Inverse & Joint Variations. Unit 4 Day 9

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1 Inverse & Joint Variations Unit 4 Da 9

2 Warm-Up: Released Eam Items & Practice. Show our work to complete these problems. Do NOT just circle an answer! 1. The equation 2 5 can be used to estimate speed, s, of a car in miles per hour, given the length in feet,, of the tire marks it leaves on the ground. A car traveling 90 miles per hour came to a sudden stop. According to the equation, how long would the tire marks be for this car? A. 355 feet B. 380 feet C. 405 feet D. 430 feet 2. Which function is even? A. f() = ( + 2)( 2) B. f() = ( + 2) C. f() = ( + 1)( 2) D. f() = ( 1)( 1) 3. A marathon is roughl 26.2 miles long. Which equation could be used to determine the time, t, it takes to run a marathon as a function of the average speed, s, of the runner where t is in hours and s is in miles per hour? A. t = s B. t = 26.2 s /26.2 C. t = 26.2 s D. t = 26.2 / s Warm Up Continues

3 4. Warm-Up Continues.Practice Graphing Inverse Variation. Do a table for each branch and completel graph the function! Also, indicate the horizontal and vertical asmptotes, domain, and range for each function. 1 ¼ 4 ½ ½ 4 ¼ 5. 2 HINT: You ma want to tr these b hand good test preparation! Looks familiar, right? -¼ -4 -½ ½ -4 -¼ 6. 5

4 Warm-Up: Released Eam Items & Practice. Show our work to complete these problems. Do NOT just circle an answer! 1. The equation can be used to estimate speed, s, of a car in miles per hour, given the length in feet,, of the tire marks it leaves on the ground. A car traveling 90 miles per hour came to a sudden stop. According to the equation, how long would the tire marks be for this car? A. 355 feet B. 380 feet C. 405 feet D. 430 feet 2. Which function is even? A. f() = ( + 2)( 2) B. f() = ( + 2) C. f() = ( + 1)( 2) D. f() = ( 1)( 1) 3. A marathon is roughl 26.2 miles long. Which equation could be used to determine the time, t, it takes to run a marathon as a function of the average speed, s, of the runner where t is in hours and s is in miles per hour? A. t = s B. t = 26.2 s /26.2 C. t = 26.2 s D. t = 26.2 / s Warm Up Continues

5 4. Graphing Inverse Variation 1 Make a table of values for each branch! ¼ 4 ½ ½ 4 ¼ -¼ -4 -½ ½ -4 -¼ Looks familiar, right? HA: = 0 VA: = 0 Domain: (-, 0) U (0, ) Range: (-, 0) U (0, )

6 Plot at least 5 eact points per branch! Graphing Inverse Variation.. 2 HA: = VA: = 0 Domain: (-, 0) U (0, ) Range: (-, 0) U (0, ) 5 ¼ 8 ½ ½ ½ ½

7 Tonight s Homework Packet p Remember: Quiz Tomorrow! We also have a lesson, so tonight *check HW online AND *work the warm-up in the notes

8 Homework Answers 1. = Direct because miles 2. = Indirect because people OR = 144/ 3. = 9 1 Direct because hours 4. = 5 4 Smmetr over -ais Even Function Most similar to = 2 because positive even eponent 5. = 3-6 Smmetr over -ais Even Function Most similar to = -2 because negative even eponent 6. = ½ -5 Smmetr about origin Odd Function Most similar to = -1 because negative odd eponent 7. = 1/6 8 Smmetr over -ais Even Function Most similar to = 2 because positive even eponent

9 Homework Answers 8. = 8 9 Smmetr about the origin Even Function Most similar to = 3 because positive odd eponent 9. = 4( - 3)( + 3) Note: This is a quadratic with roots at = 3, -3 NOT a power function Smmetr over -ais Even Function Most similar to = 2 because positive even eponent 10. = -2( + 3) Note: This is a quadratic with roots at = 0, -3 NOT a power function No Smmetr Neither even nor odd Function Most similar to = 2 because positive even eponent

10 Notes Da 9 Inverse & Joint Variations

11 Inverse Variations A relationship that can be written in the form, where k is a nonzero constant and 0, is an inverse variation. The constant k is the constant of variation. Multipling both sides of b gives. = k So, the product of and in an inverse variation is. k, the constant k

12 There are two methods to determine whether a relationship between data is an inverse variation. You can write a function rule in constant for each ordered pair. k form, or ou can check whether is a Eample: Tell whether the relationship is an inverse variation. Eplain. If it is an inverse variation, write the equation = Yes, inverse variation = 30 = k (constant) NOT inverse variation k (constant) Yes, inverse variation = 14 = k (constant) 14

13 You Tr!! Remember: You can write a function rule in form, or ou can check whether is a constant for each ordered pair. Tell whether the relationship is an inverse variation. Eplain. If it is an inverse variation, write the equation = X NOT inverse variation k (constant) Yes, inverse variation = 9 = k (constant) 9 k NOT inverse variation k (constant)

