Unit Activity Unit: Advanced Functions This activity will help you meet these educational goals: Mathematical PracticesYou will make sense of problems and solve them, reason abstractly and quantitatively, construct arguments, and model with mathematics. Introduction In this unit, you learned about advanced functions such as quadratic functions and composite functions. You modeled scenarios using these more advanced functions. In this activity, you will analyze real-world situations by modeling them as functions and interpreting their graphs. __________________________________________________________________________ Directions and Analysis Task 1: Electronics Manufacturer An electronics manufacturer recently created a new version of a popular device. It also created this function to represent the profit, P(x), in tens of thousands of dollars, that the company will earn based on manufacturing x thousand devices: P(x) = -0.16x2 + 21.6x - 400. a. The profit function for the first version of the device was very similar to the profit function for the new version. As a matter of fact, the profit function for the first version is a transformation of the profit function for the new version. For the value x = 40, the original profit function is half the size of the new profit function. Write two function transformations in terms of P(x) that could represent the original profit function. Type your response here: b. Write the two possible functions from part a in simplified form. Type your response here: c. Which of the two possible functions would represent a better situation for the company at the time the original version was released? Explain. Type your response here: d. After a natural disaster, the company decided to give 2,000 of the new devices to organizations to use in auctions or raffles to raise money to help those affected. In terms 1 2014 EDMENTUM, INC. of function transformations and P(x), write a function that would represent the new profit function based on this. Type your response here: e. Based on this new function, approximately how many devices would the company have to manufacture to avoid losing money? Type your response here: f. The average rate of change of a function over a given interval can be determined by finding the slope of the line that connects the two points on the function representing the endpoints of the interval. Find the average rate of change of the function from part d over the interval [60, 80]. Type your response here: Task 2: Population of a Species The population of a species in the wild in Asia has been decreasing exponentially since 1980. This function represents the population of the species t years after 1980: W(t) = 20,000(0.95t). The Wildlife Refuge Foundation noticed this trend before 1980 and decided to try to help reduce the population decline by breeding the species in captivity. The graph shows the number of these animals that have been born in captivity since 1980. a. About how long will it take for the number of the species born in captivity to be equal to the number of the species remaining in the wild? 2 Type your response here: b. Create a function to represent the number of the species that have been born in captivity, C(t). Type your response here: c. Assuming that the species has a long life span and that all those born in captivity are still alive today, create a function, P(t), to represent the total number of the species that are alive, both in the wild and in captivity. Type your response here: d. At what point would you say that the program to breed the species in captivity stops the decline in population? Explain. Type your response here: e. If this trend continues, about what will be the total population of the species in the world in 2020? Type your response here: f. How many more of the species will be alive in 2020 because of the breeding program? Type your response here: __________________________________________________________________________ Resources Document any references you used for this project below. At minimum, include a title and URL for any Internet resource: 3 Evaluation Your teacher will use this rubric to evaluate the completeness of your work as well as the clarity of thinking you exhibit. Total Points: 10 Task 1: Electronics Manufacturer Task points: 5 a. Write two transformations of a function that pass through a given point in terms of the function. 1 b. Write the simplified form of the two transformations. 1 c. Decide which transformation would be better for the company. 1 d. Write a transformation of a function in terms of the function. 1 e. Solve a function for a point. 0.5 f. 0.5 Find the rate of change of a function over a given interval. Task 2: Population of a Species Task points: 5 a. Estimate where an exponential function and a linear function intersect. 1 b. Write a function from a graph. 1 c. Combine an exponential function and a linear function. 1 d. Interpret a graph of the function in terms of the situation. 1 e. Solve the function at a certain value. 0.5 f. 0.5 Solve a linear function at a certain value. 4