Graph The Function F ( X ) = 12 ( 1 3 ) X F(x) = 12\left(\frac{1}{3}\right)^{x} F ( X ) = 12 ( 3 1 ) X By Hand. Then, Use A Graphing Calculator To Verify Your Graph. State The Domain And The Range Of The Function.A. The Range Of F ( X ) = 12 ( 1 3 ) X F(x) = 12\left(\frac{1}{3}\right)^{x} F ( X ) = 12 ( 3 1 ) X Is (Type

Mar 12, 2025 by ADMIN 326 views

$**Graphing the Function $f(x) = 12\left(\frac{1}{3}\right)^{x}$ by Hand and Verifying with a Graphing Calculator**$

Introduction

Graphing functions by hand can be a challenging task, especially when dealing with exponential functions. However, with a clear understanding of the function's behavior and properties, we can create an accurate graph. In this article, we will graph the function $f(x) = 12\left(\frac{1}{3}\right)^{x}$ by hand and then verify our graph using a graphing calculator. We will also determine the domain and range of the function.

Graphing the Function by Hand

To graph the function $f(x) = 12\left(\frac{1}{3}\right)^{x}$ by hand, we need to understand the behavior of the function as $x$ varies. The function is an exponential function with a base of $\frac{1}{3}$ and a coefficient of $12$ . As $x$ increases, the value of the function decreases, and as $x$ decreases, the value of the function increases.

Asymptotes

The function has a horizontal asymptote at $y = 0$ . This is because as $x$ approaches infinity, the value of the function approaches $0$ .

Intercepts

The function has a $y$ -intercept at $(0, 12)$ . This is because when $x = 0$ , the value of the function is $12$ .

Increasing and Decreasing Intervals

The function is decreasing on the interval $(-\infty, \infty)$ .

Graphing the Function

To graph the function, we can start by plotting the $y$ -intercept at $(0, 12)$ . Then, we can use the asymptote at $y = 0$ to guide the graph. As $x$ increases, the value of the function decreases, so we can plot points on the graph that reflect this behavior.

Graphing the Function with a Graphing Calculator

To verify our graph, we can use a graphing calculator to graph the function. We can enter the function into the calculator and adjust the window settings to get a clear view of the graph.

Graphing Calculator Settings

To get a clear view of the graph, we can adjust the window settings to the following:

Xmin: -10
Xmax: 10
Ymin: 0
Ymax: 20

With these settings, we can see that the graph of the function is a decreasing exponential curve that approaches the $x$ -axis as $x$ increases.

Domain and Range of the Function

The domain of the function is all real numbers, $(-\infty, \infty)$ . The range of the function is all non-negative real numbers, $(0, \infty)$ .

Domain

The domain of the function is all real numbers because the function is defined for all values of $x$ .

Range

The range of the function is all non-negative real numbers because the function approaches $0$ as $x$ approaches infinity.

Conclusion

Graphing the function $f(x) = 12\left(\frac{1}{3}\right)^{x}$ by hand requires a clear understanding of the function's behavior and properties. By using asymptotes, intercepts, and increasing and decreasing intervals, we can create an accurate graph. Verifying our graph with a graphing calculator confirms that the graph is a decreasing exponential curve that approaches the $x$ -axis as $x$ increases. The domain of the function is all real numbers, and the range is all non-negative real numbers.

References

Additional Resources

Graphing Functions
Exponential Functions
Graphing Calculators
Q&A: Graphing the Function $f(x) = 12\left(\frac{1}{3}\right)^{x}$ ===========================================================

Frequently Asked Questions

Q: What is the domain of the function $f(x) = 12\left(\frac{1}{3}\right)^{x}$ ? A: The domain of the function is all real numbers, $(-\infty, \infty)$ .

Q: What is the range of the function $f(x) = 12\left(\frac{1}{3}\right)^{x}$ ? A: The range of the function is all non-negative real numbers, $(0, \infty)$ .

Q: How do I graph the function $f(x) = 12\left(\frac{1}{3}\right)^{x}$ by hand? A: To graph the function by hand, you need to understand the behavior of the function as $x$ varies. The function has a horizontal asymptote at $y = 0$ , a $y$ -intercept at $(0, 12)$ , and is decreasing on the interval $(-\infty, \infty)$ .

Q: How do I verify my graph using a graphing calculator? A: To verify your graph using a graphing calculator, enter the function into the calculator and adjust the window settings to get a clear view of the graph. You can adjust the window settings to the following:

Xmin: -10
Xmax: 10
Ymin: 0
Ymax: 20

Q: What is the significance of the horizontal asymptote at $y = 0$ ? A: The horizontal asymptote at $y = 0$ indicates that as $x$ approaches infinity, the value of the function approaches $0$ .

Q: How do I determine the $y$ -intercept of the function? A: To determine the $y$ -intercept of the function, substitute $x = 0$ into the function and evaluate the result. In this case, the $y$ -intercept is $(0, 12)$ .

Q: What is the significance of the decreasing interval $(-\infty, \infty)$ ? A: The decreasing interval $(-\infty, \infty)$ indicates that as $x$ increases, the value of the function decreases.

Q: Can I use a graphing calculator to graph other types of functions? A: Yes, you can use a graphing calculator to graph other types of functions, such as polynomial functions, rational functions, and trigonometric functions.

Q: How do I adjust the window settings on a graphing calculator? A: To adjust the window settings on a graphing calculator, follow these steps:

Press the "Window" button on the calculator.
Adjust the Xmin, Xmax, Ymin, and Ymax settings to the desired values.
Press the "Graph" button to view the graph.

Conclusion

Graphing the function $f(x) = 12\left(\frac{1}{3}\right)^{x}$ by hand and verifying it with a graphing calculator requires a clear understanding of the function's behavior and properties. By using asymptotes, intercepts, and increasing and decreasing intervals, we can create an accurate graph. The domain of the function is all real numbers, and the range is all non-negative real numbers.