QR 1 Lesson Notes 8 - Motivation for Modelling with Linear Functions PP Show
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The document analyzes linear functions and their applications, particularly focusing on cab fares, population growth, and shoe sales profits. It illustrates how to express relationships in functional notation, derive unit rates, and model real-world scenarios using linear equations. Additionally, it explores different pricing plans for cell phone usage and offers guidance on determining the best plan based on usage.
QR 1 Lesson Notes 8 - Motivation for Modelling with Linear Functions PP Show
- 1. 1 Exploring Relationships Between Different Quantities and Measurements: Introduction to Linear Functions and Their Graphs Quantitative Reasoning I Written by Ross Flek Revisiting Rates Payments based on a rate and duration Linear functions (y = m x + b) Slope Intercept Graphs of lines Linear Functions in Excel Review of solving linear equations
- 2. Cab Fare In New York 2 According to the NYC Taxi and Limousine Commission, the current fare is calculated as follows: • The initial charge is $2.50. • Plus 50 cents per 1/5 mile. • Or 50 cents per 60 seconds in slow traffic or when the vehicle is stopped. In moving traffic on Manhattan streets, the meter should “click” approximately every four downtown blocks, or one block going cross-town (East-West). Let’s say you just took a taxi ride, from Parsons t0 Columbia, having travelled approximately 7 miles, and there was no significant traffic. How much would you fare be? This a rate. It would be more convenient to express it as a unit rate; so if you must pay 50 cents for 1/5 of a mile, how much would you have to pay for 1 mile? This a fixed amount. Multiplying by 5 we obtain a rate of $2.50 per mile; this is the unit rate. Fare ($) = Initial Charge ($) + Rate ( $ 𝑚𝑖𝑙𝑒 ) Distance (miles) 2.50$ + 2.50 $ 𝒎𝒊𝒍𝒆 7 miles = 2.50$ + 2.50 7 ( $ 𝒎𝒊𝒍𝒆 )(miles) = $20.00 Note that while the initial charge and price rate are fixed, the fare varies and depends on the distance travelled. Hence fare is the dependent variable (commonly referred to by the letter y) and distance is the independent variable (commonly referred to by the letter x). Quantitative Reasoning I Written by Ross Flek
- 3. Cab Fare In New York 3 Recalling the algebraic expression for a linear relationship, y = m x + b, the rate in this example is the __________ symbolized algebraically by m, and the fixed charge is the _____________ symbolized by b. Furthermore, the fare (y) is a function of distance (x). In functional notation, we convey this by writing Linear functions, f(x) = m x + b, are quite pervasive and pop-up frequently in everyday applications: A cell phone plan may consist of a fixed monthly amount (b) plus usage charges that depend on minutes used, - cost per minute (m) number of minutes (x). More succinctly, (monthly fee) = (fixed amount) + (rate per minute) (number of minutes) A salesperson’s salary might consist of a base amount and a 10% commission on all sales (salary) = (base amount) + (0.10) (total sales) slope y-intercept Quantitative Reasoning I Written by Ross Flek f(x) = 2.50 x + 2.50
- 4. An (Unlikely) Linear Model of Population Growth 4 In 1990, the population in Summersville was 30,000. The population has increased 250 people each year since then, and it is expected to continue to do so. (a) Express the population as a function P(x), where x is the number of years since 1990. (b) Obtain values for the function when x = 0, x = 8, and x = 12. (c) Graph the population function. Let x = the number of years since 1990 Population = P(x) = 30,000 + 250x Quantitative Reasoning I Written by Ross Flek P(0) = 30,000 + 250(0) = 30,000 + 0 = 30,000 P(8) = 30,000 + 250(8) = 30,000 + 2,000 = 32,000 P(12) = 30,000 + 250(12) = 30,000 + 3,000 = 33,000
- 5. An (Unlikely) Linear Model of Population Growth 5 In 1990, the population in Summersville was 30,000. The population has increased 250 people each year since then, and it is expected to continue to grow at the same rate. Population = P(x) = 30,000 + 250x Quantitative Reasoning I Written by Ross Flek For graphing purposes let’s change the units to “thousands of people” Adjusted: P(x) = 30 + 0.25 x (in thousands of people)
