Lesson 3 Be the Lizard Wizard Develop Understanding

Jump Start

1.

Graph the exponential function using transformations:

$g (x) = - 3 + 2^{x + 5}$

2.

Complete the table for the exponential function:

$x$	$0$	$\frac{1}{2}$	$1$	$\frac{3}{2}$	$2$	$\frac{5}{2}$	$3$	$\frac{7}{2}$	$4$
$f (x) = 4 (2^{x})$	$4$

Learning Focus

Model the growth of an invasive species.

Explain why exponential functions are good models for unchecked population growth.

Relate expressions with rational exponents to the growth of a population.

How can we describe the growth of an invasive species?

Which is a more efficient way to change the form of a radical expression: using radicals or using exponents?

How do the properties of exponents help explain equivalent forms of exponential functions?

Open Up the Math: Launch, Explore, Discuss

The residents of the state of Florida have recently found themselves in an environmental crisis because of the growth of the iguana population. According to scientists at the University of Florida, the population of green iguanas has grown in response to increasing temperatures in the state. The scientists tell us that iguanas are a problem because, “They will destroy agriculture, undermine roads, cause electrical transformers to fail, they can transmit salmonella and can be a FAA safety hazard.” (Source: John Bacon, USA Today, July 3, 2019) These not-so-cute reptiles can be up to $5 feet$ long, are voracious eaters, and can lay up to $76 eggs$ a year, making them a real problem for homeowners and a serious threat to the Florida ecosystem.

Researchers have been tracking the iguana population in one area since 2009, when residents first noticed them regularly. At that time, it was estimated that the population in the area was about $200 iguanas$ . In 2019, the iguana population was up to about $3,000$ and growing exponentially.

1.

What is the rate of the growth (or growth factor) from 2009 to 2019? Show how you got your answer and explain why it makes sense.

2.

The data that scientists have collected each year are shown in the table.

# of Years Since 2009	# of Iguanas
$0$	$200$
$1$	$262$
$2$	$343$
$3$	$451$
$4$	$591$
$5$	$775$
$6$	$1,015$
$7$	$1,331$
$8$	$1,745$
$9$	$2,288$
$10$	$3,000$

a.

What is the annual ( $1 -year$ ) growth factor for the iguana population?

b.

How is the annual growth factor related to the $10 -year$ growth rate?

c.

What will be the $5 -year$ growth factor, and how is it related to the $10 -year$ growth factor?

3.

Write an equation to model the exponential growth of the iguana population each year from 2009 to 2019. Use technology to graph your model and verify that the graph contains the given points. Make any adjustments necessary.

4.

What does the graph tell us about the nature of the growth of this population over time?

5.

a.

What is the average rate of change in the iguana population from 2018 to 2019?

b.

What is the growth factor from 2018 to 2019?

c.

How are these two measures different? What does each measure tell us about the population of iguanas?

6.

Adjust your original model to show the monthly change in the iguana population from 2009 to 2019. Use technology to verify that your monthly model is equivalent to your yearly model. Make any adjustments needed so that they match.

7.

Explain how you adjusted your annual model to make it a monthly model and why the two equations are equivalent.

8.

Explain the meaning of each of these expressions as they relate to the iguana population in Florida and use exponent rules to write an equivalent expression.

a.

$200 (15^{\frac{1}{10}}) (15^{\frac{1}{10}}) (15^{\frac{1}{10}})$

b.

$200 (15^{- \frac{1}{10}})$

Ready for More?

We have one more expression to interpret. Explain the meaning of this expression as it relates to the iguana population in Florida, and use your work from the earlier problems to evaluate the expression.

$\log_{1.31} (\frac{1,015}{200})$

Takeaways

Strategies for rewriting exponential functions in equivalent forms: $P = A r^{t}$

Vocabulary

radical
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned how to change the form of exponential functions to reflect different rates of growth over different intervals of time. We used the rules of exponents to make equivalent functions. We interpreted exponential expressions in terms of a context by interpreting both the base of the expression and the exponent.

Retrieval

Write the following with an exponent.

1.

$\sqrt[3]{x^{5}}$

2.

$\sqrt{27 x^{5}}$

Rewrite with a fractional exponent. Then find the answer.

3.

$\log_{3} \sqrt[5]{243} =$

4.

$\log_{7} \sqrt[3]{2,401} =$