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

5.7.3 Describing Graphs

5.7.3 • Describing Graphs

Activity

Your teacher will give you a set of cards containing descriptions of situations and graphs. Match each situation with a graph that represents it. Record your matches and be prepared to explain your reasoning.

Use the cards to answer questions 1 – 4. Match each graph representation to the description of the situation characterized by exponential change. Then choose the relationship between the number of years since purchase and the value.

1. Choose the description of exponential change shown in the graph.

The value of a stock doubles approximately every 4 years.
A car loses $\frac{1}{4}$ of its value every year after purchase.
The value of a stock triples roughly every 8 years.
A laptop loses $\frac{2}{5}$ of its value every year after purchase.

A car loses $\frac{1}{4}$ of its value every year after purchase.

2. What is the equation for this graph?

Compare your answer:

$v = p (1 - \frac{1}{4})^{t}$

$v = p (\frac{3}{4})^{t}$

Where:

$v =$ value of the car after $t$ years.

$p =$ initial price of the car.

$t =$ time in years.

$14 =$ fraction by which the car loses its value every year.

3. Choose the description of exponential change shown in the graph.

The value of a stock doubles approximately every 4 years.
A car loses $\frac{1}{4}$ of its value every year after purchase.
The value of a stock triples roughly every 8 years.
A laptop loses $\frac{2}{5}$ of its value every year after purchase.

The value of a stock triples roughly every 8 years.

4. What is the equation for this graph?

Compare your answer:

$v = p (3)^{\frac{t}{8}}$

Where:

$v$ is the current value of the stock after t years.

$p$ is the initial value of the stock.

$t$ is the number of years since the initial value.

In this equation, we raise 3 to the power of $(\frac{t}{8})$ to calculate the value after $t$ years. This represents the stock tripling in value approximately every 8 years.

5. Choose the description of exponential change shown in the graph.

The value of a stock doubles approximately every 4 years.
A car loses $\frac{1}{4}$ of its value every year after purchase.
The value of a stock triples roughly every 8 years.
A laptop loses $\frac{2}{5}$ of its value every year after purchase.

The value of a stock doubles approximately every 4 years.

6. What is the equation for this graph?

Compare your answer:

$v = s (2)^{\frac{t}{4}}$

$v =$ value of the stock after $t$ years.

$s =$ initial price of the stock.

$t =$ time in years.

$\frac{t}{4} =$ the number of doubling event for every four years.

7. Choose the description of exponential change shown in the graph.

The value of a stock doubles approximately every 4 years.
A car loses $\frac{1}{4}$ of its value every year after purchase.
The value of a stock triples roughly every 8 years.
A laptop loses $\frac{2}{5}$ of its value every year after purchase.

A laptop loses $\frac{2}{5}$ of its value every year after purchase.

8. What is the equation for this graph?

Compare your answer:

$v = p (1 - \frac{2}{5})^{t}$

Where:

$v$ is the current value of the laptop after t years.

$p$ is the initial purchase value of the laptop.

$t$ is the number of years since the purchase.

In this equation, the laptop loses $\frac{2}{5}$ (or $\frac{3}{5}$ of its remaining value) every year. We raise $(\frac{3}{5})$ to the power of $t$ to calculate the value after $t$ years.

Video: Analyzing Graphs of Exponential Scenarios

Watch the following video to learn more about how to analyze a graph to determine which exponential scenario it represents.

6.7.3

Self Check

Which scenario could be represented by the following graph?

A recent graduate's credit card balance started around $500 and decreases at a rate of approximately 20% each month.
The population of sea birds started at 518 and is increasing exponentially each month.
The population of sea birds started at 518 and is decreasing exponentially each month.
A recent graduate's credit card balance started around $500 and increases at a rate of approximately 20% each month.

Additional Resources

Connecting Exponential Situations and Graphs

Match each graph representation to the description of the situation characterized by exponential change.

Graph 1

Graph 2

Situation 1: A dangerous bacterial compound forms in a closed environment but is immediately detected. This bacteria is known to double in concentration in a closed environment every hour and can be modeled by the function $P (t) = 100 \cdot 2^{t}$ , where $t$ is measured in hours.

Situation 2: Loggerhead turtles reproduce every 2 to 4 years, laying approximately 120 eggs in a clutch. Using this information, we can derive an approximate equation to model the turtle population. As is often the case in biological studies, we will count only the female turtles. If we start with a population of one female turtle in a protected area and assume that all turtles survive, we can roughly approximate the population of female turtles by $P (t) = 5^{t}$ .

Since each graph represents exponential growth, and they both have the same labels, you have to look more closely at each. The biggest difference is the $y$ -intercept or initial value. Graph 1 has an initial value of 100, and Graph 2 has a very small initial value. When you reread the situations, you can see that Situation 1 starts with 100 bacteria, and Situation 2 starts with 1 female turtle. This helps you see that Situation 1 belongs to Graph 1 and Situation 2 belongs to Graph 2.

Try it

Connecting Exponential Situations and Graphs

Write a possible situation for each graph. Make sure to include vocabulary about exponential functions.

Compare your answer:

Here is how to write an exponential situation when given a graph:

First, determine if the function is growth or decay. If any points are provided, you can use them to determine the growth factor or initial value. Be sure to include the labels of the graph. Here are sample answers.

First, graph the number of people infected growing exponentially each week.

Second, graph the amount of medicine in the body starting at 80 mg and decaying by a factor of $\frac{1}{4}$ each hour.