Graph The Exponential Function:${ F(x)=\left(\frac{1}{3}\right)^x }$Plot Five Points On The Graph Of The Function And Draw The Asymptote.

Mar 12, 2025 by ADMIN 139 views

**Graphing the Exponential Function: A Comprehensive Guide**

Introduction

In mathematics, an exponential function is a function that can be written in the form $f(x) = a^x$ , where $a$ is a positive real number. The graph of an exponential function is a fundamental concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on graphing the exponential function $f(x) = \left(\frac{1}{3}\right)^x$ and discuss its properties.

Understanding the Exponential Function

The exponential function $f(x) = \left(\frac{1}{3}\right)^x$ is a decreasing function, meaning that as $x$ increases, the value of $f(x)$ decreases. This is because the base $\frac{1}{3}$ is less than 1, which causes the function to decay exponentially. The graph of this function will have a horizontal asymptote at $y = 0$ , which means that as $x$ approaches infinity, the value of $f(x)$ approaches 0.

Plotting Five Points on the Graph

To plot five points on the graph of the function, we need to choose five values of $x$ and calculate the corresponding values of $f(x)$ . Let's choose the following values of $x$ : $-2$ , $-1$ , $0$ , $1$ , and $2$ . We can then calculate the corresponding values of $f(x)$ using the formula $f(x) = \left(\frac{1}{3}\right)^x$ .

$x$	$f(x)$
-2	$\left(\frac{1}{3}\right)^{-2} = 9$
-1	$\left(\frac{1}{3}\right)^{-1} = 3$
0	$\left(\frac{1}{3}\right)^0 = 1$
1	$\left(\frac{1}{3}\right)^1 = \frac{1}{3}$
2	$\left(\frac{1}{3}\right)^2 = \frac{1}{9}$

Drawing the Asymptote

The horizontal asymptote of the graph of the function is the line $y = 0$ . This means that as $x$ approaches infinity, the value of $f(x)$ approaches 0. To draw the asymptote, we can plot the line $y = 0$ on the graph and extend it to the right and left indefinitely.

Properties of the Graph

The graph of the exponential function $f(x) = \left(\frac{1}{3}\right)^x$ has several properties that are worth noting. These properties include:

Horizontal asymptote: The graph has a horizontal asymptote at $y = 0$ .
Decreasing function: The function is decreasing, meaning that as $x$ increases, the value of $f(x)$ decreases.
Exponential decay: The function decays exponentially, meaning that the value of $f(x)$ decreases rapidly as $x$ increases.
No maximum or minimum: The function has no maximum or minimum value, meaning that it continues to decrease indefinitely.

Real-World Applications

The exponential function $f(x) = \left(\frac{1}{3}\right)^x$ has numerous real-world applications in various fields, including:

Physics: The function is used to model the decay of radioactive substances.
Engineering: The function is used to model the growth of populations and the decay of electronic components.
Economics: The function is used to model the growth of economies and the decay of investments.

Conclusion

In conclusion, the graph of the exponential function $f(x) = \left(\frac{1}{3}\right)^x$ is a fundamental concept in mathematics that has numerous applications in various fields. The graph has a horizontal asymptote at $y = 0$ , is a decreasing function, and decays exponentially. The function has no maximum or minimum value, and it continues to decrease indefinitely. The graph of the function has numerous real-world applications, including modeling the decay of radioactive substances, the growth of populations, and the decay of electronic components.

References

[1] "Exponential Functions." MathWorld, Wolfram Research, 2023.
[2] "Graphing Exponential Functions." Math Open Reference, 2023.
[3] "Exponential Decay." Physics Classroom, 2023.

Additional Resources

[1] "Exponential Functions." Khan Academy, 2023.
[2] "Graphing Exponential Functions." Mathway, 2023.
[3] "Exponential Decay." Wolfram Alpha, 2023.
Graphing the Exponential Function: A Comprehensive Guide ===========================================================

Q&A: Graphing the Exponential Function

Q: What is an exponential function?

A: An exponential function is a function that can be written in the form $f(x) = a^x$ , where $a$ is a positive real number. The graph of an exponential function is a fundamental concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics.

Q: What is the graph of the exponential function $f(x) = \left(\frac{1}{3}\right)^x$ ?

