Creating the Table of Values

Oct 13, 2025 by recoveryshake.com 99 views

Introduction

In mathematics, functions are used to describe the relationship between variables. A function $f(x)$ is a rule that assigns to each input value $x$ a unique output value $f(x)$ . In this article, we will discuss the function $f(x)=\frac{1}{3}(6)^x$ and its reflection across the $x$ -axis, denoted as $g(x)$ .

The Function $f(x)$

The function $f(x)$ is defined as $f(x)=\frac{1}{3}(6)^x$ . This function is an exponential function, where the base is $6$ and the exponent is $x$ . The coefficient $\frac{1}{3}$ is a constant that scales the output of the function.

Reflection Across the $x$ -Axis

A reflection across the $x$ -axis is a transformation that flips a function over the $x$ -axis. This means that for every point $(x, y)$ on the original function, the reflected function has the point $(x, -y)$ . In other words, the $y$ -coordinate of each point is negated.

Table of Values for $g(x)$

To graph the reflection of $f(x)$ across the $x$ -axis, we need to create a table of values for $g(x)$ . Since $g(x)$ is a reflection of $f(x)$ , we can use the same $x$ -values as $f(x)$ , but we need to negate the corresponding $y$ -values.

Creating the Table of Values

To create the table of values for $g(x)$ , we can use the following steps:

Choose a set of $x$ -values.
Calculate the corresponding $y$ -values for $f(x)$ using the equation $f(x)=\frac{1}{3}(6)^x$ .
Negate the $y$ -values to obtain the corresponding $y$ -values for $g(x)$ .

Example

Let's choose the following $x$ -values: $x = -2, -1, 0, 1, 2$ . We can calculate the corresponding $y$ -values for $f(x)$ using the equation $f(x)=\frac{1}{3}(6)^x$ .

$x$	$f(x)$
-2	$\frac{1}{3}(6)^{-2}$
-1	$\frac{1}{3}(6)^{-1}$
0	$\frac{1}{3}(6)^0$
1	$\frac{1}{3}(6)^1$
2	$\frac{1}{3}(6)^2$

Now, we can negate the $y$ -values to obtain the corresponding $y$ -values for $g(x)$ .

$x$	$g(x)$
-2	$-\frac{1}{3}(6)^{-2}$
-1	$-\frac{1}{3}(6)^{-1}$
0	$-\frac{1}{3}(6)^0$
1	$-\frac{1}{3}(6)^1$
2	$-\frac{1}{3}(6)^2$

Conclusion

In this article, we discussed the function $f(x)=\frac{1}{3}(6)^x$ and its reflection across the $x$ -axis, denoted as $g(x)$ . We created a table of values for $g(x)$ by negating the corresponding $y$ -values of $f(x)$ . This table of values can be used to graph $g(x)$ .

Discussion

The function $f(x)$ is an exponential function with a base of $6$ and an exponent of $x$ . The coefficient $\frac{1}{3}$ scales the output of the function. The reflection of $f(x)$ across the $x$ -axis is denoted as $g(x)$ , and we created a table of values for $g(x)$ by negating the corresponding $y$ -values of $f(x)$ .

Key Takeaways

The function $f(x)$ is defined as $f(x)=\frac{1}{3}(6)^x$ .
The reflection of $f(x)$ across the $x$ -axis is denoted as $g(x)$ .
To create a table of values for $g(x)$ , we can use the same $x$ -values as $f(x)$ , but we need to negate the corresponding $y$ -values.

Final Thoughts

In conclusion, the function $f(x)=\frac{1}{3}(6)^x$ and its reflection across the $x$ -axis, denoted as $g(x)$ , are important concepts in mathematics. By creating a table of values for $g(x)$ , we can graph the reflection of $f(x)$ across the $x$ -axis. This article provides a step-by-step guide on how to create a table of values for $g(x)$ and graph the reflection of $f(x)$ across the $x$ -axis.

Introduction

In our previous article, we discussed the function $f(x)=\frac{1}{3}(6)^x$ and its reflection across the $x$ -axis, denoted as $g(x)$ . We created a table of values for $g(x)$ by negating the corresponding $y$ -values of $f(x)$ . In this article, we will answer some frequently asked questions about the function $f(x)$ and its reflection across the $x$ -axis.

Q1: What is the base of the exponential function $f(x)$ ?

A1: The base of the exponential function $f(x)$ is $6$ .

Q2: What is the coefficient of the exponential function $f(x)$ ?

A2: The coefficient of the exponential function $f(x)$ is $\frac{1}{3}$ .

Q3: How do you create a table of values for $g(x)$ ?

A3: To create a table of values for $g(x)$ , you can use the same $x$ -values as $f(x)$ , but you need to negate the corresponding $y$ -values.

Q4: What is the relationship between $f(x)$ and $g(x)$ ?

A4: $g(x)$ is the reflection of $f(x)$ across the $x$ -axis.

Q5: How do you graph $g(x)$ ?

A5: To graph $g(x)$ , you can use the table of values we created earlier and plot the points on a coordinate plane. Since $g(x)$ is a reflection of $f(x)$ across the $x$ -axis, the graph of $g(x)$ will be a mirror image of the graph of $f(x)$ .

Q6: What is the domain of the function $f(x)$ ?

A6: The domain of the function $f(x)$ is all real numbers, or $(-\infty, \infty)$ .

Q7: What is the range of the function $f(x)$ ?

A7: The range of the function $f(x)$ is all positive real numbers, or $(0, \infty)$ .

Q8: How do you find the $y$ -intercept of the function $f(x)$ ?

A8: To find the $y$ -intercept of the function $f(x)$ , you can plug in $x=0$ into the equation $f(x)=\frac{1}{3}(6)^x$ . This will give you the $y$ -intercept of the function.

Q9: How do you find the $x$ -intercept of the function $f(x)$ ?

A9: To find the $x$ -intercept of the function $f(x)$ , you can set $f(x)=0$ and solve for $x$ . This will give you the $x$ -intercept of the function.

Q10: What is the difference between the functions $f(x)$ and $g(x)$ ?

A10: The main difference between the functions $f(x)$ and $g(x)$ is that $g(x)$ is a reflection of $f(x)$ across the $x$ -axis. This means that the graph of $g(x)$ will be a mirror image of the graph of $f(x)$ .

Conclusion

In this article, we answered some frequently asked questions about the function $f(x)=\frac{1}{3}(6)^x$ and its reflection across the $x$ -axis, denoted as $g(x)$ . We hope that this article has provided you with a better understanding of the function $f(x)$ and its reflection across the $x$ -axis.

Discussion

The function $f(x)=\frac{1}{3}(6)^x$ is an exponential function with a base of $6$ and an exponent of $x$ . The coefficient $\frac{1}{3}$ scales the output of the function. The reflection of $f(x)$ across the $x$ -axis is denoted as $g(x)$ , and we created a table of values for $g(x)$ by negating the corresponding $y$ -values of $f(x)$ .

Key Takeaways

The function $f(x)$ is defined as $f(x)=\frac{1}{3}(6)^x$ .
The reflection of $f(x)$ across the $x$ -axis is denoted as $g(x)$ .
To create a table of values for $g(x)$ , you can use the same $x$ -values as $f(x)$ , but you need to negate the corresponding $y$ -values.

Final Thoughts

In conclusion, the function $f(x)=\frac{1}{3}(6)^x$ and its reflection across the $x$ -axis, denoted as $g(x)$ , are important concepts in mathematics. By understanding the properties of the function $f(x)$ and its reflection across the $x$ -axis, you can better understand the behavior of exponential functions and their reflections.