Inverse & Joint Variations
Inverse & Joint VariationsUnit 4 Day 9
Warm-Up: Released Exam Items & Practice.Show your work to complete these problems. Do NOT just circle an answer!1. The equation 2 5x can be used to estimate speed, s, of a car in miles perhour, given the length in feet, x, of the tire marks it leaves on the ground. Acar traveling 90 miles per hour came to a sudden stop. According to theequation, how long would the tire marks be for this car?A. 355 feetB. 380 feetC. 405 feetD. 430 feet2. Which function is even?A. f(x) (x 2)(x – 2)C. f(x) (x 1)(x – 2)B. f(x) x(x 2)D. f(x) (x – 1)(x – 1)3. A marathon is roughly 26.2 miles long. Which equation could be used todetermine the time, t, it takes to run a marathon as a function of the averagespeed, s, of the runner where t is in hours and s is in miles per hour?A. t 26.2 – 26.2 sB. t 26.2 – s /26.2C. t 26.2 sD. t 26.2 / sWarm – UpContinues
Warm-Up Continues .Practice Graphing Inverse Variation.Do a table for each branch and completely graph the function!Also, indicate the horizontal and vertical asymptotes, domain, andrange for each function.1HINT: You may24. y 5. y want to se by hand good testpreparation! 421½¼y-4-2-1-½-¼6. 5y x
Warm-Up: Released Exam Items & Practice.Show your 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, x, of the tire marks it leaves on the ground. A cartraveling 90 miles per hour came to a sudden stop. According to theequation, how long would the tire marks be for this car?A. 355 feetB. 380 feetC. 405 feetD. 430 feet2. Which function is even?A. f(x) (x 2)(x – 2)C. f(x) (x 1)(x – 2)B. f(x) x(x 2)D. f(x) (x – 1)(x – 1)3. A marathon is roughly 26.2 miles long. Which equation could be used todetermine the time, t, it takes to run a marathon as a function of the averagespeed, s, of the runner where t is in hours and s is in miles per hour?A. t 26.2 – 26.2 sB. t 26.2 – s /26.2C. t 26.2 sD. t 26.2 / sWarm – UpContinues
4. Graphing Inverse Variation 1y xMake a table of valuesfor each ooks familiar, right?HA: y 0VA: x 0Domain: (- , 0) U (0, )Range: (- , 0) U (0, )
Plot at least 5exact pointsper branch!5.Graphing Inverse Variation .2y xHA: y 0VA: x 06. 5y xDomain:(- , 0) U (0, )Range:(- , 0) U (0, )x¼½124yx8421½½12510y-10-5-2.5-1-½
Tonight’s HomeworkPacket p. 13 – 15Remember:Quiz Tomorrow!We also have a lesson, so tonight*check HW online AND*work the warm-up in the notes
Homework Answers1. y 7.25 x12. y 144 x-1OR y 144/x3. y 9 x14.Direct because x1Indirect because x-16.25 miles2.4 peopleDirect because x1111.1 hoursy 5x4Symmetry over y-axisEven FunctionMost similar to y x2 because positive even exponent5.y 3x-6Symmetry over y-axisEven FunctionMost similar to y x-2 because negative even exponent6.y ½ x-5Symmetry about originOdd FunctionMost similar to y x-1 because negative odd exponent7.y 1/6x8Symmetry over y-axisEven FunctionMost similar to y x2 because positive even exponent
Homework Answers8.y 8x9Symmetry about the origin Even FunctionMost similar to y x3 because positive odd exponent9.y 4(x - 3)(x 3)Note: This is a quadratic withroots at x 3, -3 NOT a power functionSymmetry over y-axisEven FunctionMost similar to y x2 because positive even exponent10.y -2x(x 3)Note: This is a quadratic withroots at x 0, -3 NOT a power functionNo SymmetryNeither even nor odd FunctionMost similar to y x2 because positive even exponent
Notes Day 9Inverse & Joint Variations
Inverse VariationskA relationship that can be written in the form y , wherexk is a nonzero constant and x 0, is an inverse variation. Theconstant k is the constant of variation.y x kMultiplying both sides of by x gives .So, the product of x and y in an inverse variation isk , the constant.
