Practice With Exponential Growth Functions:A Table Representing The Function $f(x)=2\left(\frac{3}{2}\right)^x$ Is Shown Below.$\[ \begin{tabular}{|c|c|} \hline $x$ & $f(x)$ \\ \hline 0 & 2 \\ \hline 1 & 3 \\ \hline 2 & 4.5 \\ \hline 3 &

Understanding Exponential Growth Functions

Exponential growth functions are a type of mathematical function that describes how a quantity changes over time. These functions are characterized by a constant rate of growth, which can be represented by the equation $f(x) = ab^x$, where $a$ is the initial value, $b$ is the growth factor, and $x$ is the time or input variable. In this article, we will practice working with exponential growth functions using a table representation.

A Table Representing the Function $f(x)=2\left(\frac{3}{2}\right)^x$

The table below represents the function $f(x)=2\left(\frac{3}{2}\right)^x$.

Analyzing the Table

Let's analyze the table and understand how the function $f(x)=2\left(\frac{3}{2}\right)^x$ behaves.

• For $x=0$, the value of $f(x)$ is 2. This is the initial value of the function.
• For $x=1$, the value of $f(x)$ is 3. This is obtained by multiplying the initial value (2) by the growth factor $\left(\frac{3}{2}\right)^1$.
• For $x=2$, the value of $f(x)$ is 4.5. This is obtained by multiplying the previous value (3) by the growth factor $\left(\frac{3}{2}\right)^1$.
• For $x=3$, the value of $f(x)$ is 6.75. This is obtained by multiplying the previous value (4.5) by the growth factor $\left(\frac{3}{2}\right)^1$.

Calculating the Growth Factor

The growth factor is the constant ratio by which the function increases at each step. In this case, the growth factor is $\left(\frac{3}{2}\right)^1 = \frac{3}{2}$.

Calculating the Value of $f(x)$ for $x=4$

Using the table, we can calculate the value of $f(x)$ for $x=4$.

$f(4) = f(3) \times \left(\frac{3}{2}\right)^1$ $f(4) = 6.75 \times \frac{3}{2}$ $f(4) = 10.125$

Calculating the Value of $f(x)$ for $x=5$

Using the table, we can calculate the value of $f(x)$ for $x=5$.

$f(5) = f(4) \times \left(\frac{3}{2}\right)^1$ $f(5) = 10.125 \times \frac{3}{2}$ $f(5) = 15.1875$

Conclusion

In this article, we practiced working with exponential growth functions using a table representation. We analyzed the table, calculated the growth factor, and calculated the value of $f(x)$ for $x=4$ and $x=5$. Exponential growth functions are a powerful tool for modeling real-world phenomena, and understanding how to work with them is essential for solving problems in mathematics and other fields.

Exercises

1. Create a table representing the function $f(x)=3\left(\frac{4}{3}\right)^x$.
2. Calculate the value of $f(x)$ for $x=0, 1, 2, 3, 4, 5$ using the table.
3. Calculate the growth factor for the function $f(x)=3\left(\frac{4}{3}\right)^x$.
4. Calculate the value of $f(x)$ for $x=6$ using the table.

Answer Key

1. $x$	$f(x)$
   0	3
   1	4
   2	5.33
   3	7.11
   4	9.48
   5	12.66
   $x$	$f(x)$
   ---	---
   0	3
   1	4
   2	5.33
   3	7.11
   4	9.48
   5	12.66

2. The growth factor is $\left(\frac{4}{3}\right)^1 = \frac{4}{3}$.
3. $f(6) = f(5) \times \left(\frac{4}{3}\right)^1$ $f(6) = 12.66 \times \frac{4}{3}$ $f(6) = 16.88$
   Q&A: Exponential Growth Functions =====================================

Q: What is an exponential growth function?

A: An exponential growth function is a type of mathematical function that describes how a quantity changes over time. These functions are characterized by a constant rate of growth, which can be represented by the equation $f(x) = ab^x$, where $a$ is the initial value, $b$ is the growth factor, and $x$ is the time or input variable.

Q: What is the growth factor in an exponential growth function?

A: The growth factor is the constant ratio by which the function increases at each step. In the equation $f(x) = ab^x$, the growth factor is represented by $b$. For example, if $b = \frac{3}{2}$, the function will increase by a factor of $\frac{3}{2}$ at each step.

Q: How do I calculate the value of an exponential growth function?

A: To calculate the value of an exponential growth function, you can use the equation $f(x) = ab^x$. For example, if $a = 2$, $b = \frac{3}{2}$, and $x = 2$, the value of the function would be:

$f(2) = 2 \times \left(\frac{3}{2}\right)^2$ $f(2) = 2 \times \frac{9}{4}$ $f(2) = 4.5$

Q: What is the difference between exponential growth and linear growth?

A: Exponential growth and linear growth are two different types of growth patterns. Linear growth is a constant rate of change, where the function increases or decreases by a fixed amount at each step. Exponential growth, on the other hand, is a constant rate of change, where the function increases or decreases by a fixed ratio at each step.

Q: Can you give an example of an exponential growth function in real life?

A: Yes, a common example of an exponential growth function is population growth. For example, if a population of bacteria doubles every hour, the population can be modeled using an exponential growth function. The function would describe how the population changes over time, with the population increasing by a factor of 2 at each hour.

Q: How do I determine the growth factor in a real-world problem?

A: To determine the growth factor in a real-world problem, you need to analyze the data and identify the constant ratio by which the quantity is increasing or decreasing. For example, if a company's sales are increasing by 20% each year, the growth factor would be 1.20.

Q: Can you give an example of a problem that involves exponential growth?

A: Yes, here's an example:

A bank account earns an interest rate of 5% per year. If the account starts with a balance of $1000, how much will it be worth after 5 years?

To solve this problem, you can use an exponential growth function. The function would describe how the balance changes over time, with the balance increasing by a factor of 1.05 at each year.

Q: How do I calculate the value of an exponential growth function with a negative growth factor?

A: To calculate the value of an exponential growth function with a negative growth factor, you can use the equation $f(x) = ab^x$. However, you need to be careful when working with negative growth factors, as they can lead to exponential decay.

For example, if $a = 2$, $b = -\frac{3}{2}$, and $x = 2$, the value of the function would be:

$f(2) = 2 \times \left(-\frac{3}{2}\right)^2$ $f(2) = 2 \times \frac{9}{4}$ $f(2) = -4.5$

Q: Can you give an example of a problem that involves exponential decay?

A: Yes, here's an example:

A radioactive substance decays at a rate of 10% per year. If the substance starts with a mass of 100 grams, how much will it be left after 5 years?

To solve this problem, you can use an exponential decay function. The function would describe how the mass changes over time, with the mass decreasing by a factor of 0.90 at each year.

Conclusion

In this Q&A article, we covered some common questions and topics related to exponential growth functions. We discussed what exponential growth functions are, how to calculate their values, and how to determine the growth factor in real-world problems. We also covered some examples of exponential growth and decay, and how to calculate their values.