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4. Direct Variation Inverse Variation Combined Variation Variation and Problem-Solving Polynomial and Rational Functions > Variation and Problem-Solving Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_section_34&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dir ect&utm_source=boundless

5. The ratio of variables in direct variation is always constant Direct variation between variables is easily modeled using a linear graph. The equation relating directly varying variables to a constant can be rearranged to slope-intercept form. Direct Variation Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/polynomial-and-rational-functions-4/variation-and-problem-solving-34/direct- variation-162- 7360?campaign_content=book_196_section_34&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Polynomial and Rational Functions > Variation and Problem-Solving

6. The ratio of variables in direct variation is always constant. Direct variation between variables is depicted by an hyperbola. The equation relating indirectly varying variables to a constant can be rearranged to hyberbolic form. Inverse Variation Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/polynomial-and-rational-functions-4/variation-and-problem-solving-34/inverse- variation-163- 5870?campaign_content=book_196_section_34&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Polynomial and Rational Functions > Variation and Problem-Solving

7. There must be a minimum of three related variables for their relationship to be one of combined variation. Among the three or more related variables, one must directly vary with another and inversely vary with a third in order for the relationship to be one of combined variation. An example of combined variation in the physical world is the Combined Gas Law, which relates pressure, temperature, volume, and moles (amount of molecules) of a gas. Combined Variation Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/polynomial-and-rational-functions-4/variation-and-problem-solving- 34/combined-variation-164- 5888?campaign_content=book_196_section_34&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Illustration of Gay-Lussac's Law, derived from the Combined Gas Law View on Boundless.com Polynomial and Rational Functions > Variation and Problem-Solving

8. Free to share, print, make copies and changes. Get yours at www.boundless.com Appendix

9. Key terms constant An identifier that is bound to an invariant value. directly proportional If one variable is always the product of the other and a constant, the two are said to be directly proportional. hyperbola A conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone. proportional At a constant ratio (to). Two magnitudes (numbers) are said to be proportional if the second varies in a direct relation arithmetically to the first. y-intercept A point at which a line crosses the y-axis of a Cartesian grid. Free to share, print, make copies and changes. Get yours at www.boundless.com Polynomial and Rational Functions

10. Interactive Graph: Direct Variation Graph of direct variation with the linear equation y=0.8x. The line y=kx is an example of direct variation between variables x and y. For all points on the line, y/x=k. Notice what happens when you change the "k" term. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Direct Variation." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-direct-variation View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-direct-variationView on Boundless.com Polynomial and Rational Functions

11. Interactive Graph: Graph of Indirect Variation Graph of indirect variation with the equation y=1/x. This hyperbola shows the indirect variation of variables x and y. Notice what happens when you change the variants. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Graph of Indirect Variation." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-graph-of-indirect-variation View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-graph-of-indirect-variationView on Boundless.com Polynomial and Rational Functions

12. Illustration of Gay-Lussac's Law, derived from the Combined Gas Law A constant amount of gas will exert pressure that varies directly with temperature. In this illustration, volume is held constant by an increased mass weighing down the lid of the container. If not for that extra mass, the lid would raise, increasing the volume and relieving the pressure. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Temperature Pressure law." CC BY-SA http://en.wikipedia.org/wiki/File:Temperature_Pressure_law.svg View on Boundless.comCC BY-SAhttp://en.wikipedia.org/wiki/File:Temperature_Pressure_law.svgView on Boundless.com Polynomial and Rational Functions

13. Free to share, print, make copies and changes. Get yours at www.boundless.com Polynomial and Rational Functions If y varies directly as x, and y = 24 when x = 16, find y when x = 12. A) 20 B) 18 C) 16 D) 28

14. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Polynomial and Rational Functions If y varies directly as x, and y = 24 when x = 16, find y when x = 12. A) 20 B) 18 C) 16 D) 28

15. Free to share, print, make copies and changes. Get yours at www.boundless.com Polynomial and Rational Functions If y varies directly as x, and y = 15 when x = 10, then what is the value of k, the constant of variation? A) 9 B) 2/3 C) 3/2 D) 3

16. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Polynomial and Rational Functions If y varies directly as x, and y = 15 when x = 10, then what is the value of k, the constant of variation? A) 9 B) 2/3 C) 3/2 D) 3

