Chapter 9: Rational Equations and Functions

Transcription

1 Chapter 9: Rational Equations and Functions

2 Chapter 9: Rational Equations and Functions Assignment Sheet Date Topic Assignment Completed 9.: Inverse and Joint Variation pg. 57 # - 4 odd, : Graphing Simple Rational Function pg. 54 # - 9 Be sure to state the domain and range 9..: Graphing Simple Rational Function pg. 54 # 0-, - 8 Be sure to state the domain and range 9..: Graphing Rational Functions pg. 550 #- 8 (even), 0-5, 8, 9,, 4 9..: Graphing Rational Worksheet Functions with Slant Asmptotes Review pg. 576 #- 5 pg. 579 #- 5 Quiz None 9.4: Multipling and Dividing Rational Epressions pg. 558 #,, 7, 9-49 (odd) 9.5.: Addition and Subtraction of Rational Epressions pg. 566 # : Addition, Subtraction, and Comple Fractions pg. 566 #8-40, : Solving Rational Equations pg. 57 #- 0 (even),, 4-50 (even) Review Review Review Review Review Review Test None

3 9. Direct, Inverse, & Joint Variation Direct Variation Two variables and show direct variation provided = k and k 0. The nonzero constant k is called the constant of variation and is said to var directl with. The graph of = k is a line through the origin. E: The variables and var directl and = when = 4. a.) Write an equation that relates and. b.) Find the value of when = 0. E: Tell whether the data show direct variation. If so, write an equation relating and. Direct variation? If es, what s the equation? If es, find the price for a 8 inch gold chain. Direct variation? If es, what s the equation? If es, find the price for.8 carats of loose diamonds.

4 Inverse Variation Two variables and show inverse variation if the are related as follows: As above, the nonzero constant k is called the constant of variation and is said to var inversel with. E: The variables and var inversel, and = 8 when =. a.) Write an equation that relates and. b.) Find the value of when = - 4. E: Tell whether and show direct variation, inverse variation, or neither. Given Equation Rewritten Equation Tpe of Variation a) = 5 b) = c) = 4 Word Problems. The speed of the current in a whirlpool varies inversel with the distance from the whirlpool s center. The Lofoten Maelstrom is a whirlpool located off the coast of Norwa. At a distance of kilometers (000 meters) from the center, the speed of the current is about 0. meter per second. Describe the change in the speed of the current as ou move closer to the whirlpool s center.

5 . The table below compares the wing flapping rate r (in beats per second) to the wing length l (in centimeters) for several birds. Do these data show inverse variation? If so, find a model for the relationship between r and l. Inverse variation? If es, what s the equation? Joint Variation Joint variation occurs when a quantit varies directl as the product of two or more other quantities. For instance, if z = k where k 0, then z varies jointl with and. Additionall, if inversel with., z varies directl with and E: z varies jointl with and. Given that =, = 8 and z = 6. a.) Write an equation relating,, and z. b.) Then find z when = - and = 4. E: Write an equation for the given relationship. a) varies inversel with and directl with z. b) varies jointl with z and the square root of. c) w varies inversel with and jointl with and z. d) varies inversel with the square of. e) z varies directl with and inversel with. Homework: p # - 4 odd, 54

6 9.. Graphing Simple Rational Functions in the form: = a h k = graphs a hperbola Vertical Asmptote: Horizontal Asmptote: Domain: Range: = Vertical Asmptote: Horizontal Asmptote: Domain: Range:

7 a Rational Functions in the form = k have graphs that are hperbolas h with asmptotes and asmptotes = Vertical Asmptote: Horizontal Asmptote: Domain: Range: Homework: p #-9 (Be sure to state the domain and range)

8 9.. Graphing Rational Functions in the Form = a b c d = Vertical Asmptote: Horizontal Asmptote: Domain: Range: a b Rational Functions in the form = have graphs that are hperbolas c d with asmptotes and asmptotes = Vertical Asmptote: Horizontal Asmptote: Domain: Range:

9 = 4 Vertical Asmptote: Horizontal Asmptote: Domain: Range: Homework: p. 54 #0- all, -8 all (Be sure to state the domain and range)

10 9.. Graphing General Rational Functions Graphs of Rational Functions: Let p ( ) and q( ) be polnomials with NO common factors other than. The graph of f ( ) = p( ) has the following characteristics: q( ) Eample : Graph = 9 a) Find the - int (zeros of the numerator) - int: b) Find the VERTICAL asmptotes (zeros of the denominator) Vertical Asmptote(s): c) Find the HORIZONTAL asmptotes (compare the degree of the numerator and the degree of the denominator) d) Use a table and above information to graph Horizontal Asmptote(s):

11 Eample : Graph = 4 a) Find the - int (zeros of the numerator) - int: b) Find the VERTICAL asmptotes (zeros of the denominator) Vertical Asmptote(s): c) Find the HORIZONTAL asmptotes (compare the degree of the numerator and the degree of the denominator) d) Use a table and above information to graph Horizontal Asmptote(s): Eample : Graph = 4 a) Find the - int (zeros of the numerator) - int: b) Find the VERTICAL asmptotes (zeros of the denominator) Vertical Asmptote(s): c) Find the HORIZONTAL asmptotes (compare the degree of the numerator and the degree of the denominator) d) Use a table and above information to graph Horizontal Asmptote(s):

