Variation

1. Variation Mrs. Curia Math for College Readiness February 11, 2013

2. Learning Targets By the end of class, students will be able to: • Define and identify direct and indirect variation • Classify functions as increasing or decreasing

3. Essential Questions • What is the difference between direct and inverse variation? • What are some real world examples of direct and inverse variation? • How can you determine if a function is increasing or decreasing?

4. What Are Variation Models? • VARIATION models show how one quantity (number) varies in proportion to another. • Often variation models are used comparing for real world quantities • Example: Circumference • Example: Area • Example: Calculating wages earned

5. What are DIRECT and INVERSE Variation? Direct Variation Inverse Variation y varies inversely as x OR y is inversely proportional to x y varies directly as x OR y is directly proportional to x * Note: “k” is known as the CONSTANT OF VARIATION

6. Can you think of any “real world” examples of direct or inverse variation?

7. Real World Examples Direct Variation Inverse Variation Area varies inversely as height Area = Volume / height Rate (speed) varies inversely as time Rate = distance/ time • Circumference (C = 2πr; C = πd) • Circumference = 2 ×π× radius • Circumference = π× diameter • Distance varies directly as time (d = rt) • Distance = rate × time • Paycheck totals vary directly as hours worked • Total = (hourly rate) × (hours worked)

8. Joint Variation y varies jointly as w and z OR y is jointly proportional to w and z Example: Volume varies jointly as radius squared and height V = πr2 h

9. Translating In “English” In “Math” a = k × b × c a = k × c a = k / c • “a varies jointlyasb and c” • “a varies directlyasb” • “a variesinverselyasb”

10. Now you try! Please complete the practice exercise in your notes. You may work with the people in your table groups. Follow class rules for SMALL GROUP DISCUSSION Conversations should not occur with students at other table groups

11. Translating In “English” In “Math” r = k / t A = k r2 x = k y z • Rate “r” varies inversely as time “t” • Area “A” varies directly as radius squared “r2” • x varies jointly as y and z

12. Increasing and Decreasing Increasing Functions Decreasing Functions As x increases, y decreases As x decreases, y increases INVERSE variation Examples: f(x) = -x3from (-∞, ∞) f(x) = x2 from (-∞, 0) • As x increases, y increases • As x decreases, y decreases • DIRECT variation • Examples: • f(x) = x3 from (-∞, ∞) • f(x) = x2 from (0, ∞)

13. Remember Quick Sketches? Look back to your notes on Quick Sketches for this activity

14. For the following functions • Draw a quick sketch of the function • Based on the graph, determine if the function is INCREASING or DECREASING • You may work with the people in your table groups • Conversations should not occur with students at other table groups

15. Is it INCREASING or DECREASING? • f(x) = |x| from (0, ∞) • f(x) = 1/x from (-∞, 0) • f(x) = -x2 from (-∞, 0) • f(x) = -x2 from (0, ∞) • f(x) = √x from (0, ∞)

16. Individual Practice • HW Packet 4.4 #9-24 all • You are to complete these problems INDIVIDUALLY and follow class rules for INDIVIDUAL WORK • Raise your hand if you have questions; Mrs. Curia will come by to answer any questions