Question

Each person on the team has a different opinion based on their diverse backgrounds:

Robert’s family has been struggling financially.
They are counting on Robert to bring home money from his new business sooner rather than later. From his perspective, the cheaper the car, the better.

Samantha knows that first impressions are important. Her top priority is that their clients see them as a luxury business. She would prefer a fancier car.

Lyna has always been cautious. She remembers her uncle started a new business a few years ago that didn’t end up working out, and he lost a lot of money. She doesn’t want the same thing to happen to her. She’s not sure whether they should even be purchasing a car at all.

1. Research the cost of two different new cars that might be good options for Robert, Samantha, and Lyna. Create sequences to show how much each car would be worth each year after purchase until the 5th year. Also create sequences to show how much they will still owe on each car each year after purchase until the 5th year. What type of sequences are these?

2.Write a recursive and an explicit formula for each of the four sequences identified in question 1.

3.Write a function to represent each of the four sequences. Identify whether each function shows linear change, exponential growth, or exponential decay. Explain how you know.

4.You have learned how to prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals. Show that these facts hold true for the functions you wrote in question 3.

5.For each car you chose, graph the function that shows the amount the car is worth and the function that shows the amount that is still owed on the car on the same coordinate grid. Discuss the key features of the graphs and connect these features to whether the graphs are showing linear change, exponential growth, or exponential decay.

6.If you were on Robert, Samantha, and Lyna’s team, how would you take into account all their opinions to come to a decision about what to do? Does purchasing a car and trading in after 5 years make sense, or is there a better option? Propose a plan that would satisfy everyone. Use your sequences and graphs to justify your proposal.

Answer 1

The question involves creating mathematical sequences, formulas, functions, and graphs to help decide on the best car purchase for a team with diverse backgrounds. It includes considering depreciation, loan repayment, and the impacts on financial planning, using algebraic models to justify the eventual decision.

Step-by-step explanation:

The student's multi-part question revolves around helping a team decide on the best car purchase considering their varied circumstances. This analysis involves researching car costs, creating sequences for the car's value and amount owed over five years, developing formulas and functions to represent these sequences, and graphing these functions. Finally, the student is asked to take into account the team's opinions and propose a plan that might include purchasing a car and trading it after five years, supported by mathematical justification.

To address this question, we must select two cars that represent the different perspectives of the team members. For Robert, we'd select a more cost-effective car, and for Samantha, a luxury vehicle. Then, it's necessary to establish depreciation rates and loan terms for both vehicles to create sequences showing their values and the amounts owed over five years. These sequences are typically arithmetic sequences for car values (due to linear depreciation) and possibly geometric sequences for the loan amounts if the interest is compounded.

Once the sequences are created, we can write the recursive and explicit formulas, as well as functions to represent each sequence. To illustrate growth rates, we'd utilize graphical representations. As for making a decision that considers the concerns of each team member, a compromise might be leasing a mid-range car, which would allow for lower upfront costs while maintaining a professional image. This would involve using the sequences and graphs to argue the financial viability of the plan.

In summary, we would combine linear and possibly exponential models to reflect the car value and loan sequences. For each car choice, we would show the function displaying the car's worth and loan balance. Visually, the key features of the graphs will depict if the car's value decreases linearly or exponentially over time, while the loan amount graph will indicate if the team is approaching a payoff.