17. Free to share, print, make copies and changes. Get yours at www.boundless.com Polynomial and Rational Functions The number of hours, h, it takes for a block of ice to melt varies inversely as the temperature, t. If it takes 2 hours for a square inch of ice to melt at 65º, find the constant of proportionality. A) 32.5 B) 90 C) 130 D) 56.5

18. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Polynomial and Rational Functions The number of hours, h, it takes for a block of ice to melt varies inversely as the temperature, t. If it takes 2 hours for a square inch of ice to melt at 65º, find the constant of proportionality. A) 32.5 B) 90 C) 130 D) 56.5

19. Free to share, print, make copies and changes. Get yours at www.boundless.com Polynomial and Rational Functions In a formula, Z varies inversely as p. If Z is 200 when p = 4, find Z when p = 10. A) 800 B) 500 C) 50 D) 80

20. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Polynomial and Rational Functions In a formula, Z varies inversely as p. If Z is 200 when p = 4, find Z when p = 10. A) 800 B) 500 C) 50 D) 80

21. Free to share, print, make copies and changes. Get yours at www.boundless.com Polynomial and Rational Functions If y varies directly as x and inversely as z, and y = 24 when x = 48 and z = 4, find x when y = 44 and z = 6. A) 2 B) 264 C) 4 D) 132

22. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Polynomial and Rational Functions If y varies directly as x and inversely as z, and y = 24 when x = 48 and z = 4, find x when y = 44 and z = 6. A) 2 B) 264 C) 4 D) 132

23. Free to share, print, make copies and changes. Get yours at www.boundless.com Polynomial and Rational Functions The resistance of a wire varies directly as its length and inversely as the square of its diameter. A wire with length 200 in and a diameter of 1/4 in has a resistance of 20 ohms. Find the resistance in a 500 in wire with the same diameter. A) 0.000625 ohms B) 25 ohms C) 50 ohms D) 1.25 ohms

24. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Polynomial and Rational Functions The resistance of a wire varies directly as its length and inversely as the square of its diameter. A wire with length 200 in and a diameter of 1/4 in has a resistance of 20 ohms. Find the resistance in a 500 in wire with the same diameter. A) 0.000625 ohms B) 25 ohms C) 50 ohms D) 1.25 ohms

25. Attribution Wikipedia. "Proportionality (mathematics)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Proportionality_(mathematics)CC BY-SA 3.0http://en.wikipedia.org/wiki/Proportionality_(mathematics) Wiktionary. "constant." CC BY-SA 3.0 http://en.wiktionary.org/wiki/constantCC BY-SA 3.0http://en.wiktionary.org/wiki/constant Wiktionary. "hyperbola." CC BY-SA 3.0 http://en.wiktionary.org/wiki/hyperbolaCC BY-SA 3.0http://en.wiktionary.org/wiki/hyperbola Connexions. "Data Concepts -- Variations." CC BY 3.0 http://cnx.org/content/m18281/latest/CC BY 3.0http://cnx.org/content/m18281/latest/ Wikipedia. "Combined gas law." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Combined_gas_lawCC BY-SA 3.0http://en.wikipedia.org/wiki/Combined_gas_law Wiktionary. "constant." CC BY-SA 3.0 http://en.wiktionary.org/wiki/constantCC BY-SA 3.0http://en.wiktionary.org/wiki/constant Wikipedia. "directly proportional." CC BY-SA 3.0 http://en.wikipedia.org/wiki/directly%20proportionalCC BY-SA 3.0http://en.wikipedia.org/wiki/directly%20proportional Wikipedia. "Direct variation." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Direct_variation#Direct_proportionCC BY-SA 3.0http://en.wikipedia.org/wiki/Direct_variation#Direct_proportion Wiktionary. "proportional." CC BY-SA 3.0 http://en.wiktionary.org/wiki/proportionalCC BY-SA 3.0http://en.wiktionary.org/wiki/proportional Wiktionary. "y-intercept." CC BY-SA 3.0 http://en.wiktionary.org/wiki/y-interceptCC BY-SA 3.0http://en.wiktionary.org/wiki/y-intercept Wiktionary. "constant." CC BY-SA 3.0 http://en.wiktionary.org/wiki/constantCC BY-SA 3.0http://en.wiktionary.org/wiki/constant Free to share, print, make copies and changes. Get yours at www.boundless.com Polynomial and Rational Functions