A: The graph of the exponential function $f(x) = \left(\frac{1}{3}\right)^x$ is a decreasing function, meaning that as $x$ increases, the value of $f(x)$ decreases. The graph has a horizontal asymptote at $y = 0$ , which means that as $x$ approaches infinity, the value of $f(x)$ approaches 0.

Q: How do I plot five points on the graph of the function?

A: To plot five points on the graph of the function, you need to choose five values of $x$ and calculate the corresponding values of $f(x)$ . Let's choose the following values of $x$ : $-2$ , $-1$ , $0$ , $1$ , and $2$ . We can then calculate the corresponding values of $f(x)$ using the formula $f(x) = \left(\frac{1}{3}\right)^x$ .

$x$	$f(x)$
-2	$\left(\frac{1}{3}\right)^{-2} = 9$
-1	$\left(\frac{1}{3}\right)^{-1} = 3$
0	$\left(\frac{1}{3}\right)^0 = 1$
1	$\left(\frac{1}{3}\right)^1 = \frac{1}{3}$
2	$\left(\frac{1}{3}\right)^2 = \frac{1}{9}$

Q: How do I draw the asymptote?

A: The horizontal asymptote of the graph of the function is the line $y = 0$ . This means that as $x$ approaches infinity, the value of $f(x)$ approaches 0. To draw the asymptote, you can plot the line $y = 0$ on the graph and extend it to the right and left indefinitely.

Q: What are the properties of the graph?

A: The graph of the exponential function $f(x) = \left(\frac{1}{3}\right)^x$ has several properties that are worth noting. These properties include:

Horizontal asymptote: The graph has a horizontal asymptote at $y = 0$ .
Decreasing function: The function is decreasing, meaning that as $x$ increases, the value of $f(x)$ decreases.
Exponential decay: The function decays exponentially, meaning that the value of $f(x)$ decreases rapidly as $x$ increases.
No maximum or minimum: The function has no maximum or minimum value, meaning that it continues to decrease indefinitely.

Q: What are the real-world applications of the graph?

A: The exponential function $f(x) = \left(\frac{1}{3}\right)^x$ has numerous real-world applications in various fields, including:

Physics: The function is used to model the decay of radioactive substances.
Engineering: The function is used to model the growth of populations and the decay of electronic components.
Economics: The function is used to model the growth of economies and the decay of investments.

Q: How can I use the graph in real-world applications?

A: The graph of the exponential function $f(x) = \left(\frac{1}{3}\right)^x$ can be used to model various real-world phenomena, including:

Radioactive decay: The function can be used to model the decay of radioactive substances, such as uranium or plutonium.
Population growth: The function can be used to model the growth of populations, such as the growth of a city or a country.
Electronic component decay: The function can be used to model the decay of electronic components, such as batteries or capacitors.

Q: What are some common mistakes to avoid when graphing the exponential function?

A: Some common mistakes to avoid when graphing the exponential function include:

Not using a horizontal asymptote: The graph of the exponential function has a horizontal asymptote at $y = 0$ , which means that as $x$ approaches infinity, the value of $f(x)$ approaches 0.
Not using a decreasing function: The function is decreasing, meaning that as $x$ increases, the value of $f(x)$ decreases.
Not using exponential decay: The function decays exponentially, meaning that the value of $f(x)$ decreases rapidly as $x$ increases.

Introduction

Understanding the Exponential Function

Plotting Five Points on the Graph

Drawing the Asymptote

Properties of the Graph

Real-World Applications

Conclusion

References

Additional Resources

Q&A: Graphing the Exponential Function

Q: What is an exponential function?

Q: What is the graph of the exponential function f(x)=(13)xf(x) = \left(\frac{1}{3}\right)^xf(x)=(31​)x?

Q: How do I plot five points on the graph of the function?

Q: How do I draw the asymptote?

Q: What are the properties of the graph?

Q: What are the real-world applications of the graph?

Q: How can I use the graph in real-world applications?

Q: What are some common mistakes to avoid when graphing the exponential function?

Conclusion

References

Additional Resources

Q: What is the graph of the exponential function $f(x) = \left(\frac{1}{3}\right)^x$ ?