Pi in the Sky Period 3
Nicolas Fache, Vuk Radovanovic, Matthew Lu
1 Table of Contents: 1­7: Properties Provable from the Axioms​
……..…………….….....………...…3 3­2: Properties of Linear Function ​
……………………………………....……..4 4­11: Linear Program​
ming​
………………………………….……….………..5­6 6­9: Logarithms with Other Bases​
………….…………………………...…..…..7 6­13: The Add­Multiply Property of Exponential Functions​
……………….......8 Practice Problems​
…………………………………………………..…….……..9 Appendix & Answers​
………………………....…………….…...……..…..10­11 2 1­7: Properties Provable from the Axioms Axioms to explain: Reflexive. if x is a real number, x = x. Symmetry: if x = y, then y = x. Transitivity for equality ­ if x = y and y = z, then x = z. Transitivity for order ­ if x > y and y > z, then x > z, as well as for the other side. Trichotomy ­ if x and y are two real numbers, then exactly one of the following must be true ­ y < x, y > x, or y = x. Explain the theorem­proving techniques 1. Start with one member of the desired equation and transform it to the other member. 2. Start with a given equation and transform it to the desired equation. 3. Start somewhere else and use a clever series of transformations or a clever argument. 3 3­2: Properties of Linear Function Graphs Intercepts: The y­intercept of a function is the value of y when x=0 The x­intercept of a function is the value of x when y=0 To easily find the slope of a linear function use the rise/run, or the equation form (y​
­y​
)/(x​
­x​
) 2​
1​
2​
1​
Slope Intercept Form: y=mx+b m equals the slope of the graph, and b equals the y­intercept Horizontal and Vertical Lines: If y=constant, then the graph is a horizontal straight line. Slope=0 If x=constant, then the graph is a vertical straight line. There is no number for slope because it is infinitely large. Example Problem Plot the graph of 5x+7y=14 7y=­5x+14 y=­5/7 x + 2 m=­5/7 b=2 Plot the graph. Use the slope and y­intercept, where possible. Y=⅖ x+3 Y=­5 4 4­11: Linear Programming Linear programming is most often used in real­world problems. You use the data given to you to plot a graph of a system. Ordered pairs within the system which satisfy all the restrictions are called “feasible”. The region of the graph in which a set of ordered pairs could be considered feasible is called the feasible region. When doing linear programming, you must graph each inequality and then solve to find the corner points ­ the extreme boundaries of the feasible region where multiple points intersect. For linear programming problems, you can choose to use a calculator, but these problems are possible without one. Let’s start with an economics problem in which you attempt to maximize the profit your company can make. This is sure to earn you a promotion! EXAMPLE PROBLEM The manager of your local football stadium is considering adding new seats to allow more people to watch the game. He can add seats to the lower section, to accommodate for people who are afraid of heights, or add seats to the higher sections, for people who want a better aerial view of the game. Aerial seats have tickets that cost $40 and lower seats have tickets that cost $30. Where should the manager install new seats to make the highest profit? Also, the manager needs to make $250,000 during the upcoming Super Bowl to keep his team funded. In Linear Programming you must define the variables before you set up the equations, so let’s do this first. Let the number of high seats purchased = H. Let the number of low seats purchased = L. Now, we can finally take a look at the restrictions. 1. City ordinances state that the height of the stadium can only go so high, so the manager can only add a total of 8,000 high seats. 2. There is not much room left on the bottom floors, so he can only add 4,000 low seats. 3. The manager estimates that he must have at least 7,000 new seats to accommodate for the upcoming Super Bowl. Now we need to write a system of inequalities according to these restrictions. H <= 8,000 L <= 4,000 H + L >= 7,000 Now, we need to plot the graph of the system. 5 (in this graph, y represents H and x represents L.) The feasibility region is the area in which all of the systems overlap. Now, we can use the graph to find things like the maximum and minimum of the feasible region. Simply find the coordinates of the vertices of the polygon that is created and plug these into the equation that gives you the total income: 40H + 30L = C These coordinates can be found by using the given information seen on the graph, as well as solving the system of equations. It is much easier if we set up a table to find the maximum profit, using the vertices as our points. H L C 7000 0 280,000 0 4000 120,000 3000 4000 240,000 0 0 0 Therefore, the manager will be able to make the required profit, but only if he invests in some new high­view seating. 6 6­9: Logarithms with Other Bases A logarithm is an exponent. y​
Base b Logarithm:​
log​
x = y if and only b​
=x b ​
Where x>0, b>0, and b​
≠​
1 y​
is the logarithm b ​
is the base x ​
is the argument log​
x and log x are the same thing 10​
Example Problems: log​
8=x 2​
x​
2​
=8 x =​3 2​2​
x=3 log​
x =­4 3 ​
­4​
3​
=x 1/81 =x 7 6­13: The Add­Multiply Property of Exponential Functions x​
The general equation of an exponential function is f(x)= a x b​
(Adding a constant to the value of x multiplies f(x) by a (different) constant) Where a and b stands for constants x​
c
The Add­Multiply Property of Exponential Functions is If f(x)= a x b​
, then f(x+c)=f(x) x b​
The Add­Add Property of Linear Functions: If f(x)= mx+b, then f(x+c)=f(x)+mc (Adding a constant to the value of x adds a different constant to the value of f(x)) 8 Practice Problems 1.7: Example of each axiom 1. transitive for equality 2. transitivity for order 3. symmetry for equality 4. reflexive axiom for equality 5. trichotomy 3.2: Plot the graph of.. y=2/5x+2 y=­4x+½ 14x+4y=20 4.11: Mr. Nguyen is selling used video games to fund his city council candidacy. He will sell his PS3 games for $20.00 each, and his PS4 games for $30.00 each. He has 40 PS3 games and 80 PS4 games (Jeez! How is he going to find the time to campaign?) to sell. However, he still wants to keep some games. I mean, who wouldn’t? He wants to have at least 20 games left at the end of the sale. Also, the number of PS4 games sold will be no more than 6 times the number of PS3 games. Don’t ask me why, ask Mr. Nguyen! How many of each kind should Mr. Nguyen sell to make the greatest amount of money for his candidacy? How much money is this? 6.9: Solve for X. log​
9=x 3 ​
log​
81=4/3 x ​
log​
x=3 2​
log​
16=x 4​
6.13: 1. f(2)=36 and f(5)=54 2. f(4)=100 and f(6)=70 3. f(­2)=1 and f(3)=2 9 Appendix & Answers 1.7 1. ​
If a=b, and b=c, then a=c 2. ​
If x<y, y<z, then x<z 3. ​
If a=b, then b=a 4. ​
a=a 5. ​
Either x<y, x=y, or x>y 3.2 1. 2. 3. 10 4.11 Let x = the number of PS4 games sold. Let y = the number of PS3 games sold. Let c = the total cost. c = 30x+20y Restrictions: ● He must sell at most 40 PS3 games, and at most 80 PS4 games. ● 40+80 games = 120 games, and if he wants to keep 20, that means he must sell at most 100 games. ● The number of PS4 games sold must be at most 6 times the number of PS3 games. x > 0 y > 0 x <= 80 y <= 40 x + y <= 100 x <= 6y Graph: c = 30x + 20y x y c ($) 80 39/3 2,660 80 20 2,800 60 40 2,600 11 Mr. Nguyen must sell 80 PS4 games and 20 PS3 games to get a maximum profit of $2,800. 6.9 x=3 x=27 x=8 x=2 6.13 1. 2. 3. 12