There are two methods to determine whether a relationshipbetween data is an inverse variation. You can write afunction rule in y kform, or you can check whether xy is axconstant for each ordered pair.Example: Tell whether the relationship is an inverse variation.Explain. If it is an inverse variation, write the equation.1.x y1 302 153 10xy303030 y x30Yes, inverse variationxy 30 k (constant)2.x124y51020xy52080NOT inverse variationxy k (constant)3. 2xy 28Yes, inverse variationxy 14 k (constant)14y x
ky Remember: You can write a function rule inx form,You Try!!or you can check whether xy is a constant for each ordered pair.Tell whether the relationship is an inverse variation. Explain. If it isan inverse variation, write the equation.4.xy-12 241-28 -16xy-288-2-128NOT inverse variationxy k (constant)5.xy339118 0.5xy999Yes, inverse variationxy 9 k (constant)9y x6. 2x y 10y 10 2 Xx y1 8xy82 63 41212500NOT inverse variationxy k (constant)
Examples:Write and graph the inverse variation in which y 0.5 when x –12.Steps:1. Find K. (k xy)2. Write new equation as y k/x3. Graph on grid.ky x0.5k 1 12k 6 6y x
You Try!Write and graph the inverse variation in which y 1/2 when x 10Steps:1. Find K. (k xy)2. Write new equation as y k/x3. Graph on grid.ky x0.5 k 1 10k 55y x
ExamplesThe inverse variation xy 350 relates the constant speed x inmi/h to the time y in hours that it takes to travel 350 miles.Determine a reasonable domain and range and then graph thisinverse variation.350y xPractical Domain: (0, )Practical Range: 0, X1y3502175510107050501001007353.5
You Try!The inverse variation xy 100 represents the relationshipbetween the pressure x in atmospheres (atm) and the volumey in mm3 of a certain gas. Determine a reasonable domain andrange and then graph this inverse variation.100y xPractical Domain:(0, )Practical Range: 0, X1Yy1002505101020252550504102
Examples5. Let x1 5, x2 3, and y2 10. Let y vary inversely as x. Find y1.x1 y1 x2 y25 y1 3 105 y1 30y1 6So xy k every time!
You Try!6. Let x1 2, y1 -6, and x2 -4. Let y vary inversely as x. Find y2.7. Boyle’s law states that the pressure of a quantity of gas x variesinversely as the volume of the gas y. The volume of gas inside acontainer is 400 in3 and the pressure is 25 psi. What is the pressurewhen the volume is compressed to 125 in3?8. On a balanced lever, weight varies inversely as the distance fromthe fulcrum to the weight.The diagram shows a balancedlever. How much does thechild weigh?
ANSWERS!6. Let x1 2, y1 -6, and x2 -4. Let y vary inversely as x. Find y2.x1 y1 x2 y2 2 6 4 y2y2 3 12 4y27. Boyle’s law states that the pressure of a quantity of gas x variesinversely as the volume of the gas y. The volume of gas inside acontainer is 400 in3 and the pressure is 25 psi. What is the pressurewhen the volume is compressed to 125 in3?x1 y1 x2 y2400 25 125 y210000 125y2y2 80 psi8. On a balanced lever, weight varies inversely as the distance fromthe fulcrum to the weight.The diagram shows a balancedlever. How much does the3.2 y1 4.3(60)child weigh?x1 y1 x2 y23.2 y1 258y1 80.625 lbs
Occurs when 1 quantity varies directly as theproductof2 or more other quantities.z kxyForm ,x 0, z 0Ex: The area of a trapezoid varies jointly as theheight h and the sum of its bases b1 and b2.Find the equation of joint variation if A 48in2, h 8 in, b1 5in, and b2 7 in.A k h (b1 b2)48 k (96) ½ k48 k (8) (5 7)A ½ h (b1 b2)
y varies directly with x and inversely with z2.y kx2y kz3 y varies inversely with x .x3 y varies directly with x2 and inversely with z.y k x2z22 z varies jointly with x and y. z kx y y varies inversely with x and z.y kxz
Tell whether x and y show direct variation, inverse variation, orneither.1.) xy 142.) 2x y 4Inversek 1/4Neithery3.) 12xDirectk 1214.) y xInversek 1Write the function that models each relationship. Find z whenx 6 and y 4.5. z varies jointly with x and y. When x 7 and y 2, z 28.z kxy28 k (7) (2)2 kz 2xyz 2 (6) (4)z 486. z varies directly with x and inversely with the cube of y.When x 8 and y 2, z 3.z kxy33 k (8)(2)33 kz 3xy3z 3 (6) z 9(4)332
