Determining the Next Point on the Graph

Oct 13, 2025 by windailys.com 85 views

Introduction to Exponential Functions

Exponential functions are a fundamental concept in mathematics, describing the behavior of quantities that grow or decay at a constant rate. The function $f(x) = 2(3)^x$ is a classic example of an exponential function, where the base is 3 and the coefficient is 2. In this article, we will explore the behavior of this function and determine the next point on Pherris's graph.

Understanding the Function $f(x) = 2(3)^x$

The function $f(x) = 2(3)^x$ can be rewritten as $f(x) = 2 \cdot 3^x$ . This means that for every value of $x$ , the function will multiply 2 by the result of $3^x$ . The base 3 is raised to the power of $x$ , indicating that the function will grow or decay exponentially.

Graphing the Function

To graph the function $f(x) = 2(3)^x$ , we can start by plotting the point $(1, 6)$ . This means that when $x = 1$ , the value of the function is 6. We can then use this point as a reference to determine the next point on the graph.

Determining the Next Point on the Graph

To determine the next point on the graph, we need to find the value of the function when $x = 2$ . We can do this by substituting $x = 2$ into the function:

$f(2) = 2(3)^2$

$f(2) = 2(9)$

$f(2) = 18$

Therefore, the next point on the graph is $(2, 18)$ .

Comparing the Options

Let's compare the options given to us:

A. $(2, 12)$ B. $(2, 18)$ C. $(2, 7)$ D. $(3, 7)$

Based on our calculation, the correct answer is:

B. $(2, 18)$

This is because we found that the value of the function when $x = 2$ is 18.

Conclusion

Understanding Exponential Growth and Decay

Exponential growth and decay are fundamental concepts in mathematics and science. Exponential growth occurs when a quantity increases at a constant rate, while exponential decay occurs when a quantity decreases at a constant rate. The function $f(x) = 2(3)^x$ is an example of exponential growth, where the base 3 is raised to the power of $x$ .

Real-World Applications of Exponential Functions

Exponential functions have many real-world applications, including population growth, chemical reactions, and financial modeling. For example, the population of a city may grow exponentially, while the amount of money in a savings account may decay exponentially due to interest rates.

Graphing Exponential Functions

Graphing exponential functions can be a challenging task, but it is essential to understand the behavior of these functions. By plotting points on the graph, we can determine the next point on the graph and understand the behavior of the function.

Tips for Graphing Exponential Functions

Here are some tips for graphing exponential functions:

Start by plotting the point $(1, f(1))$ .
Use the point $(1, f(1))$ as a reference to determine the next point on the graph.
Use a calculator or computer software to graph the function.
Plot multiple points on the graph to understand the behavior of the function.

Conclusion

In conclusion, the function $f(x) = 2(3)^x$ is an exponential function that grows or decays at a constant rate. By understanding the behavior of this function, we can determine the next point on Pherris's graph. We found that the next point on the graph is $(2, 18)$ . Exponential functions have many real-world applications, and graphing these functions can be a challenging task, but it is essential to understand the behavior of these functions.

Q: What is an exponential function?

A: An exponential function is a mathematical function that describes the behavior of a quantity that grows or decays at a constant rate. The general form of an exponential function is $f(x) = a \cdot b^x$ , where $a$ is the coefficient and $b$ is the base.

Q: How do I graph an exponential function?

A: To graph an exponential function, start by plotting the point $(1, f(1))$ . Then, use the point $(1, f(1))$ as a reference to determine the next point on the graph. You can also use a calculator or computer software to graph the function.

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

A: Exponential growth occurs when a quantity increases at a constant rate, while exponential decay occurs when a quantity decreases at a constant rate. The function $f(x) = 2(3)^x$ is an example of exponential growth, while the function $f(x) = 2(1/3)^x$ is an example of exponential decay.

Q: How do I determine the next point on the graph of an exponential function?

A: To determine the next point on the graph of an exponential function, substitute the next value of $x$ into the function and calculate the corresponding value of $f(x)$ .

Q: Can I use a calculator or computer software to graph an exponential function?

A: Yes, you can use a calculator or computer software to graph an exponential function. Many graphing calculators and computer software programs have built-in functions for graphing exponential functions.

Q: What are some real-world applications of exponential functions?

A: Exponential functions have many real-world applications, including population growth, chemical reactions, and financial modeling. For example, the population of a city may grow exponentially, while the amount of money in a savings account may decay exponentially due to interest rates.

Q: How do I understand the behavior of an exponential function?

A: To understand the behavior of an exponential function, plot multiple points on the graph and observe the pattern. You can also use a calculator or computer software to graph the function and analyze its behavior.

Q: Can I use exponential functions to model real-world phenomena?

A: Yes, you can use exponential functions to model real-world phenomena. Exponential functions can be used to model population growth, chemical reactions, and financial modeling, among other things.

Q: How do I determine the coefficient and base of an exponential function?

A: To determine the coefficient and base of an exponential function, look at the general form of the function, $f(x) = a \cdot b^x$ . The coefficient is the value of $a$ , and the base is the value of $b$ .

Q: Can I use exponential functions to solve problems in science and engineering?

A: Yes, you can use exponential functions to solve problems in science and engineering. Exponential functions can be used to model population growth, chemical reactions, and financial modeling, among other things.

Q: How do I graph an exponential function with a negative base?

A: To graph an exponential function with a negative base, use the same process as graphing an exponential function with a positive base. However, the graph will be reflected across the x-axis.

Q: Can I use exponential functions to model real-world phenomena that involve negative growth?

A: Yes, you can use exponential functions to model real-world phenomena that involve negative growth. Exponential functions can be used to model population decline, chemical reactions, and financial modeling, among other things.

Q: How do I determine the domain and range of an exponential function?

A: To determine the domain and range of an exponential function, look at the general form of the function, $f(x) = a \cdot b^x$ . The domain is all real numbers, and the range is all positive real numbers if the base is positive, and all negative real numbers if the base is negative.

Q: Can I use exponential functions to solve problems in finance?

A: Yes, you can use exponential functions to solve problems in finance. Exponential functions can be used to model population growth, chemical reactions, and financial modeling, among other things.

Q: How do I graph an exponential function with a fractional base?

A: To graph an exponential function with a fractional base, use the same process as graphing an exponential function with a positive base. However, the graph will be reflected across the x-axis.

Q: Can I use exponential functions to model real-world phenomena that involve fractional growth?

A: Yes, you can use exponential functions to model real-world phenomena that involve fractional growth. Exponential functions can be used to model population growth, chemical reactions, and financial modeling, among other things.