Complete The Missing Parts Of The Table For The Following Function.$\[ Y = 6^x \\]$\[ \begin{tabular}{c|cccccc} x & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline y & \frac{1}{36} & \frac{1}{6} & 1 & 6 & 36 & 216

Introduction

In mathematics, exponential functions are a fundamental concept that describe the relationship between two variables, often representing growth or decay. The given function, $y = 6^x$, is an example of an exponential function where the base is 6. In this article, we will focus on completing the missing parts of the table for the given function.

Understanding Exponential Functions

Before we dive into solving the problem, let's briefly review the concept of exponential functions. An exponential function is a function of the form $y = a^x$, where $a$ is the base and $x$ is the exponent. The value of $y$ is obtained by raising the base $a$ to the power of $x$. In the given function, $y = 6^x$, the base is 6, and the exponent is $x$.

Completing the Missing Parts of the Table

To complete the missing parts of the table, we need to calculate the values of $y$ for the given values of $x$. We can do this by substituting the values of $x$ into the function $y = 6^x$.

Calculating y for x = -3

To calculate the value of $y$ for $x = -3$, we substitute $x = -3$ into the function $y = 6^x$.

$$y = 6^{-3} = \frac{1}{6^3} = \frac{1}{216}$$

Calculating y for x = -1.5

To calculate the value of $y$ for $x = -1.5$, we substitute $x = -1.5$ into the function $y = 6^x$.

$$y = 6^{-1.5} = \frac{1}{6^{1.5}} = \frac{1}{\sqrt{6^3}} = \frac{1}{6\sqrt{6}}$$

Calculating y for x = 0.5

To calculate the value of $y$ for $x = 0.5$, we substitute $x = 0.5$ into the function $y = 6^x$.

$$y = 6^{0.5} = \sqrt{6^2} = \sqrt{36} = 6$$

Calculating y for x = 2.5

To calculate the value of $y$ for $x = 2.5$, we substitute $x = 2.5$ into the function $y = 6^x$.

$$y = 6^{2.5} = 6^2 \cdot 6^{0.5} = 36 \cdot \sqrt{6}$$

Calculating y for x = 3.5

To calculate the value of $y$ for $x = 3.5$, we substitute $x = 3.5$ into the function $y = 6^x$.

$$y = 6^{3.5} = 6^3 \cdot 6^{0.5} = 216 \cdot \sqrt{6}$$

The Completed Table

x	y
-3	1/216
-2	1/36
-1.5	1/(6*sqrt(6))
-1	1/6
0	1
0.5	6
1	6
1.5	6*sqrt(6)
2	36
2.5	36*sqrt(6)
3	216
3.5	216*sqrt(6)

Conclusion

In this article, we completed the missing parts of the table for the given function $y = 6^x$. We calculated the values of $y$ for the given values of $x$ by substituting the values of $x$ into the function. The completed table provides a clear understanding of the relationship between the variables $x$ and $y$ in the given function.

Discussion

The given function $y = 6^x$ is an example of an exponential function where the base is 6. The values of $y$ are obtained by raising the base 6 to the power of $x$. The completed table provides a clear understanding of the relationship between the variables $x$ and $y$ in the given function.

Exercises

1. Calculate the value of $y$ for $x = -4$.
2. Calculate the value of $y$ for $x = 4$.
3. Calculate the value of $y$ for $x = 1.5$.
4. Calculate the value of $y$ for $x = 2.5$.
5. Calculate the value of $y$ for $x = 3.5$.

Answers

1. $y = 1/1296$
2. $y = 1296$
3. $y = 6\sqrt{6}$
4. $y = 36\sqrt{6}$
5. $y = 216\sqrt{6}$
   Q&A: Completing the Missing Parts of the Table for the Exponential Function ====================================================================

Introduction

In our previous article, we completed the missing parts of the table for the given function $y = 6^x$. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the base of the exponential function $y = 6^x$?

A: The base of the exponential function $y = 6^x$ is 6.

Q: What is the exponent of the exponential function $y = 6^x$?

A: The exponent of the exponential function $y = 6^x$ is $x$.

Q: How do I calculate the value of $y$ for a given value of $x$ in the exponential function $y = 6^x$?

A: To calculate the value of $y$ for a given value of $x$ in the exponential function $y = 6^x$, you can substitute the value of $x$ into the function and evaluate the expression.

Q: What is the relationship between the variables $x$ and $y$ in the exponential function $y = 6^x$?

A: The relationship between the variables $x$ and $y$ in the exponential function $y = 6^x$ is that $y$ is obtained by raising the base 6 to the power of $x$.

Q: How do I complete the missing parts of the table for the exponential function $y = 6^x$?

A: To complete the missing parts of the table for the exponential function $y = 6^x$, you can calculate the values of $y$ for the given values of $x$ by substituting the values of $x$ into the function and evaluating the expression.

Q: What are some common mistakes to avoid when completing the missing parts of the table for the exponential function $y = 6^x$?

A: Some common mistakes to avoid when completing the missing parts of the table for the exponential function $y = 6^x$ include:

• Not substituting the values of $x$ into the function correctly
• Not evaluating the expression correctly
• Not using the correct base and exponent

Q: How do I check my work when completing the missing parts of the table for the exponential function $y = 6^x$?

A: To check your work when completing the missing parts of the table for the exponential function $y = 6^x$, you can:

• Use a calculator to evaluate the expression
• Check your work by plugging in different values of $x$ and verifying that the values of $y$ are correct
• Use a graphing calculator to visualize the relationship between the variables $x$ and $y$

Q: What are some real-world applications of the exponential function $y = 6^x$?

A: Some real-world applications of the exponential function $y = 6^x$ include:

• Modeling population growth
• Modeling financial growth
• Modeling chemical reactions

Conclusion

In this article, we answered some frequently asked questions related to completing the missing parts of the table for the exponential function $y = 6^x$. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the topic.

Exercises

1. Calculate the value of $y$ for $x = -5$.
2. Calculate the value of $y$ for $x = 5$.
3. Calculate the value of $y$ for $x = 1.8$.
4. Calculate the value of $y$ for $x = 2.8$.
5. Calculate the value of $y$ for $x = 3.8$.

Answers

1. $y = 1/7776$
2. $y = 7776$
3. $y = 6^{1.8} = 6 \cdot 6^{0.8} = 6 \cdot \sqrt{6^2} = 6 \cdot \sqrt{36} = 6 \cdot 6 = 36$
4. $y = 6^{2.8} = 6^2 \cdot 6^{0.8} = 36 \cdot \sqrt{6^2} = 36 \cdot \sqrt{36} = 36 \cdot 6 = 216$
5. $y = 6^{3.8} = 6^3 \cdot 6^{0.8} = 216 \cdot \sqrt{6^2} = 216 \cdot \sqrt{36} = 216 \cdot 6 = 1296$