1 / 32

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.

wade-mcintosh
Télécharger la présentation

Everything You Always Wanted to Know About Math*

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related