2. Everything You Always Wanted to Know About Math* * But were afraid to ask.

3. Outline 1.Graphs and Equations 2.Depicting and Solving Systems of Equations 3.Levels, Changes and Percentage Change 4.Non-linear relationships and Elasticities

5. Depicting 2-dimensional relationships 1.Depicting the association between pairs of variables using a Cartesian Plane. 2.Depicting Bivariate (2-variable) functions.

6. 0 1 1 34285 67 2 3 4 5 6 7 8 The Cartesian Plane y x

7. 0 1 1 34285 67 2 3 4 5 6 7 8 (1,1) (2,6) (5,5) (7,2) Points in a Cartesian Plane

8. Equation for a straight-line function y = mx + b Dependent Variable Independent Variable SlopeIntercept Examples:SlopeIntercept y = x 1 0 y = 3 + 0.25 x 0.25 3 y = 6 – 2 x - 2 6

9. 0 1 1 34285 67 2 3 4 5 6 7 8 run ( = 2) rise ( = 1) Slope = rise / run (In this case = 1/2)

10. 0 1 1 34285 67 2 3 4 5 6 7 8 y x Plotting the Function y = x

11. 0 1 1 34285 67 2 3 4 5 6 7 8 Plotting the Function y = 2x

12. 0 1 1 34285 67 2 3 4 5 6 7 8 y x Plotting the Function y = 3+0.25x

13. 0 1 1 34285 67 2 3 4 5 6 7 8 Plotting the Function y = 6 - 2x

14. 0 1 1 34285 67 2 3 4 5 6 7 8 Plotting the Function x = 4

15. 0 1 1 34285 67 2 3 4 5 6 7 8 Shifting Lines: Changing the Intercept y = 8 – 2x y = 6 – 2x y = 4 – 2x

17. Solving Systems of Equations y = 6 - 2x y = 3 + x 1. Solve out for y 6 - 2x = 3 + x 2. Isolate x so 3 = 3x so x = 1 3. Solve for y using either equation: y = 6-2 = 3+1 = 4

18. 0 1 1 34285 67 2 3 4 5 6 7 8 Solving Systems of Equation y = 6 - 2x & y = 3 + x Solution: (1,4)

19. 0 1 1 34285 67 2 3 4 5 6 7 8 Equilibriums with Different Intercepts: Demand Shift P = 8 – 2Q P = 4 – 2Q P = 6 – 2Q 8-2Q = 3+Q P = 3 + Q Q = 5/3 P = 14/3 6-2Q = 3+Q 4-2Q = 3+Q Q = 1 P = 4 Q = 1/3 P = 10/3

20. 0 1 1 34285 67 2 3 4 5 6 7 8 Equilibriums with Different Intercepts: Supply Shift P = 6 – 2Q 6 – 2Q = 5+Q P = 3 + Q Q = 1/3 P = 16/3 6 – 2Q = 3+Q 6 – 2Q = Q Q = 1 P = 4 Q = 2 P = 2 P = 5 + Q P = Q

22. Levels, Changes, and Percentage Changes LevelChange Percentage Change x t x t = x t – x t-1 % x t = 100[(x t – x t-1 ) / x t-1 ] 100 1202020% 1402016.67% Formula for Percentage Change % x t = 100[(x t – x t-1 ) / x t-1 ] = 100[(x t / x t-1 ) - 1]

23. Some General Rules For z = xy, with small percentage changes % z % x + % y For z = y/x, with small percentage changes % z % y - % x

24. Examples x t = 10 x t+1 = 11 % x t = 10% y t = 20 y t+1 = 24 % y t = 20% z = xyz t = 200 z t+1 = 264 % z t = 100([264/200]-1) = 32 % z = y/xz t = 2 z t+1 = 2.18182 % z t = 100([2.18182 /2]-1) = 9.091 %

26. Non-Linear Relationships May want to associate percentage change with percentage change, rather than change with change. y = 2x + 3 => y = 2 x, but not % y = 2% x One function that relates % y to a constant % x takes the form y = bx a Where a & b are constant parameters

27. Rules of Exponents x 0 = 1 x 1 = x x -1 = 1 / x (x a ) b = (x b ) a = x ab x a x b = x a+b x a / x b = x a – b x a y a = (xy) a x a / y a = (x/y) a x 1/a = a x 2 0 = 1 2 1 = 2 2 -1 = 1 / 2 (2 1 ) 3 = (2 3 ) 1 = 8 2 2 2 3 = 2 5 = 32 2 3 / 2 2 = 2 1 = 2 2 2 3 2 = 6 2 = 36 4 2 / 2 2 = (4/2) 2 = 4 9 1/2 = 9 = 3 Rule Example

28. 0 1 1 34285 67 2 3 4 5 6 7 8 Plotting y = 8 / x

29. 0 1 1 34285 67 2 3 4 5 6 7 8 A Tangent Line to a Hyperbola Shows the Slope at a Point Tangent line at x = 1.5 Slope = - 3.56 Tangent line at x = 5.5 Slope = - 0.26

30. Elasticities An elasticity relates the percent change in one variable to the percent change in another variable; Elasticity between x & y = % y / % x = ( y / y) / ( x / x ) = ( y / x) (x / y )

31. 0 1 1 34285 67 2 3 4 5 6 7 8 Constant Elasticities with a Hyperbola y/y / x/x = ( y/ x)(x/y) = slope (x/y) = -3.56 (1.5/5.33) = - 1 y/y / x/x = ( y/ x)(x/y) = slope (x/y) = -0.26 (5.33/1.5) = - 1

33. 1.In a Cartesian plane, plot the following points: (0,5), (4,2), (6,1), (3,3) 2.Graph the following linear equations 1.y = 2x + 3 2.y = 21 – 4x 3.Solve the system of two equations given by the equations in question (2) above. Also solve the system for the case where equation 2.1 changes to y = 15 – 4x and show how this change in equation 2.1 is represented in a graph. 4.Graph the equation y = x 0.5. Calculate the percentage change in the dependent variable between the points where x=4 and x=4.41. Determine the elasticity between these two points.