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Math Refresher Unit II: Graphing, Exponents, and Logs RPAD Welcome Week 2016 Elizabeth A.M. Searing, Ph.D. 2 Who am I? • Elizabeth Searing, PhD, CNP • Most of my training is in economics, though now I conduct research primarily in nonprofits and social enterprise • MPA Courses at U Albany: – RPAD 501 (Public and Nonprofit Financial Management) – RPAD 613 (Issues in Nonprofit Management) – Elsewhere taught MPA courses in program evaluation, nonprofit financial management, and undergraduate microeconomics and nonprofit management 3 Agenda • Overview • Graphing – Slopes – Intercepts – Areas • Exponents – Radicals – Logarithms 4 Overview (1) • There is a lot of math in an MPA (and the world) • Why in the MPA? – Policy AND management are data driven – CANNOT evaluate evidence without some basic numeracy – In this program, math is crucial to success in RPAD 501 (budgets and accounting), 503 (economics), 504 (data), and 505 (statistics) • Luckily, EVERYONE can do math 5 Overview (2) • • • • Key to math: practice (Almost) nobody gets this stuff the first time If you don’t use it, you tend to forget it So we’re going to review some stuff you probably saw in high school 6 Overview (3) • Suggested book: Barron’s Forgotten Algebra • On order at Mary Jane Books (see ad in Welcome Week booklet) or buy from favorite online retailer • Today: chapters 7, 12, 23, 24, 30, plus areas 7 Graphing • One of the most powerful tools of mathematics is the GRAPH • Powerful means of summarizing relationships between two variables • You MUST be able to graph linear equations 8 Graphing Vocab (1) • We graph in “Cartesian” coordinates (named after Descartes) • Two axes, the horizontal and the vertical – Usually called X and Y, but be flexible – In 503 (or any other economics course), for example, you’ll usually graph Q and P • The axes meet at the ORIGIN 9 Graphing Vocab (2) • A coordinate on a graph is a pair of numbers, such as (x,y) or (Q,P) • The first number is the HORIZONTAL position • The second number is the VERTICAL position • Example: graph (2,3), (5,-4), and (0,7) • The coordinates for the origin are (0,0) 10 Graphing Vocab (3) • Terminology: when we draw a point on a graph, we call it “plotting” the point • Aside: some students like to be very neat when drawing graphs, and break out rulers and evenly measure the distances • This is NOT a good use of time on a test • Just sketch it reasonably 11 Graphing Vocab (4) • A LINE shows a relationship between two variables • A line shows, for every x value, what y value corresponds to it • Thus, a line consists of one equation with two variables 12 Graphing a Line (1) • You may have seen many forms for a line in high school • Very common and popular form is: Y = mX + b • That is, I prefer to write lines so that – The vertical-axis variable is all by itself on the lefthand side – The horizontal-axis variable has been multiplied by something and added to something 13 Graphing a Line (2) • Y = mX + b – b is the “Y-intercept” • It tells us what value Y has when X=0 – m is the “slope” • Slope: Change in Y over Change in X • How much does Y change by when X changes by 1 unit? • Aka, “rise over run” 14 Graphing a Line (3) • To graph a linear equation Y = mX + b 1. Draw and label your axes 2. Plot the Y-intercept (b) 3. Move over 1 unit in X and m units in Y and plot a second point 4. Draw a line through these two points 15 Graphing Hints (1) • Note: the greater the slope, the steeper the line • A slope of 0 means a line is completely flat • A slope of ∞ means a line is completely vertical • A positive slope means a line is upward sloping • A negative slope means a line is downward sloping 16 Graphing Hints (2) • Note Well: you may not be given a line in this form • You may need to solve the equation to get it into the correct form • Example: 2X + 3Y = 9 • 3Y = 9 – 2X • Y = 3 – 2/3 X 17 Graphing Hints (3) • The SLOPE can be used to quickly estimate the change in Y for any change in X • Example: Y = 5 – 4X • If X increases by 2 units, Y DECREASES by 8 units 18 Graphing: Peer Support • P = 100 – 0.25Q – If Q goes up by 8 units, what happens to P? • A = -5 + 4B – If B goes DOWN by 3 units, what happens to A? • A=5 – If B goes UP by 1 unit, what happens to A? • Y = 12 + 2X – If Y goes UP by 3 units, what happens to X? 19 Graphing: Solutions (1) 20 Graphing: Solutions (2) 21 Areas (1) • In 503, we very often need to calculate the areas of regions we’ve graphed – Area of a Rectangle: base•height (aka b•h) – Area of a Triange: 0.5b•h 22 Areas (2) • What is the height of a triangle? – The distance from any corner to the opposite side, if it hits the opposite side at a RIGHT ANGLE • The base is then the length of that side – Very easy to see in a Right Triangle – Example: area between X-axis, Y-axis, and line Y = 10 – 0.5X 23 Areas (3) • Example: area between X-axis, Y-axis, and line Y = 10 – 0.5X • We need to know the distances from the origin to the Y-int and the X-int • Y-intercept: (0,10) (read right off formula) • X-int: 0 = 10 – 0.5X; 10 = 