- 6. Determining Linear Functions from Data 6 The market manager of a shoe company compiled the data in the table. a. Plot the data values and connect the points to see the graph of the underlying linear function. b. Express the profit as a function of pairs of shoes sold. c. From the graph, determine the profit from selling 4000 pairs of shoes in one month. d. Verify using the functional expression. e. What kind of profit would you expect to make from selling 0 pairs of shoes in a month? What value do you obtain on the graph for x = 0? What does this mean? x Pairs of Shoes Sold in a Month p(x) Monthly Profit from the Sales of Shoes 3 5 5 9 7 13 Quantitative Reasoning I Written by Ross Flek
- 7. Determining Linear Functions from Data 7 The market manager of a shoe company compiled the data in the table. a. Plot the data values and connect the points to see the graph of the underlying function. x Pairs of Shoes Sold in a Month p(x) Monthly Profit from the Sales of Shoes 3 5 5 9 7 13 We plot the ordered pairs and connect with a straight line, verifying the linear relationship between the variables. b. Now chose any two of the three points and find the slope of the line. Recall that slope = rate of change in y with respect to x: 𝒎 = 𝒚 𝟐−𝒚 𝟏 𝒙 𝟐−𝒙 𝟏 𝒚 𝟐 𝒚 𝟏𝒙 𝟏 𝒙 𝟐 𝒎 = 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 = 𝟏𝟑 − 𝟗 𝟕 − 𝟓 = 𝟐 Finally, using the point-slope form: 𝒎 = 𝒚 − 𝒚 𝟏 𝒙 − 𝒙 𝟏 = 𝒚 − 𝟗 𝒙 − 𝟓 = 𝟐 𝒚 − 𝟗 = 𝟐 𝒙 − 𝟓 → 𝒚 − 𝟗 = 𝟐𝒙 − 𝟏𝟎 y = 2x - 1 → p(x) = 2x - 1 Quantitative Reasoning I Written by Ross Flek
- 8. 8Determining Linear Functions from Data The market manager of a shoe company compiled the data in the table. c. From the graph, determine the profit from selling 4000 pairs of shoes in one month. We find the value x = 4 on the graph and the corresponding y value is 7. So, if we sold 4000 pairs of shoes, we would expect a profit of $7000. d. Verify using the functional expression. e. What kind of profit would you expect to make from selling zero pairs of shoes in a month? What value do you obtain on the graph for x = 0? What does this mean? p(x) = 2x - 1 p(4) = 2(4) – 1 = 7 thousand dollars x=0; y= – 1000 p(0) = – 1000 The company would experience a loss of $1000. Quantitative Reasoning I Written by Ross Flek
- 9. Non-Linear Functions 9 If 𝑔 𝑥 = 3𝑥2 − 4𝑥 − 15 find each of the following. a. 𝑔 5 = 3(5)2 −4 5 − 15 = 75 − 20 − 15 = 40 b. 𝑔 1 = 3(1)2 −4 1 − 15 = 3 − 4 − 15 = −16 c. 𝑔 −2 = d. 𝑔 0 = e. 𝑔 3 = 3(−2)2−4 −2 − 15 = 12 + 8 − 15 = 5 3(0)2−4 0 − 15 = 0 − 0 − 15 = −15 3(3)2−4 3 − 15 = 27 − 12 − 15 = 0 Parabola Quadratic Functions; used to model free-fall motion in physics, cost and profit in economics, optimization in construction, geometric applications. Quantitative Reasoning I Written by Ross Flek http://www.meta-calculator.com/online/piag80xkr1qd
- 10. 10 A cell phone company has introduced a pay-as-you-go price structure, with three possibilities. Plan 1: $10 a month + 10 cents per minute Plan 2: $15 a month + 7.5 cents per minute Plan 3: $30 a month + 5 cents per minute (a) For each plan, find a linear function that describes how the total cost for one month depends on the number of minutes used. (Name the functions B(x), C(x) and D(x)) (b) Graph the three functions by hand using the graph paper provided. (Start with the y- intercept and use the slope to obtain the additional points) (c) Use the graph to determine how to advise someone about which plan they should choose. (We’ll need to recall how to solve linear equations!) For the next part, use the worksheet before entering any data into MS-Excel Comparing Telephone Calling Plans Quantitative Reasoning I Written by Ross Flek
- 11. 11 Plan 1: $10 a month + 10 cents per minute Plan 2: $15 a month + 7.5 cents per minute Plan 3: $30 a month + 5 cents per minute (d) Construct a table in Excel showing the total cost for one month for each of the three plans. Organize your data this way: Create a sequence of cells in column A for the various possible numbers of minutes. Label that column “monthly use”. What is a reasonable place to start? What's a good step to use? What's a reasonable place to stop? Use columns B, C and D for each of the three plans. Each row will show the total monthly cost for the corresponding # of minutes indicated in column A. You will need to construct three formulas based on the three functions defined earlier. This will call for clever use of the “$” to keep Excel from changing row numbers and column letters when you don't want it to. (e) Use Excel to draw one chart showing how the monthly bill (y-axis) depends on the number of minutes you use the phone (x-axis) for all three plans. (f) Write a brief paragraph explaining to your friend how she should go about choosing the plan that's best for her. Comparing Telephone Calling Plans Quantitative Reasoning I Written by Ross Flek