1/18/2012 Finite Mathematics MAT 141: Chapter 1 Notes Slopes and Equations of Lines. David J. Gisch January 10, 2012 Recall the Cartesian Plane Intercepts • The x-intercept is where a line crosses the x-axis. • The y-intercept is where a line crosses the y-axis. • The x-intercept occurs when the y-ordinate is zero. René Descartes • The x-intercept occurs 1596-1650 when the y-ordinate is zero. 1
1/18/2012 S lope S lope Example 1.1.1 Find the slope of the line. • The order of the subscripts does not matter as long as you are consistent. S lope S lope Example 1.1.2: What is the slope of the line with the following properties? (a) passes through 2, 3 and 4, �1 (b) passes through �3, 5 and has an x-intercept of 5 . 2
1/18/2012 Equations of Lines S pecial Cases S ummary of Equations Equation of a Line Example 1.1.3 Find the slope of the line. (a) � � 2� � 10 (b) 4� � 8� � 24 (c) Horizontal and through �2.2, 5.875 3
1/18/2012 Equation of a Line Parallel and Perpendicular Example 1.1.4 Find the equation of the line. (a) passing through 2, 8 and �4, �4 (b)has an x-intercept of 5 and a y-intercept of 2 (c) Horizontal and through 2, 5 Or if their slopes are opposite reciprocals � � (e.g. � and � � ) (d) Vertical and through �2, 10 Equation of a Line Equation of a Line Example 1.1.6 Find � so that the line through Example 1.1.5 Find the equation of the line. 4, �1 and ��, 2� is (a) Through �4, 6 and parallel to 3� � 2� � 13 (a) parallel to 2� � 3� � 6 (b) perpendicular to 5� � 2� � �1 (b)Through 3, �4 and perpendicular to � � � � 4 4
1/18/2012 Graphing Graphing � Example 1.1.8 Graph the line 5� � 6� � 11 Example 1.1.7 Graph the line � � � � � 5 Linear Functions Recall that � � notation just signifies that we have a function with the Linear functions and Applications variable � . There is also nothing special about the letter � . • We could have ���� a function describing height in reference to time. • We could have ���� a function describing calories burned in reference to hours. • We could have ���� a function describing your grade in reference to minutes of studying. 5
1/18/2012 S upply and Demand Example 1.2.1: Suppose that Greg Tobin, manager of a giant • Supply – the amount of “objects” a company produces. supermarket chain, has studied the supply and demand for • Dem and – the number of people wanting to buy those watermelons. He has noticed that the demand increases as “objects.” the price decreases. He has determined that the quantity (in • Equilibrium – where supply and demand are equal. Best thousands) demanded weekly, � , and the price (in dollars) scenario for the company (not the consumer). per watermelon, � , are related by the linear function � � � � � 9 � 0.74� (a) Find the demand at a price of $5.25 and at $3.75 (b) Greg also noticed that the supply decreased as the price decreased. Price p and supply q are related by � � � � � 0.75� Find the supply at a price of $5.25. (c) Graph both functions on the same axes. S upply and Demand Equilibrium Point • We saw graphically that the equilibrium point is when the quantity is 6000 watermelons and the price is $4.50. • Solve for this algebraically. � � � � � 9 � 0.74� � � � � � 0.75� 6
1/18/2012 Cost Function Marginal Cost Marginal cost is the rate of change of cost ���� at a level of production � (i.e. slope). Example 1.2.2: The marginal cost to make � batches of a prescription is $10 per batch, while the cost to produce 100 • Give an example of a linear cost function. batches is $1500. Find the cost function, given that it is linear. Cost and Revenue Cost and Revenue • Just like supply and demand we can analyze the Cost > Revenue Revenue > Cost relationship of cost and revenue. • Spent more than we took in. • Took in more than we spent • Cost – The amount you spend to make/provide a service • A loss of money • A profit or item. • Revenue - the amount of money you make for selling that service or item. 7
1/18/2012 Critical Thinking Cost and Revenue Example 1.2.4: The cost of producing � iPhone 4s’s (32 Example 1.2.3: For supply and demand you have the GB) is equilibrium point. For cost and revenue you have the � � � 300� � 1500000 break-even point. Compare and contrast the two (from the And the revenue is companies point-of-view). � � � 849�. What is the break-even quantity? *Apple sold about 15 million iPhones just in the 4 th quarter. Least S quares Line • The table below lists the number of accidental deaths in the U.S. through the past century. The Least Squares Line 8
1/18/2012 Least S quares Line Least S quares Line • Now it is obvious that the data has a linear trend. We would like to create a line, as accurately as possible, which would help predict future data. But how? Least S quares Least S quares on TI-84 • We will not do this by hand • How do we get the line below or even know • We will use the TI calculator to calculate the line of best when it is the “best fit”? fit using least squares. 1. Stat Edit 2. Input your list. (check if the columns are L1, L2) 3. Stat CALC LineReg(ax+b) 4. Enter, Enter 9
1/18/2012 Graphing Y our Results Correlation Coefficient 1. 2nd Stat Plot (Top Left), Enter • This gives us the best possible line but it would be nice to know how linear the data really is. 2. Turn plot on, check xlist, ylist • The correlation coefficient, r, tells us how strong our 3. Y=, and type the results of LinReg(ax+b) conviction can be in saying the data is linear. ▫ In this case type in -.5597x+90.3333 ▫ Clear any other equations 4. Zoom Zoom Stat • Again, we will not do this by hand. We will let the calculator do it. 1. 2 nd , Catalog 2. Find DiagnosticOn 3. Enter, Enter 4. Run LinReg(ax+b) again. Correlation Coefficient Linear Regression (Least S quares) • The closer � is to 1 or �1 , the more linear it is. Example 1.3.1: An economist wants to estimate a line that • The sign tells you whether it is sloping up or down. relates personal consumption expenditures � and disposable income � . Both � and � are in thousands of dollars. She interviews eight heads of households for families of size 3 and obtains the data show below. (a)Us linear regression to find a prediction equation. (b)State your level of confidence in your equation. (c)Using your equation, predict the amount of This is close to -1 so the data is very linear and consumption if a family had a disposable we can feel confident about our equation income of $42,000. � � �0.5597� � 90.3333 10
1/18/2012 Linear Regression (Least S quares) Linear Regression (Least S quares) Example 1.3.2: The marketing manager at Levi-Strauss Example 1.3.3: As a class gather data on your height and wishes to find a function that relates the demand � for the length of your wingspan. Make your height the men’s jeans and � , the price of the jeans. The following independent variable. data was obtained. (a)Us linear regression to find a prediction equation. (b)State your level of confidence in your equation. (a)Us linear regression to find a prediction (c)If I was 73” tall, what would my estimated wingspan be? equation. (b)State your level of confidence in your equation. (c)Using your equation, predict the amount of demand if the price of jeans were $21. (d)Based on the data, what is the optimal price? FIRS T TES T • We will take a test over the Review and Chapter 1 next Wednesday ▫ You should be able to Simplify, solve, factor, use the quadratic equation Anything from Chapter 1 ▫ No notes. ▫ Bring your calculator! ▫ We will review for the first 20 minutes, then you get the rest of the period. 11
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