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9.1 Inverse and Joint Variation Goals p Write and use inverse variation models. p Write and use joint variation models. Your Notes VOCABULARY Inverse variation Two variables x and y show inverse k variation provided y ! "" where k is a nonzero x constant. Constant of variation The nonzero constant in a direct variation equation, inverse variation equation, or joint variation equation Joint variation A relationship that occurs when a quantity varies directly as the product of two or more other quantities Example 1 Classifying Direct and Inverse Variation Tell whether x and y show direct variation, inverse variation, or neither. Given Equation Rewritten Equation Type of Variation Neither a. y ! x # 2 b. 2 ! yx 2 y ! "" x Inverse y c. x ! "" 7 y ! 7x Direct Checkpoint Tell whether x and y show direct variation, inverse variation, or neither. 1. yx ! 1 inverse variation 204 Algebra 2 Notetaking Guide • Chapter 9 y 2. "" ! x 1.3 direct variation 3. y ! x $ 1 neither M2ng_nt.09 2/18/05 9:32 AM Page 205 Your Notes Writing an Inverse Variation Equation Example 2 The variables x and y vary inversely, and y ! 2 when x ! 3. a. Write an equation that relates x and y. b. Find y when x ! "2. Solution a. k y ! ## x k 2 ! 3 6 !k Write general equation for inverse variation. Substitute for y and for x. Solve for k. 6 The inverse variation equation is y ! ## . x b. When x ! "2, the value of y is: 6 y ! ## ! "3 "2 Checking Data for Inverse Variation Example 3 Tell whether the following data show inverse variation. If they do, find a model for the relationship between a and b. a 5 10 15 20 25 b 21 10.5 7 5.25 4.2 Solution Each product ab is equal to 105 . For instance, (5)(21) ! 105 and (20)(5.25) ! 105 . So, the data do show inverse variation. A model for the 105 relationship is b ! ## . a Lesson 9.1 • Algebra 2 Notetaking Guide 205 M2ng_nt.09 2/18/05 9:33 AM Page 206 Your Notes Comparing Different Types of Variation Example 4 Write an equation for the given relationship. Relationship Equation a. v varies directly with u. v ! ku b. v varies inversely with u. k v ! "" u c. w varies jointly with t, u, and v. w ! ktuv d. v varies inversely with the cube of u. k v ! "" u3 e. w varies directly with u and inversely with v. ku w ! "" v Checkpoint Complete the following exercises. 4. The variables x and y vary inversely, and y ! 3.5 when x ! 4. Find an equation that relates x and y. Find y when x ! 6. 14 7 y ! ""; "" x 3 5. Do the data below show inverse variation? If so, find a model for the relationship between x and y. x 2 4 6 8 10 y 12 6 4 3 2.4 24 yes; y ! "" x Homework 6. Write an equation for the given relationship: y varies jointly with x and z and varies inversely with the square of v. kxz y ! "" v2 206 Algebra 2 Notetaking Guide • Chapter 9