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SPECIAL TYPES OF FUNCTIONS Written by: Coryn Wilson Warren, Ohio Part One: Direct and Inverse Variation Slide 2 Instructor Notes Subject Area(s): Math Patterns, Functions and Algebra Standard Grade level: 8 Lesson Length: about 40 minutes Synopsis: This lesson is over two special types of functions, direct and inverse variations. In this presentation you will find an explanation along with graphical and algebraic representations of each. Objective/goals: Understand what a direct variation is Identify a constant of variation, given an equation or graph Differentiate between both equations and graphs for direct and inverse variations Write and use direct and inverse variation equations. Slide 3 Standards: Patterns, Functions, and Algebra Relate the various representations of a relationship; i.e., relate a table to graph, description and symbolic form Extend the uses of variables to include covariants where y depends on x Differentiate and explain types of changes in mathematical relationships, such as direct variation vs. inverse variation. Use symbolic algebra, graphs, and tables Pre-requisite skills: Students must be able to evaluate algebraic expressions and recognize and graph linear and nonlinear equations. TurningPoint functions: Standard question slides Materials: none *Instructional delivery notes can be found in the notes section of the slide. Instructor Notes Slide 4 Direct Vs. Inverse Variation Linear Vs. Nonlinear Functions Quadratic Function S PECIAL T YPES OF F UNCTIONS Slide 5 D IRECT V S. I NVERSE V ARIATION Direct Variation an equation of the form y = kx, where k 0. A direct variation represents a constant rate of change or constant variation (k) and a y-intercept of 0. Example: The distance formula (d = rt) is an example of a direct variation. In the formula, distance d varies directly as time t, and the rate r is the constant of variation, or rate of change. How fast is the car driving according to the graph? 45 MPH Slide 6 E XAMPLES OF D IRECT V ARIATION Name the constant of variation. Slope = -2 Slope = 3 Slope = 3 / 2 Slide 7 E XAMPLES OF D IRECT V ARIATION Write a direct variation equation that relates x to y. Assume that y varies directly as x. Then solve. If y = 4 when x = 2, find y when x = 16. If y = 9 when x = -3, find x when y = 6. y = kx 4 = k(2) 2 = k So, y = 2x y = 2 (16) y = 32 y = kx 9 = k(-3) -3 = k y = -3x 6 = -3x -2 = x Slide 8 D IRECT V S. I NVERSE V ARIATION Inverse Variation - an equation in the form xy = k, where ko. When the product of two variables remains constant, the relationship forms an inverse variation. Example: Suppose you travel 200 miles without stopping. The time it takes to get to your destination varies inversely as the rate at which you travel. Let x= speed and y = time. Use various speeds to make a table to graph the function. xy = 200 Slide 9 E XAMPLES OF I NVERSE V ARIATION Graph an inverse variation in which y varies inversely as x and y = 3 when x = 12. xy = k (3)(12) = k 36 = k Next, choose values of x and y that when multiplied, equals 36. Make a table of points XY 218 312 66 -2-18 -3-12 -6 xy = 36 Slide 10 E XAMPLES OF I NVERSE V ARIATION Graph each variation if y varies inversely as x. Slide 11 CLOSURE Get your clickers ready! Slide 12 Which of the following is an example of a direct variation equation. 1234567891011121314151617181920 21222324252627282930 1. y = 2x + 5 2. y = x 3. y = 5x 2 4. y = 2 Slide 13 Write a direct variation equation that relates x to y if y = -15 when x = -5. 1234567891011121314151617181920 21222324252627282930 1. y = 2x 2. y = 3x 3. y = -3x 4. y = 75x Slide 14 Use the direct variation equation y=3x to find x when y=-54 1234567891011121314151617181920 21222324252627282930 1. 18 2. -162 3. -18 4. 162 Slide 15 Write a direct variation equation that relates x and y if y = 4 when x = 20. Then find x when y = -25 1234567891011121314151617181920 21222324252627282930 1. y = 1 / 5 x ; - 499 / 4 2. y = 1 / 5 x ; -125 3. y = - 1 / 5 x ; -125 4. y = 3 / 10 x ; - 250 / 3 Slide 16 Write an inverse variation equation that relates x and y if y = 10 when x = 12. 1234567891011121314151617181920 21222324252627282930 1. y = 120x 2. xy = 120 3. xy = 1.2 4. 10 = k 12 Slide 17 If y = -14 when x = -7, find x when y = 7. Assume y varies inversely as x. 1234567891011121314151617181920 21222324252627282930 1. 686 2. 105 3. -28 4. 14 Slide 18 BEGIN INDIVIDUALIZED PRACTICE ON DIRECT AND INVERSE VARIATIONS END OF LESSON