12 Eample 4: Graph = a) Find the - int (zeros of the numerator) - int: b) Find the VERTICAL asmptotes (zeros of the denominator) Vertical Asmptote(s): c) Find the HORIZONTAL asmptotes (compare the degree of the numerator and the degree of the denominator) d) Use a table and above information to graph Horizontal Asmptote(s): Eample 5: Graph = a) Find the - int (zeros of the numerator) - int: b) Find the VERTICAL asmptotes (zeros of the denominator) Vertical Asmptote(s): c) Find the HORIZONTAL asmptotes (compare the degree of the numerator and the degree of the denominator) d) Use a table and above information to graph Horizontal Asmptote(s):

13 Eample 6: Graph = 4 a) Find the - int (zeros of the numerator) - int: b) Find the VERTICAL asmptotes (zeros of the denominator) Vertical Asmptote(s): c) Find the HORIZONTAL (compare the degree of the numerator and the degree of the denominator) d) Use a table and above information to graph Horizontal Asmptote(s): Homework: Pg #-8 even, 0-5 all, 8, 9,, 4

14 9..: SLANT ASYMPTOTES Use long division to determine the slant asmptote for each problem below. Then graph. Your complete graph should contain: ) slant asmptote (when degree of numerator > degree of denominator BY ) ) vertical asmptote or horizontal asmptote ) - and - intercepts listed as coordinate pairs 4) maimum and minimum values where appropriate listed as coordinate pairs 5) a minimum of points per branch of the graph ) f () = ) f () = 5 ) f () = 6 8 4) f () = 5

15 5) f () = 7 6) f () = 9 7) f () = 8) f () =

16 9..: Graphing Rational Functions Homework Worksheet Graph each of the following. Be sure to include: a) Slant asmptote b) Vertical asmptote or Horizontal asmptote c) - and - intercepts listed as coordinate pairs d) Maimum and minimum values where appropriate listed as coordinate pairs e) A minimum of points per branch of the graph ) f () = 5 5 ) g() = ) t() = 9

17 4 4) f () = ( ) 5) f () =

18 9.4 Multipling and Dividing Rational Epressions Factor the following perfect cube binomials:.) 5.) 8 Which of the following can ou simplif and wh?.) ( ) ( ).).) Simplifing, Multipling, and Dividing Polnomials In order to full simplif a polnomial ou must factor ever numerator and denominator and then see what ou can divide out..) ) ) ) ) (6 4 ) 64 Remember ou NEVER DIVIDE fractions ou alwas INVERT the fraction and MULTIPLY!

19 .) ) ) ( ) ) ( 5 4.) Homework: pg. 559 # 6-48 even Skdiving

20 A falling skdiver accelerates until reaching a constant falling speed called the terminal velocit. Because of air resistance, the ratio of the skdiver volume to his or her cross-sectional surface area affects the terminal velocit. A.) The diagram shows a simplified geometric model of a skdiver with maimum cross sectional surface area. Use the diagram to write a model for the ratio of volume to cross sectional surface area for a skdiver. Bod Part Volume Cross sectional Surface Area Arm or leg Head Trunk Volume SurfaceArea = B.) Use the result from part A to compare the terminal velocities of two skdivers: One who is 60 inches tall and one who is 7 inches tall.

21 9.5 Addition and Subtraction of Comple Fractions.) = 4 5.) = 4.) = 4.) ) ) 4 6

22 7.) ) ) 5 ( ) ( ) Homework: Pg. 56 #6-7 Simplif the following:

23 .) = 4.) = In order to Simplif COMPLEX fractions..) Simplif the numerator and denominator separatel..) Simplif the fraction b multipling b the reciprocal..).) 4 4.)

24 4.) Using Comple Fractions: Monthl pament for items that are financed such as a car, house, or steno sstem will var with the rate of interest, the amount of time, the loan and the length of the loan. The formula for a given monthl pament is the following: r P P= r= t= r M = t A.) Find the monthl pament when P= $0, 000, r = 0.75, and t = B.) Find the monthl pament for financing a $,500 car for ears using the following interest rates. a.) 8% b.) % c.) 8% For each of the interest rates in part b, how much more would ou be paing for the car? Homework: pg. 566 #8-40, 4-45

25 9.6 Solving Rational Equations Solve the following equation b getting rid of ALL the fractions: 5 = 6 In order to get rid of the fractions ou multiplied b the Least Common Denominator. To solve rational equations ou want to get RID of the fractions b multipling all pieces of the equations b the LCD. Equations with one solution: 4 5 = LCD: Check: Equations with an Etraneous Solution: 5 0 = 7 LCD: Check: Equations with Two Solutions: 4 = LCD: Check: Solving b cross multipling: = LCD: Check:

26 Solve the following and be sure to check for etraneous solutions:.) 4 = Check:.) = Check:.) = Check:

27 4.) 5 5 = 4 Check: 6 5.) = 4 6 Check: Homework: Pg #-0 even,, 4-50 even