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Chapter 8

Chapter 8. Rational Functions. Chapter Summary. 8.1: Model Inverse and Joint Variation 8.2: Graph Simple Rational Functions Quiz 8.3: Graph General Rational Functions 8.4: Multiply and Divide Rational Expressions Quiz 8.5: Add and Subtract Rational Functions 8.6: Solve Rational Equations

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Chapter 8

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  1. Chapter 8 Rational Functions

  2. Chapter Summary • 8.1: Model Inverse and Joint Variation • 8.2: Graph Simple Rational Functions • Quiz • 8.3: Graph General Rational Functions • 8.4: Multiply and Divide Rational Expressions • Quiz • 8.5: Add and Subtract Rational Functions • 8.6: Solve Rational Equations • Quiz • Chapter 8 Test

  3. Section 8.1 Model Inverse and Joint Variation

  4. Direct and Inverse Variation • Two variables x and y vary directly if there is a nonzero number a such that the following is true: • Two variables x and y show inverse variation if they are related as follows: Constant of Variation

  5. y = y c. = x 4 7 x Classify Tell whether xand yshow direct variation, inverse variation, or neither. Type of Variation Given Equation Rewritten Equation a.xy = 7 Inverse b.y = x + 3 Neither Direct y = 4x 1. 3x = y 2.xy = 0.75 3.y = x –5 Direct Inverse Neither

  6. y= 7= 28 The inverse variation equation is y = x a a 12 x 4 = 6. Whenx = 2,y = 2 12 –14. Whenx = –2, y = y = x Write Inverse Variation Equation The variables xand yvary inversely, and y = 7 when x=4. Write an equation that relates xand y. Then find ywhen x = –2 . Write general equation for inverse variation. Substitute 7 for yand 4 for x. 28 = a Solve for a. ANSWER 4.x = 4,y = 3 ANSWER

  7. Joint Variation • Joint variation occurs when a quantity varies directly with the product of two or more quantities. In the equations below, a is a nonzero constant. z = axy z varies jointly with x and y. p = aqrs p varies jointly with q, r and s.

  8. STEP 1 Write a general joint variation equation. STEP2 Use the given values of z, x, and y to find the constant of variation a. Write a Joint Variation Equation The variable zvaries jointly with xand y. Also, z= –75 when x = 3 and y = –5. Write an equation that relates x, y, and z. Then find zwhen x = 2 and y = 6. SOLUTION z= axy –75 = a(3)(–5) Substitute 75 for z, 3 for x, and 25 for y. –75 = –15a Simplify. 5 = a Solve for a.

  9. ANSWER ; – 35 ANSWER z = – 2xy ; 20 7 z = xy 2 Write a Joint Variation Equation STEP 3 Rewrite the joint variation equation with the value of afrom Step 2. z = 5xy STEP 4 Calculate zwhen x = 2 and y = 6 using substitution. z = 5xy= 5(2)(6) = 60 The variable zvaries jointly with xand y. Use the given values to write an equation relating x, y, and z. Then find zwhen x = –2 and y = 5. 9.x = 1,y = 2,z = 7 10.x = 4,y = –3,z =24

  10. y = y = z = atr x = s ay a a x x2 x Compare Different Types of Variation Write an equation for the given relationship. Relationship Equation a.yvaries inversely with x. b.zvaries jointly with x, y, and r. z = axyr c.y varies inversely with the square of x. d.zvaries directly with yand inversely with x. e.xvaries jointly with tand rand inversely with s.

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