0.5X; X = 20 – X-int = (20,0) 24 Areas (4) • TWO CHOICES for what the “height” is – The height could be the line from (0,10) to (0,0). The base would be the line from (0,0) to (20,0). Area = 0.5*20*10 = 100 – The height could be the line from (20,0) to (0,0). The base would be the line from (0,0) to (0,10). Area = 0.5*10*20 = 100 25 Areas: Peer Support • Examples: Find the area between – X-axis, Y-axis, and line Y = 8 – X – X-axis, Y-axis, and line Y = -5 + 0.2X – Y = 4, Y-axis, and Y = 20 – 2X 26 Areas: Solutions (1) 27 Areas: Solutions (2) 28 Areas: Peer Support • What is the area formed by – Y-axis – Y = 20 – 0.5X – Y = 5 + 2.5X – Intersection (a taste of Session III) at (5,17.5) 29 Areas: Solution 30 Exponents (1) • Exponents are a notation for doing multiplication over and over and over again • 24 means “multiply 2 by itself 4 times” • 24 = 2•2•2•2 = 16 • We call this “raising 2 to the power of 4” 31 Exponents (2) • Special terms: – x2 is “x squared” – x3 is “x cubed” – The thing being multiplied is the BASE – The number of times you multiply is the POWER or the exponent • Special rule: – x0 = 1, as long as x isn’t 0 32 Exponents: Peer Support • What is – 43 – 25 – 124 – (-1)3 – (-5)2 33 Exponents: Solutions 34 What CAN’T you do with exponents? • Things you CANNOT do with exponents: – ADD terms with different powers • x2 + x 5 ≠ x 7 – ADD terms with different bases • 43 + 63 ≠ 103 – Basically, addition and exponents do not mix – The best you can do is add them together when the BASE and the EXPONENT are both the same • 4x3 + 6x3 = 10x3 35 What CAN you do with exponents? • Simplifying is a useful step towards solving an equation. Example: x2•x2 = 16 is much easier to solve if you can rewrite it to x4 = 16 • Why? – Rewrite the exponents as a bunch of multiplications, and then re-group terms • x2•x4 = (x•x)•( x•x•x•x) = (x•x)•( x•x•x•x) = x6 36 What ELSE can you do with exponents? • Here are some basic rules that often come in handy (see p51) – x2•x4 = x2+4 = x6 – (x2)4 = x8 – (x•y)3 = x3•y3 – (x/y)5 = x5 / y5 37 Exponents: Peer Support Simplify: (That is, rewrite without any parentheses, and with each variable used as few times as possible) 1. (5y)2 2. 4Q0 3. x2 • (x5)2 4. (-3c)2 5. 2P3 + 3P4 38 Exponents: Solutions 39 Powers, Roots, and Logs (1) • We’ve been looking at mathematical statements like bm = x • So far, we’ve taken b and m and found x • Two other processes: – Radicals/Roots: Given x and m, find b – Logarithms: Given x and b, find m 40 Powers, Roots, and Logs (2) • • • • Example: 23 = 8 “Two cubed is 8” “The cube root of 8 is 2” “The log-base-2 of 8 is 3” Roots: 3 8 2 Logarithms: log 2 8 3 41 Roots (1) • Even-powered roots have two possible solutions: the positive or the negative • The square root of 25 could be either 5 or -5 • We call the positive square root the “principal square root” 42 Roots (2) • Negative numbers do NOT have even-powered roots • The square root of -16 is NOT a real number (There’s a whole system of math, called “complex algebra,” that uses “imaginary numbers” based on the square root of negative numbers. This has proved surprisingly useful in many settings. Just not here.) 43 Roots (3) • It turns out that roots follow the EXACT same rules as exponents • Actually, roots are a special case of exponents • We can re-write roots as fractional exponents • √25 = 251/2 44 Cool Things You Can Do with Roots a b a b 1/2 (ab) a 1/2 b 1/2 x y xy y 2 3 3 2= 3 2 45 Roots: Peer Support • Simplify: 1. ab a 2. 2 10 3. x 2 4. 3 15 3 15 3 15 5. x 3 x 46 Roots: Solutions • Simplify: 47 Logs (1) • Logs are one of the least intuitive subjects in algebra • Unfortunately, they’re very important • My goal here is to remind you that they exist and that they have some (incredibly unintuitive) rules 48 Logs (2) • logbN = x • Say it: “The log-base-b of N is x” • It means: “What power do you need to raise b to, to get N?” • 43 = 64, so log4 64 = 3 • Say it! 49 Logs: Peer Support • Problems: 1. log2 64 2. log5 125 3. log90 1 4. log8 2 50 Logs: Solutions 51 Natural Logs • “Natural log”, aka “log-base-e” or ln • ln (x): “What power do we need to take e to in order to get x?” • It turns out that there’s a number, Euler’s constant, that we call e, that is super useful to use with logs • e is like pi, in that it has an infinite number of decimals. It’s about 2.72 • Allows us to compare elasticities in statistics 52 Log Rules logb AC logb A logb C logb A C logb A logb C logb A k logb A k 53 Example of Logs Making Life Easier Y AK L • Economists often model an economy’s output using the CobbDouglas form • Suppose you had data on output (Y), capital (K), and labor (L), and wanted to estimate A (productivity/technology), α (returns to capital), and β (returns to labor.) • Statistically, this would be a real @%@#$% • It’s more complicated to estimate multiplicative models in statistics 54 An Example of Logs Making Life Easier ln Y ln A ln K ln L • If you log both sides, you can use log laws to change the multiplicative model into an additive one • This is a much easier model to estimate statistically • Yay logs!