Lesson 4 Getting Down to Business Solidify Understanding

Learning Focus

Make modeling decisions about business plans.

Interpret mathematical models to make business decisions.

Determine which type of function grows faster and make arguments about why.

Which type of function increases faster—linear or exponential?

Which model is best for a given situation, discrete or continuous?

How can mathematical models help to make business decisions?

Technology guidance for today’s lesson:

Basic Graphing of Linear Functions: Casio ClassPad Casio fx-9750GIII
Basic Graphing of Exponential Functions: Casio ClassPad Casio fx-9750GIII

Open Up the Math: Launch, Explore, Discuss

Calcu-rama had a net income of $$ 5 million$ in 2020, while a small competing company, Computafest, had a net income of $$ 2 million$ . The management of Calcu-rama develops a business plan for future growth that projects an increase in net income of $$ 0.5 million$ per year, while the management of Computafest develops a plan aimed at increasing its net income by $15 %$ each year.

1.

Create standard mathematical models (table, graph, and equations) for the projected net income over time for both companies.

Calcu-rama Model:

Equation:

Table:

Computafest Model:

Equation:

Table:

2.

Compare the two companies.

3.

If both companies were able to meet their net income growth goals, which company would you choose to invest in? Why?

4.

When, if ever, would your projections suggest that the two companies have the same net income? How did you find this? Will they ever have the same net income again?

5.

Since we are creating the models for these companies, we can choose to have a discrete model or a continuous model. What are the advantages or disadvantages for each type of model?

6.

Write the domain of each function, based on your model.

Ready for More?

In this task, we compared a linear and an exponential function and found that the exponential function was greater for large values of $t$ .

Can you find a linear function that exceeds an increasing exponential function for very large values of $t$ ? Use technology to investigate and record your results here.

Takeaways

Linear Function

Definition:

It’s not a linear function if:

Characteristics:

Equations:

Explicit:
Recursive:

Graphs:

Tables:

Contexts:

Other:

Exponential Function

Definition:

It’s not an exponential function if:

Characteristics:

Equations:

Explicit:
Recursive:

Tables:

Contexts:

Other:

Graphs:

Lesson Summary

In this lesson, we modeled the growth of two businesses and made comparisons. We used our representation to find when the two businesses had the same net income and to justify which business was the best investment. We found that exponential functions exceed linear functions for large values of $x$ and that the point of intersection of the two functions has meaning in a realistic context.

Retrieval

1.

The first and fourth terms of both an arithmetic and geometric sequence are given. Find the missing values for each sequence.

Arithmetic	$6$			$48$
Geometric	$6$			$48$

For problems 2–4, find the equation for the relationship represented.

2.

$x$	$f (x)$
$7$ $8$ $9$ $10$	$27$ $21$ $15$ $9$

$x$

$f (x)$

$7$

$8$

$9$

$10$

$27$

$21$

$15$

$9$

3.

4.

$x$	$f (x)$
$- 7$ $- 6$ $- 5$ $- 4$	$- 35$ $- 32$ $- 29$ $- 26$

$x$

$f (x)$

$- 7$

$- 6$

$- 5$

$- 4$

$- 35$

$- 32$

$- 29$

$- 26$