MLC at Boise State Polynomials Activity 3 Week #5

Transcription

1 Polynomials Activity 3 Week #5 This activity will be discuss maximums, minimums and zeros of a quadratic function and its application to business, specifically maximizing profit, minimizing cost and break-even points. Introduction to the Math of Maximums, Minimums and X-intercepts Below are two different quadratic functions and their graphs. In order to be successful in this activity, you must understand where the maximums, minimums and x-intercepts are located on the graph, if they exist, and what they represent. In algebra, it is important that you know how to calculate them mathematically in additon to locating them on the gragh. A. B. 1. Which graph has a maximum? Explain and indicate where it is on the graph. Describe a situation in which locating a maximum would be necessary. 2. Describe a situation in business where a maximum would be useful. 3. Which graph has a minimum? Explain and indicate where it is on the graph. Describe a situation in which locating a minimum would be necessary. pg. 1

2 4. Describe a situation in business where a minimum would be useful. Next, locate the x-intercepts on both graphs. 5. What is the value of the y-coordinate at each of the x-intercepts? In a quadratic function, the maximum and minimum values are located at the vertex of the parabola. If we wanted to find the vertex algebraically, we would use the vertex formula: b b b, f. The maximum/minimum VALUE is the y-coordinate of the vertex, f, and 2a 2a 2 a b is located at the x-coordinate,. 2a In a quadratic function, the x-intercepts of a function are also called the zeros. The zeros of a function are values that make the equation equal to zero (ie. y = 0 or f x 0 ); therefore, x- intercepts are also the zeros of a function. Creating the Functions Country Motorbikes Incorporated finds that it costs $200 to produce each motorbike, and that fixed costs are $1500 per day. 6. What is the cost function for producing n motorbikes per day? Recall that cost is equal to the fixed cost plus the variable cost. p n motorbikes will be sold. The price function is n, where p is the price (in dollars) at which exactly n 7. What is the revenue function for selling n motorbikes? Recall that revenue is equal to the price of each item times the number of items sold. Hopefully you got C n n for the cost function and R n 600 5n n 2 R n 600n 5n for the revenue function. or 8. Describe how you would find the profit function using revenue and cost. 9. What is the equation for the profit function? Now let s examine these three functions in Excel; starting with revenue. You will use column A for the number of motorbikes. First we must determine the values that we will use for n. 10. What values are appropriate for n in this business model? Justify your selection. pg. 2

3 Hopefully you chose values between zero and 200. Discuss why this is an appropriate interval (domain) for this situation. Creating the Revenue Values and Graph An employee recognized the limitiations of the revenue graph created above, where revenue is price time the number of motorcycles sold. The employee feels that this process is too simple and ignores many other issues of the related to revenue. This employee collected actual data on revenue and created the following table: Number of Revenue R( n ) motorbikes sold ( n ) 20 $ $15, $15, $11, $7,500 Use EXCEL to create a graph create a trendline and find a quadratic equation for this set of data, using the following steps: The Graph Enter the labels into A1 and B1. Enter the values in A2:A6 and B2:B6 Highlight all cells Click on insert and choose the scatter graph The Trendline In the graph, left click on a data point Choose add trendline In pop-up window Choose polynomial and order 2 Click on box to display the equation You will have a Revenue equation which better fits your observed data. Please write the equation to 3 decimal points: Examining the Revenue Graph Based on graph the employee created pg. 3

4 11. If this was your store, how many motorbikes would you want to sell? Explain your answer. 12. What would be the number of motorbikes that you could sell to have a maximum revenue? What is that revenue? Where is this value located on the graph? 13. For what number of motorbikes sold will the revenue be zero? Indicate where this is located on the graph. 14. What will be your revenue for selling 35 motorbikes? Explain how you found this value both graphically and algebraically. 15. If you sell 20 motorbikes, will you make $10,000? Explain why or why not. Creating the Cost Values and Graph Again the employee decided to collect data on the actual costs of the moterbikes built. Number of Cost C( n ) motorbikes Manufactured ( n ) 20 $ $11, $16, $20, $24,000 Follow the instructions to create the Cost function graph and to create a Trendline for the cost functions (as done for the revenue function). In this case the trendline would be linear. Write the equation down to three digits. Examining the Cost Graph 16. What is the fixed cost? Indicate where it is located on the graph. Explain what is meant by fixed cost. 17. What is the cost of producing 60 motorbikes? Where is this located on the graph? Creating the Profit Values and Graph To create a profit equation, subtract the Cost equation from the Revenue equation ( ) ( ) ( ) pg. 4

5 Use the two equations from your Trendlines. To create a profit equation: In a cell type N and the cell next to it, type Profit. In the cells below N pick some resonable values for N (25, 50, ) In the cell next to each value for N calculuate the profit. Use this date table to create your chart. Examining the Profit Graph 18. If this was your store, how many motorbikes would you want to sell? Explain your answer. 19. What would be the number of motorbikes that you could sell to have a maximum profit? What is that profit? Where is this value located on the graph? 20. For what number of motorbikes sold will the profit be zero? Indicate where this is located on the graph. 21. What will be your profit for selling 35 motorbikes? Explain how you found this value both graphically and algebraically. 22. If you sell 20 motorbikes, will you make $10,000? Explain why or why not. Using Excel to Find Maximum/Minimum Values and Zeros of a Graph For each graph (revenue, cost, profit) we will be using Excel to find the maximum values, minimum values and/or zeros. Let s start with a maximum value. 23. If this was your company, which of these quantities would you want to maximize? (revenue, cost or profit) Why? Use Excel to find the maximum of the function(s) you chose. To do this, you will need the Week 4 EXCEL printout. You can place these value(s) in any cell, but make sure that you label the cells. 24. Interpret the result(s) from Excel; be sure to use correct units on all answers. 25. What is the number of motorbikes you need to sell in order for the profit to be as large as possible? pg. 5

6 26. What is the number of motorbikes you need to sell in order for the revenue to be as large as possible? Let s look at a minimum value. 27. If this was your company, which of these quantities would you want to minimize? (revenue, cost or profit) Why? Use Excel to find the minimum of the function you chose. 28. What is the number of motorbikes you need to make in order for the cost to be as small as possible? Think about this answer in the context of the problem. Does this make sense? Why or why not? For each of the three graphs, use the Excel printout for week 4 to find the zeros of each graph. Hint, use the Solver tool with value of zero. 29. Explain what the zeros represent for each function. 30. How would you find the zeros algebraically? Examining Break-Even Points You will want to create a graph in Excel that has all three functions on one grid. Highlight the cells to include columns A, B, C, and D as well as the rows 3 through your chosen values. Once highlighted, go up to the top, click on Insert and under the charts, find scatter plot then click on it. You should now see a scatter plot with three colored sets of points; one for each function (revenue, cost, profit). Add the Trendlines for each set of points as described above. Break-even points can be found in one of two ways. First, to break-even the total cost and total revenue must be equal. Based on this definition, using the graph that has all three functions, locate the break-even point(s). Then use the Solver to find them. 31. Explain what the break-even point(s) mean in this situation. Second, to break-even the profit is equal to zero. Based on this definition, using the graph that has all three functions, locate the break-even point(s). Then use the Solver to find them. 32. Explain what the break-even point(s) mean in this situation. pg. 6

7 33. How many motorbikes must be sold in order to break-even? 34. What do the break-even point(s) correspond to algebraically on the profit function? 35. What factors could cause a change in the cost function? 36. What factors could cause a change in the revenue function? 37. What factors could cause a change in the profit function? pg. 7