Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

Clare noticed mold on the last slice of bread in a plastic bag. The area covered by the mold was about 1 square millimeter. She left the bread alone to see how the mold would grow. The next day, the area covered by the mold had doubled, and it doubled again the day after that.

1. If the doubling pattern continues, how many square millimeters will the mold cover 4 days after she noticed the mold? Be prepared to show your reasoning.

Compare your answer:

16 square millimeters. $2 \cdot 2 \cdot 2 \cdot 2$ or $2^{4}$ is 16.

2. Represent the relationship between the area $A$ , in square millimeters, covered by the mold and the number of days $d$ since the mold was spotted using:

a. A table of values, showing the values from the day the mold was spotted through 5 days later.

b. An equation

c. A graph

Compare your answers:

Number of days, $d$	Area, $A$ , in sq mm
0	1
1	2
2	4
3	8
4	16
5	32

Equation: $A = 1 \cdot 2^{d}$ or equivalent (like $A = 2^{d}$ ).
Your graph may be discrete or continuous, this example is a discrete graph.

3. Discuss with your partner: Is the relationship between the area covered by mold and the number of days a function? If so, write ____ is a function of ____. If not, explain why it is not.

Compare your answer:

Yes.

The area covered by mold is a function of the number of days that have passed. At any given time since mold was spotted, there is a certain area of the bread that is covered in mold.
The number of days that have passed is a function of the area covered by the mold up to a certain point (e.g., when the entire piece of bread is already completely covered, the area of mold would correspond to more than one day). We do not have a way to write this relationship using an equation, but we could use a table or a graph to show the relationship.

Are you ready for more?

Extending Your Thinking

What do you think is an appropriate domain for the mold area function $A$ ? Be prepared to show your reasoning.

Compare your answer:

Your answer may vary, but here is an example.

Some negative values of $t$ could be relevant, but we don’t know how the mold was growing before Clare started watching. Positive values of $t$ will not be valid indefinitely as the bread will soon be completely covered by mold. If a piece of bread is 10 cm by 10 cm, that’s 10,000 square mm. An appropriate domain would be $0 \leq d \leq 14$ .

Self Check

Which of the following equations represents the values in the table?

$x$	0	1	2	3
$y$	1	3	9	27

$y=3x+1$
$y=1 \cdot 3^x$
$y=3 \cdot 1^x$
$y=3x$

Additional Resources

Different Exponential Representations

A table of values is shown below for a function $g (x) = (\frac{1}{2})^{x}$ , which represents exponential decay.

$x$	–3	–2	–1	0	1	2	3
$g (x) = (\frac{1}{2})^{x}$	8	4	2	1	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{8}$

When these points are graphed, they show the decay because the function graphed decreases from left to right:

A graph of the function g(x) = one half to the power of x, showing a decreasing curve passing through points (-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 1/2), (2, 1/4), (3, 1/8), with the x-axis as an asymptote.

Try it

Try It: Different Exponential Representations

Make a table for the function $f (x) = 4^{x}$ . Then graph the function.

Compare your answer:

Here is how to make a table and then graph the exponential function:

First, substitute values in for $x$ to find the values of $f (x)$ .

$x$	0	1	2	3	4
$f (x) = 4^{x}$	$4^{0} = 1$	$4^{1} = 4$	$4^{2} = 16$	$4^{3} = 64$	$4^{4} = 256$

A graph of an exponential function increasing rapidly, with three black points marked on the curve at (0,1), (1,4), and (2,16). The x-axis and y-axis are labeled, with gridlines visible.

5.8.2 Represent a Function as a Table of Values, Graph, and Equation

Activity

Are you ready for more?

Extending Your Thinking

Additional Resources

Different Exponential Representations

Try It: Different Exponential Representations