Lesson 3 The Name Game Solidify Understanding

Learning Focus

Use representations to determine if a function is discrete or continuous.

Determine the domain of a function.

How can I tell if a function is linear or exponential, given any representation?

How can I think about a constant rate of change or a constant ratio if a function is continuous?

Why are arithmetic sequences linear functions?

Why are geometric sequences exponential functions?

Open Up the Math: Launch, Explore, Discuss

In today’s lesson, you’re going to play the name game. You and your partner have been given a game board and a set of cards. Each large square on your game board has four smaller squares. Information about a function is given in the first square. In one of the squares, you are asked to produce some kind of information about the function. The last two squares will come from finding cards in your deck to match the function. By the time you finish a set, you will have filled in all four squares and each large square will have the following information:

The given function
The domain of the function and whether the function is discrete or continuous
Another representation of the function
The type of the function and the type of growth

To identify the type of function, you will need to understand the defining characteristics of linear and exponential functions:

Linear functions grow by equal differences over equal intervals.
Exponential functions grow by equal factors over equal intervals.

1.

As you discuss the game with the class, record the results in the table below so that you have a record of your work.

Function	Function Characteristics
The given function: $y = - \frac{1}{4} x + 3$	The domain of the function and whether the function is discrete or continuous:
Another representation of the function:	Type of function and the difference or growth factor:

Function

Function Characteristics

The given function:

$y = - \frac{1}{4} x + 3$

The domain of the function and whether the function is discrete or continuous:

Another representation of the function:

Type of function and the difference or growth factor:

2.

Function	Function Characteristics
The given function: $f (1) = 1$ , $f (n) = \frac{2}{3} f (n - 1)$	The domain of the function and whether the function is discrete or continuous:
Another representation of the function:	Type of function and the difference or growth factor:

Function

Function Characteristics

The given function:

$f (1) = 1$ , $f (n) = \frac{2}{3} f (n - 1)$

The domain of the function and whether the function is discrete or continuous:

Another representation of the function:

Type of function and the difference or growth factor:

3.

Function	Function Characteristics
The given function:	The domain of the function and whether the function is discrete or continuous.
Another representation of the function:	Type of function and the difference or growth factor:

4.

Function	Function Characteristics
The given function:	The domain of the function and whether the function is discrete or continuous:
Another representation of the function:	Type of function and the difference or growth factor:

5.

Function	Function Characteristics
The given function:	The domain of the function and whether the function is discrete or continuous:
Another representation of the function:	Type of function and the difference or growth factor:

6.

Function	Function Characteristics
The given function: A bubble gum machine in a grocery store is filled with $500 candies$ . Each time a person puts in money, $10 candies$ come out. This function is a model for the number of candies left in the machine after a person buys some candy.	The domain of the function and whether the function is discrete or continuous:
Another representation of the function:	Type of function and the difference or growth factor:

Function

Function Characteristics

The given function:

A bubble gum machine in a grocery store is filled with $500 candies$ . Each time a person puts in money, $10 candies$ come out. This function is a model for the number of candies left in the machine after a person buys some candy.

The domain of the function and whether the function is discrete or continuous:

Another representation of the function:

Type of function and the difference or growth factor:

7.

Function	Function Characteristics
The given function: A radioactive element decays at a rate of $5 %$ each year. The scientists begin with $40 grams$ of the element. The function is a model for the amount of the element remaining at a given time.	The domain of the function and whether the function is discrete or continuous:
Another representation of the function:	Type of function and the difference or growth factor:

Function

Function Characteristics

The given function:

A radioactive element decays at a rate of $5 %$ each year. The scientists begin with $40 grams$ of the element. The function is a model for the amount of the element remaining at a given time.

The domain of the function and whether the function is discrete or continuous:

Another representation of the function:

Type of function and the difference or growth factor:

8.

Function	Function Characteristics
The given function:	The domain of the function and whether the function is discrete or continuous:
Another representation of the function:	Type of function and the difference or growth factor:

9.

Function	Function Characteristics
The given function:	The domain of the function and whether the function is discrete or continuous:
Another representation of the function:	Type of function and the difference or growth factor:

Ready for More?

Make an additional representation for each function given in problems 1–9.

Takeaways

Identifying linear and exponential functions:

Lesson Summary

In this lesson, we used the definitions of linear and exponential functions to determine if functions were linear or exponential. We learned to identify equal intervals on a continuous function so that we could tell if there were equal differences or change factors. We practiced determining domains and whether a function is discrete or continuous.

Retrieval

Rewrite each expression using the rules of exponents.

1.

$b^{3} \cdot b^{5}$

2.

${(n^{3})}^{4}$

3.

$\frac{m^{5}}{m^{8}}$

4.

Find the missing terms of the geometric sequence.

$x$	$1$	$2$	$3$	$4$	$5$	$6$
$g (x)$	$128$					$4$

Find the rate of change using the information given.

5.

$x$	$f (x)$
$3$	$- 5$
$5$	$- 1$
$8$	$5$
$12$	$13$

6.

$(- 12, 7) (- 6, 31)$