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SPECIAL TYPES OF FUNCTIONS

SPECIAL TYPES OF FUNCTIONS. Written by: Coryn Wilson Warren, Ohio. Part One: Direct and Inverse Variation. Instructor Notes. Subject Area(s): Math – Patterns, Functions and Algebra Standard Grade level: 8 Lesson Length: about 40 minutes

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SPECIAL TYPES OF FUNCTIONS

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  1. SPECIAL TYPES OF FUNCTIONS Written by: Coryn Wilson Warren, Ohio Part One: Direct and Inverse Variation

  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.

  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

  4. Special Types of Functions Direct Vs. Inverse Variation Linear Vs. Nonlinear Functions Quadratic Function

  5. Direct Vs. Inverse Variation • 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

  6. Examples of Direct Variation • Name the constant of variation. Slope = -2 Slope = 3 Slope = 3/2

  7. Examples of Direct Variation • 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 = -3x 6 = -3x -2 = x y = kx 9 = k(-3) -3 = k

  8. Direct Vs. Inverse Variation • Inverse Variation - an equation in the form xy = k, where k≠o. 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

  9. Examples of Inverse Variation • Graph an inverse variation in which y varies inversely as x and y = 3 when x = 12. xy = k (3)(12) = k 36 = k xy = 36 Next, choose values of x and y that when multiplied, equals 36. Make a table of points

  10. Examples of Inverse Variation • Graph each variation if y varies inversely as x.

  11. CLOSURE Get your clickers ready!

  12. Which of the following is an example of a direct variation equation. y = 2x + 5 y = ½x y = 5x2 y = 2

  13. Write a direct variation equation that relates x to y if y = -15 when x = -5. y = 2x y = 3x y = -3x y = 75x

  14. Use the direct variation equation y=3x to find x when y=-54 18 -162 -18 162

  15. Write a direct variation equation that relates x and y if y = 4 when x = 20. Then find x when y = -25 y = 1/5x ; -499/4 y = 1/5x ; -125 y = -1/5x ; -125 y = 3/10x ; -250/3

  16. Write an inverse variation equation that relates x and y if y = 10 when x = 12. y = 120x xy = 120 xy = 1.2 10 = k ·12

  17. If y = -14 when x = -7, find x when y = 7. Assume y varies inversely as x. 686 105 -28 14

  18. END OF LESSON BEGIN INDIVIDUALIZED PRACTICE ON DIRECT AND INVERSE VARIATIONS

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