vertical asymptote: x = 0 Horizontal asymptote: none

115 Identify the zeros and asymptotes of the function. Then graph. f(x) = x 2 x 2 1 Zero: 2 Vertical asymptote: x = 1, x = 1 Horizontal asymptote: y = 0

116 My words: holes where there are common factors. These values are NOT x intercepts or vertical asymptotes!!! Hole at x = 3 Identify holes in the graph of f(x) = x2 9 x 3 There is a hole in the graph at x = 3. You won t see this in the graph in your calculator but what will you see in your table??

117 Process for finding key characteristics: x intercepts: Set the numerator = 0 and solve Vertical asymptotes: set the denominator = 0 and solve Horizontal asymptotes: - if degree on top>degree on bottom none - if degree on top < degree on bottom y = 0 - if degree on top = degree on bottom LC top LC bottom Domain: all real numbers x vertical asymptotes Range: all real numbers y horizontal asymptotes Holes: in factored form, set cancelled factor = 0 FOLDABLE

118 Practice: pg 204/ thru 69 odd

119 Rational Function Scavenger Hunt

120 Jeopardy