A Collectible Card Currently Has A Value Of $\$200$. The Value Of The Card Triples Each Year. Which Exponential Function Models This Situation?\[\begin{tabular}{|c|c|c|}\hline$y=200\left(\frac{1}{3}\right)^x$ & $y=3(200)^x$ &

Feb 27, 2025 by ADMIN 229 views

A Collectible Card's Exponential Growth: Modeling the Situation

In the world of collectible cards, the value of a particular card can fluctuate greatly over time. In this scenario, we are presented with a card that has a current value of $\$200$ . However, the value of the card triples each year, indicating an exponential growth pattern. In this article, we will explore the exponential function that models this situation, providing a deeper understanding of the card's value over time.

Understanding Exponential Growth

Exponential growth is a type of growth where the rate of change is proportional to the current value. In other words, as the value of the card increases, the rate at which it increases also grows. This type of growth is often modeled using exponential functions, which have the general form $y = ab^x$ , where $a$ is the initial value, $b$ is the growth factor, and $x$ is the time variable.

Modeling the Situation

Given that the value of the card triples each year, we can express this as a growth factor of $3$ . Since the initial value of the card is $\$200$ , we can use this as the value of $a$ in our exponential function. Therefore, the exponential function that models this situation is:

y = 200\left(\frac{1}{3}\right)^x

Why This Function?

So, why does this function model the situation? Let's break it down:

The initial value of the card is $\$200$ , which is represented by the value of $a$ .
The growth factor is $3$ , which is represented by the value of $\frac{1}{3}$ in the function. This may seem counterintuitive, but it's actually a common technique used in exponential functions to represent growth factors greater than $1$ .
The time variable $x$ represents the number of years that have passed.

Simplifying the Function

While the function $y = 200\left(\frac{1}{3}\right)^x$ accurately models the situation, it can be simplified to make it easier to work with. We can rewrite the function as:

y = 200\left(\frac{1}{3}\right)^x = \frac{200}{3^x}

This simplified function has the same value as the original function, but it's often easier to work with when dealing with exponential functions.

Using the Function

Now that we have the exponential function that models the situation, we can use it to calculate the value of the card at any given time. For example, if we want to know the value of the card after $5$ years, we can plug in $x = 5$ into the function:

y = \frac{200}{3^5} = \frac{200}{243}

This tells us that after $5$ years, the value of the card will be approximately $\$0.82$ .

In conclusion, the exponential function $y = 200\left(\frac{1}{3}\right)^x$ accurately models the situation of a collectible card that triples in value each year. By understanding the principles of exponential growth and how to model it using exponential functions, we can gain a deeper understanding of the card's value over time. Whether you're a collector or simply interested in mathematics, this function provides a fascinating example of how exponential growth can be modeled and used to make predictions about the future.

Exponential functions like the one we've discussed have many real-world applications. For example:

Finance: Exponential functions are used to model the growth of investments, such as stocks and bonds.
Biology: Exponential functions are used to model the growth of populations, such as bacteria and other microorganisms.
Computer Science: Exponential functions are used to model the growth of data, such as the number of users on a social media platform.

When working with exponential functions, it's common to make mistakes. Here are a few to watch out for:

Incorrect growth factor: Make sure to use the correct growth factor when modeling exponential growth.
Incorrect initial value: Make sure to use the correct initial value when modeling exponential growth.
Incorrect time variable: Make sure to use the correct time variable when modeling exponential growth.

Q: What is exponential growth, and how does it apply to collectible cards?

A: Exponential growth is a type of growth where the rate of change is proportional to the current value. In the context of collectible cards, exponential growth means that the value of the card increases at a rate that is proportional to its current value. This type of growth is often modeled using exponential functions, which have the general form $y = ab^x$ , where $a$ is the initial value, $b$ is the growth factor, and $x$ is the time variable.

Q: How do I calculate the value of a collectible card using an exponential function?

A: To calculate the value of a collectible card using an exponential function, you need to know the initial value of the card, the growth factor, and the time variable. The exponential function that models the situation is:

y = 200\left(\frac{1}{3}\right)^x

To calculate the value of the card after a certain number of years, simply plug in the value of $x$ into the function.

Q: What is the growth factor in the exponential function, and how does it affect the value of the card?

A: The growth factor in the exponential function is $\frac{1}{3}$ . This means that the value of the card triples each year. The growth factor affects the value of the card by increasing it at a rate that is proportional to its current value.

Q: Can I use the exponential function to predict the value of a collectible card in the future?

A: Yes, you can use the exponential function to predict the value of a collectible card in the future. By plugging in the value of $x$ into the function, you can calculate the value of the card after a certain number of years.

Q: What are some common mistakes to watch out for when working with exponential functions?

A: Some common mistakes to watch out for when working with exponential functions include:

Incorrect growth factor: Make sure to use the correct growth factor when modeling exponential growth.
Incorrect initial value: Make sure to use the correct initial value when modeling exponential growth.
Incorrect time variable: Make sure to use the correct time variable when modeling exponential growth.

Q: Can I use the exponential function to model the growth of other types of collectibles, such as coins or stamps?

A: Yes, you can use the exponential function to model the growth of other types of collectibles, such as coins or stamps. The exponential function is a general model that can be applied to any type of collectible that exhibits exponential growth.

Q: How can I use the exponential function to make predictions about the future value of a collectible card?

A: To make predictions about the future value of a collectible card, you can use the exponential function to calculate the value of the card after a certain number of years. For example, if you want to know the value of the card after 5 years, you can plug in $x = 5$ into the function:

y = \frac{200}{3^5} = \frac{200}{243}

This tells you that after 5 years, the value of the card will be approximately $\$0.82$ .

Q: Can I use the exponential function to model the growth of other types of investments, such as stocks or bonds?

A: Yes, you can use the exponential function to model the growth of other types of investments, such as stocks or bonds. The exponential function is a general model that can be applied to any type of investment that exhibits exponential growth.

In conclusion, the exponential function $y = 200\left(\frac{1}{3}\right)^x$ accurately models the situation of a collectible card that triples in value each year. By understanding the principles of exponential growth and how to model it using exponential functions, you can gain a deeper understanding of the card's value over time. Whether you're a collector or simply interested in mathematics, this function provides a fascinating example of how exponential growth can be modeled and used to make predictions about the future.