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Everything You Always Wanted to Know About Math*

Everything You Always Wanted to Know About Math*. * But were afraid to ask. Outline. Graphs and Equations Depicting and Solving Systems of Equations Levels, Changes and Percentage Change Non-linear relationships and Elasticities. Graphs and Equations. Depicting 2-dimensional relationships.

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Everything You Always Wanted to Know About Math*

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

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

  3. Graphs and Equations

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

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

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

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

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

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

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

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

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

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

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

  15. Systems of Equations

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

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

  18. Equilibriums with Different Intercepts: “Demand Shift” Q = 5/3 P = 14/3 P = 8 – 2Q 8 8-2Q = 3+Q 7 P = 3 + Q 6 Price Q = 1 P = 4 5 6-2Q = 3+Q P = 6 – 2Q 4 P = 4 – 2Q 3 Q = 1/3 P = 10/3 2 4-2Q = 3+Q 1 0 6 7 1 2 3 4 5 8 Quantity

  19. Equilibriums with Different Intercepts: “Supply Shift” Q = 1/3 P = 16/3 8 6 – 2Q = 5+Q P = 5 + Q 7 6 P = 3 + Q Price Q = 1 P = 4 5 6 – 2Q = 3+Q 4 3 P = Q Q = 2 P = 2 2 6 – 2Q = Q 1 P = 6 – 2Q 0 6 7 1 2 3 4 5 8 Quantity

  20. Percent Changes

  21. Levels, Changes, and Percentage Changes • Level Change Percentage Change • xt xt = xt – xt-1 % xt = 100[(xt – xt-1) / xt-1] • 100 • 120 20 20% • 140 20 16.67% • Formula for Percentage Change • % xt = 100[(xt – xt-1) / xt-1] • = 100[(xt / xt-1) - 1]

  22. Some General Rules • For z = xy, with small percentage changes • %z  %x + % y • For z = y/x, with small percentage changes • %z  %y - % x

  23. Examples xt = 10 xt+1 = 11 % xt = 10% yt = 20 yt+1 = 24 % yt = 20% z = xy zt = 200 zt+1 = 264 % zt = 100([264/200]-1) = 32 % z = y/x zt = 2 zt+1 = 2.18182 % zt = 100([2.18182 /2]-1) = 9.091 %

  24. Elasticities

  25. 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 = bxa • Where a & b are constant parameters

  26. x0 = 1 x1 = x x-1 = 1 / x (xa ) b = (xb ) a = xab xa x b = xa+b xa / x b = xa –b xa y a = (xy)a xa / y a = (x/y)a x1/a = ax 20 = 1 21 = 2 2-1 = 1 / 2 (21 ) 3 = (23 ) 1 = 8 22 23 = 25 = 32 23 / 22 = 21 = 2 22 32 = 62 = 36 42 / 22 = (4/2)2 = 4 9 1/2 = 9 = 3 Rules of Exponents Rule Example

  27. Plotting y = 8 / x 8 7 6 5 4 3 2 1 0 6 7 1 2 3 4 5 8

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

  29. 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 )

  30. ConstantElasticities with a Hyperbola 8 • y/y / x/x = (y/x)(x/y) = slope  (x/y) • = -3.56  (1.5/5.33) = - 1 7 6 5 • y/y / x/x = (y/x)(x/y) = slope  (x/y) • = -0.26  (5.33/1.5) = - 1 4 3 2 1 0 6 7 1 2 3 4 5 8

  31. Exercises

  32. In a Cartesian plane, plot the following points: (0,5), (4,2), (6,1), (3,3) • Graph the following linear equations • y = 2x + 3 • y = 21 – 4x • 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. • Graph the equation y = x0.5. 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.

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