Determine the Type of Graph

Oct 13, 2025 by thesweetdesigns.com 135 views

Introduction

When it comes to graphing functions, understanding the behavior of the function is crucial in determining the type of graph it will produce. In this case, we are given the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ , and we need to determine which graph represents this function.

Understanding the Function

The given function is $f(x)=\left(\frac{3}{2}\right)^{-x}$ . To understand the behavior of this function, let's break it down. The function is in the form of $f(x)=a^x$ , where $a$ is a constant. In this case, $a=\frac{3}{2}$ .

Properties of Exponential Functions

Exponential functions have several properties that can help us understand their behavior. One of the key properties is that the base of the exponential function determines the rate of growth or decay of the function. If the base is greater than 1, the function will grow exponentially. If the base is between 0 and 1, the function will decay exponentially.

Graphing the Function

To graph the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ , we need to consider the properties of exponential functions. Since the base is $\frac{3}{2}$ , which is greater than 1, the function will grow exponentially. However, the negative exponent will cause the function to decay as $x$ increases.

Types of Graphs

There are several types of graphs that can represent the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ . Some of the possible graphs include:

Exponential Growth Graph: This graph will show the function growing exponentially as $x$ increases.
Exponential Decay Graph: This graph will show the function decaying exponentially as $x$ increases.
Logarithmic Graph: This graph will show the function as a logarithmic curve.

Determining the Correct Graph

To determine which graph represents the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ , we need to consider the properties of the function. Since the base is $\frac{3}{2}$ , which is greater than 1, the function will grow exponentially. However, the negative exponent will cause the function to decay as $x$ increases.

Conclusion

In conclusion, the graph that represents the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ is an exponential decay graph. This graph will show the function decaying exponentially as $x$ increases.

Step-by-Step Solution

Here's a step-by-step solution to determine which graph represents the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ :

Understand the Function: The function is $f(x)=\left(\frac{3}{2}\right)^{-x}$ .
Determine the Type of Graph: Since the base is $\frac{3}{2}$ , which is greater than 1, the function will grow exponentially. However, the negative exponent will cause the function to decay as $x$ increases.
Choose the Correct Graph: Based on the properties of the function, the correct graph is an exponential decay graph.

Common Mistakes

Here are some common mistakes to avoid when graphing the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ :

Incorrectly Identifying the Type of Graph: Make sure to consider the properties of the function, including the base and the exponent.
Not Considering the Negative Exponent: The negative exponent will cause the function to decay as $x$ increases.
Not Choosing the Correct Graph: Make sure to choose the graph that represents the function, which is an exponential decay graph.

Final Answer

The final answer is: Exponential Decay Graph.

Graphs of Exponential Functions

Exponential functions can be graphed in several ways, including:

Exponential Growth Graph: This graph will show the function growing exponentially as $x$ increases.
Exponential Decay Graph: This graph will show the function decaying exponentially as $x$ increases.
Logarithmic Graph: This graph will show the function as a logarithmic curve.

Graphing Exponential Functions

To graph an exponential function, follow these steps:

Determine the Type of Graph: Exponential functions can be graphed as exponential growth or decay graphs.
Choose the Correct Graph: Based on the properties of the function, choose the correct graph.
Graph the Function: Use a graphing calculator or software to graph the function.

Conclusion

In conclusion, the graph that represents the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ is an exponential decay graph. This graph will show the function decaying exponentially as $x$ increases.

References

Exponential Functions: Exponential functions have several properties that can help us understand their behavior.
Graphing Exponential Functions: Exponential functions can be graphed in several ways, including exponential growth and decay graphs.
Logarithmic Graphs: Logarithmic graphs can be used to represent exponential functions.

Final Thoughts

In conclusion, graphing the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ requires understanding the properties of the function, including the base and the exponent. By following the steps outlined in this article, you can determine which graph represents the function and graph it correctly.

Introduction

In our previous article, we discussed how to graph the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ . We covered the properties of exponential functions, the types of graphs that can represent the function, and how to determine the correct graph. In this article, we will answer some frequently asked questions about graphing the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ .

Q&A

Q: What is the base of the exponential function $f(x)=\left(\frac{3}{2}\right)^{-x}$ ?

A: The base of the exponential function $f(x)=\left(\frac{3}{2}\right)^{-x}$ is $\frac{3}{2}$ .

Q: What is the exponent of the exponential function $f(x)=\left(\frac{3}{2}\right)^{-x}$ ?

A: The exponent of the exponential function $f(x)=\left(\frac{3}{2}\right)^{-x}$ is $-x$ .

Q: What type of graph will the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ produce?

A: The function $f(x)=\left(\frac{3}{2}\right)^{-x}$ will produce an exponential decay graph.

Q: How can I determine the correct graph for the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ ?

A: To determine the correct graph for the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ , you need to consider the properties of the function, including the base and the exponent. If the base is greater than 1, the function will grow exponentially. If the base is between 0 and 1, the function will decay exponentially. In this case, the base is $\frac{3}{2}$ , which is greater than 1, but the exponent is negative, so the function will decay exponentially.

Q: Can I use a graphing calculator to graph the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ ?

A: Yes, you can use a graphing calculator to graph the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ . Simply enter the function into the calculator and adjust the window settings as needed to see the graph.

Q: What are some common mistakes to avoid when graphing the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ ?

A: Some common mistakes to avoid when graphing the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ include:

Incorrectly identifying the type of graph
Not considering the negative exponent
Not choosing the correct graph

Conclusion

In conclusion, graphing the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ requires understanding the properties of the function, including the base and the exponent. By following the steps outlined in this article and avoiding common mistakes, you can determine which graph represents the function and graph it correctly.

Additional Resources

Exponential Functions: Exponential functions have several properties that can help us understand their behavior.
Graphing Exponential Functions: Exponential functions can be graphed in several ways, including exponential growth and decay graphs.
Logarithmic Graphs: Logarithmic graphs can be used to represent exponential functions.

Final Thoughts

In conclusion, graphing the function $f(x)=\left(\frac{3}{2}\right)^{-x}$ requires attention to detail and a understanding of the properties of exponential functions. By following the steps outlined in this article and avoiding common mistakes, you can determine which graph represents the function and graph it correctly.