Creating Exponential Functions from Graphs

5.13.2 Creating Exponential Functions from Graphs

5.13.2 • Creating Exponential Functions from Graphs

Activity

Examine the graph representing an exponential function $f$ . The function $f$ gives the value of a computer, in dollars, as a function of time, $x$ , measured in years since the time of purchase.

Graph of an exponential decay function, where value in dollars is a function of time the graph passes through the points 0 comma four hundred, 2 comma 100, and 4 comma 25.

Based on the graph, what can you say about the following?

1. The purchase price of the computer

Compare your answer:

$400

2. The value of $f$ when $x$ is 1

Compare your answer:

$f (1) = 200$

3. The meaning of $f (1)$

Compare your answer:

The value of the computer after 1 year

4. How the value of the computer is changing each year

Compare your answer:

The computer loses half of its value each year.

5. An equation that defines $f$ .

Compare your answer:

$f (x) = 400 (\frac{1}{2})^{x}$ or equivalent (i.e., $f (x) = 400 \cdot (0.5)^{x}$ ).

6. Whether the value of $f$ will reach 0 after 10 years

Compare your answer:

No, it will not. After 10 years, the value of the computer will be a little less than $0.40.

For questions 7 – 8, examine the graph of the exponential function below.

Graph of an exponential growth function The graph passes through the points 0 comma 5, 1 comma 15, and 2 comma 45.

7. Write an equation that defines the function.

Compare your answer:

$f (x) = 5 \cdot 3^{x}$

8. Find the value of the function when $x$ is 5.

Compare your answer:

$f (5) = 1, 215$

For questions 9 – 10, examine the graph of the exponential function below.

Graph of an exponential decay function. The graph passes through the points negative 1 comma 40, 0 comma 20, and 2 comma 5.

9. Write an equation that defines the function.

Compare your answer:

$f (x) = 20 \cdot (\frac{1}{2})^{x}$ (or equivalent, i.e., $f (x) = 20 \cdot (0.5)^{x}$ ).

10. Find the value of the function when $x$ is 5.

Compare your answer:

$f (5) = 0.625$

Are you ready for more?

Extending Your Thinking

Consider a function $f$ defined by $f (x) = a \cdot b^{x}$ .

If the graph of $f$ goes through the points $(2, 10)$ and $(8, 30)$ , would you expect $f (5)$ to be less than, equal to, or greater than 20?

If the graph of $f$ goes through the points $(2, 30)$ and $(8, 10)$ , would you expect $f (5)$ to be less than, equal to, or greater than 20?

Compare your answer:

The value 20 is what you would get for $f (5)$ if $f$ were a linear function through $(2, 10)$ and $(8, 30)$ (or $(2, 30)$ and $(8, 10)$ ). Looking at graphs representing our exponential functions, we can see that the line connecting two points on the graph always seems to be above the graph of the exponential function itself. Thus, we would expect $f (5) < 20$ in both cases.

Video: Interpreting an Exponential Function from a Graph

Watch the following video to learn more about interpreting exponential functions:

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Self Check

Which of the following is the function for the curve graphed below?

GRAPH OF AN EXPONENTIAL GROWTH FUNCTION. THE GRAPH PASSES THROUGH THE POINTS 0 COMMA 100, 1 COMMA THREE HUNDRED, AND 2 COMMA 900.

$f(x)=(100^3)^x$
$f(x)=100 \cdot 3^x$
$f(x)=3 \cdot 100^x$
$f(x)=100+3^x$

Additional Resources

Writing Functions from Graphs

Graph of an exponential growth function. The graph passes through the points 0 comma 4, 1 comma 12, and 2 comma 36.

Find the equation of the function graphed above. Then find the value at $f (5)$ .

1. Find the value of $a$ , the starting point.	$a = 4$
2. Find the $b$ , the growth factor.	$b = 3$
3. Write the equation of the function.	$f (x) = 4 \cdot 3^{x}$
4. Find $f (5)$ .	$f (5) = 4 \cdot 3^{5} = 972$

Step-by-step guide to defining an exponential function by identifying its parameters 'a' and 'b', writing the equation, and evaluating f(5).

Try it

Try It: Writing Functions from Graphs

Use the graph below to write the equation of the function. Then find $f (5)$ .

Graph of an exponential growth function The graph passes through the points 0 comma 2, 1 comma 8, and 2 comma 32.

Compare your answers:

Here is how to write the exponential function from the graph:

1. Find the value of $a$ , the starting point.	$a = 2$
2. Find the $b$ , the growth factor.	$b = 4$
3. Write the equation of the function.	$f (x) = 2 \cdot 4^{x}$
4. Find $f (5)$	$f (x) = 2 \cdot 4^{5} = 2048$

Steps to define an exponential function by identifying the starting point (a) and growth factor (b), writing the equation, and evaluating it at a given x-value.