14 Eamples: Write and graph the inverse variation in which = 0.5 when = 12. Steps: 1. Find K. (k = ) 2. Write new equation as =k/ 3. Graph on grid. k 0.5 k 1 12 k 6 6

15 You Tr! Write and graph the inverse variation in which =1/2 when = 10 Steps: 1. Find K. (k = ) 2. Write new equation as =k/ 3. Graph on grid. k 0.5 k 1 10 k 5 5

16 Eamples The inverse variation = 350 relates the constant speed in mi/h to the time in hours that it takes to travel 350 miles. Determine a reasonable domain and range and then graph this inverse variation. 350 Practical Domain: (0, ) Practical Range: 0, X

17 You Tr! The inverse variation = 100 represents the relationship between the pressure in atmospheres (atm) and the volume in mm 3 of a certain gas. Determine a reasonable domain and range and then graph this inverse variation. 100 Practical Domain: (0, ) Practical Range: 0, X Y

18 Eamples 5. Let 1 = 5, 2 = 3, and 2 = 10. Let var inversel as. Find So = k ever time!

19 You Tr! 6. Let 1 = 2, 1 = -6, and 2 = -4. Let var inversel as. Find Bole s law states that the pressure of a quantit of gas varies inversel as the volume of the gas. The volume of gas inside a container is 400 in 3 and the pressure is 25 psi. What is the pressure when the volume is compressed to 125 in 3? 8. On a balanced lever, weight varies inversel as the distance from the fulcrum to the weight. The diagram shows a balanced lever. How much does the child weigh?

20 ANSWERS! 6. Let 1 = 2, 1 = -6, and 2 = -4. Let var inversel as. Find Bole s law states that the pressure of a quantit of gas varies inversel as the volume of the gas. The volume of gas inside a container is 400 in 3 and the pressure is 25 psi. What is the pressure when the volume is compressed to 125 in 3? psi On a balanced lever, weight varies inversel as the distance from the fulcrum to the weight. The diagram shows a balanced lever. How much does the child weigh? (60) lbs

21 Occurs when 1 quantit varies directl as the of product 2 or more other quantities. z = k Form, 0, z 0 E: The area of a trapezoid varies jointl as the height h and the sum of its bases b 1 and b 2. Find the equation of joint variation if A = 48 in 2, h= 8 in, b 1 = 5in, and b 2 = 7 in. A = k h (b 1 + b 2 ) 48 = k (96) ½ = k 48 = k (8) (5 + 7) A = ½ h (b 1 + b 2 )

22 varies directl with and inversel with z 2. varies inversel with 3. varies directl with 2 and inversel with z. = k 2 z varies jointl with 2 and. = k 3 z = k 2 = k z 2 z varies inversel with and z. = k z

Tell whether and show direct variation, inverse variation, or neither. 1.) 1 2.) 2 4 3.) 12 4.) 4 Inverse k = 1/4 Neither Direct k = 12 Write the function that models each relationship.

23 Tell whether and show direct variation, inverse variation, or neither. 1.) 1 2.) ) 12 4.) 4 Inverse k = 1/4 Neither Direct k = 12 Write the function that models each relationship. Find z when = 6 and = z varies jointl with and. When = 7 and = 2, z = 28. z = k 28 = k (7) (2) 2 = k z = 2 6. z varies directl with and inversel with the cube of. When = 8 and = 2, z = 3. 1 Inverse k = 1 z = 2 (6) (4) z = 48 z = k 3 3 = k (8) (2) 3 3 = k z = 3 3 z = 3 (6) (4) 3 z = 9 32

24 1. 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 3000 meters from the center, the speed of the current is about 0.1 meters per second. a. Find the equation for this scenario. s = k d 0.1 = k = k s = 300 d b. What s the speed of the whirlpool when 50 meters from the center? s = 300 s = 6 meters/sec 50

25 2. In building a brick wall, the amount of time it takes to complete the wall varies directl with the number of bricks in the wall and varies inversel with the number of bricklaers that are working together. A wall containing 1200 bricks, using 3 bricklaers, takes 18 hours to build. How long would it take to build a wall of 4500 bricks if 5 bricklaers worked on it? t = k b where t = time, b = # bricks, p = # people working p 18 = k (1200) = k t = b p t = (4500) 5 t = 40.5 hours

26 Tonight s Homework Packet p Remember: Quiz Tomorrow! We also have a lesson, so tonight *check HW online AND *work the warm-up in the notes

Unit 9: Rational Functions

Date Period Unit 9: Rational Functions DAY TOPIC Direct, Inverse and Combined Variation Graphs of Inverse Variation Page 484 In Class 3 Rational Epressions Multipling and Dividing 4 Adding and Subtracting