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

Activity

Here is a table that shows some input and output values of an exponential function $g$ . The equation $g (x) = 3^{x}$ defines the function.

$x$	$g (x)$
3	27
4	81
5	243
6	729
7	2,187
8	6,561
$x$
$x + 1$

1. How does $g (x)$ change every time $x$ increases by 1? Show or explain your reasoning.

Compare your answer:

The output values increase by a factor of 3. For example: When we divide two consecutive output values, the quotient is always 3.

2. Choose two new input values that are consecutive whole numbers and find their output values. Record them in the table. How do the output values change for those two input values?

Compare your answer:

$g (10) = 3^{10}$ and $g (11) = 3^{11}$ . The output still grows by a factor of 3 when the input increases by 1 (from 10 to 11).

3. Complete the table with the output when the input is $x$ and when it is $x + 1$ .

Compare your answer:

$x$	$g (x)$
$x$	$3^{x}$
$x + 1$	$3^{x + 1}$

4. Look at the change in output values as the $x$ increases by 1. Does it still agree with your earlier findings? Show your reasoning.

Compare your answer:

Yes, the outputs still increase by a factor of 3. By rules of exponents, we know that $3 x \cdot 3 = 3^{x + 1}$ . (Or by rules of exponents, we know that $\frac{3^{x + 1}}{3^{x}} = 3^{x + 1 - x}$ , which equals 3.)

Pause here for a class discussion. Then, work with your group on the next few questions.

5. Choose two $x$ -values where one is 3 more than the other (for example, 1 and 4). How do the output values of $g$ change as $x$ increases by 3? (Each group member should choose a different pair of numbers and study the outputs.)

Compare your answer:

The output values increase by a factor of 27 or $3^{3}$ .

6. Complete this table with the output when the input is $x$ and when it is $x + 3$ . Look at the change in output values as $x$ increases by 3. Does it agree with your group’s findings in the previous question? Show your reasoning.

$x$	$g (x)$
$x$
$x + 3$

Compare your answer:

$x$	$g (x)$
$x$	$3^{x}$
$x + 3$	$3^{x + 3}$

Yes, the outputs still increase by a factor of 27. By rules of exponents, we know that $3 x \cdot 3^{3} = 3^{x + 3}$ . (Or by rules of exponents, we know that $\frac{3^{x + 3}}{3^{x}} = 3^{x + 3 - x}$ , which equals $3^{3}$ or 27.)

7. Summarize the process you used in questions 1 - 6 to analyze this data.

Compare your answer:

Answers will vary.

8. Explain if you found it helpful or not to analyze the data for simpler data (1 - 2) before analyzing it for more abstract data (questions 3 - 6).

Compare your answer:

Answers will vary.

Video: Analyzing Tables of Exponential Functions

Watch the following video to learn more about analyzing the tables of exponential functions:

Access multimedia content

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Extending Your Thinking

For integer inputs, we can think of multiplication as repeated addition and exponentiation as repeated multiplication:

Multiplication representing repeated addition	Exponentiation representing repeated multiplication
$3 \cdot 5 = 3 + 3 + 3 + 3 + 3$	$3^{5} = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$

We could continue this process with a new operation called tetration. It uses the symbol $↑ ↑$ , and it is defined as repeated exponentiation:

Tetration representing repeated exponentiation

$3 ↑ ↑ 5 = 3^{3 \land 3 \land 3 \land 3}$

1. Compute $2 ↑ ↑ 3$ and $3 ↑ ↑ 2$ .

$2 ↑ ↑ 3 = 2^{2 \land 2} = 24 = 16$

2. Compute $f (x) = 3 ↑ ↑ x$ .

$3 ↑ ↑ 2 = 3^{3} = 27$

3. What is the relationship between $f (x)$ and $f (x + 1)$ ?

Compare your answer:

The numbers output by this function get large absurdly fast, even in comparison to exponential functions. We calculated $f (2) = 27$ above, and now $f (3) = 3^{f (2)} = 3^{27} = 7, 625, 597, 484, 987$ .

Self Check

Given

f(x) = 4^x

, what is

\frac{f(x+2)}{f(x+1)}

?

16
There is not enough information to tell.
4
2

Additional Resources

Determining the Growth Rate of Exponential Functions

For the table below, assume the function $f$ is defined for all real numbers. Calculate $△ f = f (x + 1) - f (x)$ in the last column. (The symbol $△$ in this context means “change in.”) What do you notice about $△ f$ ? Could the function be linear or exponential? Write a linear or an exponential function formula that generates the same input–output pairs as given in the table.

$x$	$f (x)$	$△ f = f (x + 1) - f (x)$
0	2
1	6
2	18
3	54
4	162

For each row, you subtract $f (x)$ values. The first row would be $6 - 2 = 4$ , then $18 - 6 = 12$ , then $54 - 18 = 36$ , and then $162 - 54 = 108$ . Since the rate of change isn’t consistent, it is not a linear function. Exponential functions have a growth factor or a repeated rate that terms are being multiplied by. If you divide consecutive terms, you will find the growth rate is 3. Notice that the change between consecutive terms also has a growth rate of 3.

Since this function is exponential, an equation will have the form $f (x) = i n i t i a l v a l u e (g r o w t h \ r a t e)^{x}$ . The growth rate is 3. The initial value is the same as the $y$ -intercept, or where $x = 0$ . $f (0)$ is given in the table, so the initial value is 2. The equation for this table would be $f (x) = 2 (3)^{x}$ .

Try it

Try It: Determining the Growth Rate of Exponential Functions

For the graph provided, assume that the function is defined for all real numbers. What is the rate of change? Write an equation that would define this function.

Graph of an exponential growth function with a y-intercepts of 1 and passing through the points 1 comma 2, 2 comma 4, and 3 comma 8

Compare your answer:

Here is how to write the equation of an exponential function from a graph:

The graph seems to be exponential. If you find the rate of change between consecutive points, you find the graph is exponential and the rate of change is 2 since the $y$ -values increase by a factor of 2 every time the $x$ -values increase by 1. Since the graph crosses the $y$ -axis at $(0, 1)$ , that is the initial value. The equation for this graph is $f (x) = 2^{x}$ .

5.15.3 Growth Rate of an Exponential Function

Activity

Video: Analyzing Tables of Exponential Functions

Are you ready for more?

Extending Your Thinking

Additional Resources

Determining the Growth Rate of Exponential Functions

Try It: Determining the Growth Rate of Exponential Functions