1. The speed of the current in a whirlpool varies inverselywith the distance from the whirlpool’s center. TheLofoten Maelstrom is a whirlpool located off the coast ofNorway. 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 k0.1 ks 300300 kd3000db. What’s the speed of the whirlpool when 50 metersfrom the center?s 300s 6 meters/sec50
2. In building a brick wall, the amount of time it takes tocomplete the wall varies directly with the number ofbricks in the wall and varies inversely with the number ofbricklayers that are working together. A wall containing1200 bricks, using 3 bricklayers, takes 18 hours to build.How long would it take to build a wall of 4500 bricks if 5bricklayers worked on it?t k b where t time, b # bricks, p # people workingp18 k (1200)0.045 kt 0.045 b3pt 0.045 (4500)5t 40.5 hours
Tonight’s HomeworkPacket p. 13 – 15Remember:Quiz Tomorrow!We also have a lesson, so tonight*check HW online AND*work the warm-up in the notes
between data is an inverse variation. You can write a function rule in form, or you can check whether xy is a constant for each ordered pair. Example: Tell whether the relationship is an inverse variation. Explain. If it is an inverse variation, write the equation. 1. 2. 3. 2xy 28 k y
B.Inverse S-BOX The inverse S-Box can be defined as the inverse of the S-Box, the calculation of inverse S-Box will take place by the calculation of inverse affine transformation of an Input value that which is followed by multiplicative inverse The representation of Rijndael's inverse S-box is as follows Table 2: Inverse Sbox IV.
State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation. 1. u 5 8wz joint; 8 2. p 5 4s direct; 4 3. L 5 inverse; 5 4. xy 5 4.5 inverse; 4.5 5. 5 p 6. 2d 5 mn 7. 5 h 8. y 5 direct; p joint; inverse; 1.25 inverse; Find each value. 9. If y varies directly as x and y 5 8 when x 5 2, find y .
Inverse functions mc-TY-inverse-2009-1 An inverse function is a second function which undoes the work of the first one. In this unit we describe two methods for finding inverse functions, and we also explain that the domain of a function may need to be restricted before an inverse function can exist.
This handout defines the inverse of the sine, cosine and tangent func-tions. It then shows how these inverse functions can be used to solve trigonometric equations. 1 Inverse Trigonometric Functions 1.1 Quick Review It is assumed that the student is familiar with the concept of inverse
6.1 Solving Equations by Using Inverse Operations ' : "undo" or reverse each others results. Examples: and are inverse operations. and are inverse operations. and are inverse operations. Example 1: Writing Then Solving One-Step Equations For each statement below, write then SOIVP an pnnation to determine each number. Verify the solution.
24. Find the multiplicative inverse of 53 4 25. Find five rational numbers between 0 and -1 26. Find the sum of additive inverse and multiplicative inverse of 7. 27. Find the product of additive inverse and multiplicative inverse of -3 . 28. What should be subtracted from 5/8 to ma
Inverse Problems 28 (2012) 055002 T Bui-Thanh and O Ghattas For the forward problem, n is given and we solve the forward equations (1a) and (1b)for the scattered fieldU.For the inverse problem, on the other hand, given observation dataUobs over some compact subset dobs R , we are asked to infer the distribution of the refractive index n.One way to solve the inverse problem is to cast it .
Asset management is a critical component in helping us comply with our licence and the road investment strategy. The Department for Transport (DfT) sets out what we must do, and provides funding through 5-year ‘Road Periods’. The Office of Rail and Road monitors how we deliver our strategic business plan, including key performance indicators, and Transport Focus represents our customers .
