For The Equation $y = 8 \cdot 2^x$, Complete The Table And Plot The Ordered Pairs On The Graph.$\[ \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline $x$ & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline $y$ & & & & & & &

Introduction

In this article, we will delve into the world of exponential functions and explore the graph of the equation $y = 8 \cdot 2^x$. We will complete a table of ordered pairs and plot them on a graph to visualize the behavior of this function.

Understanding Exponential Functions

Exponential functions are a type of mathematical function that describes a relationship between two variables, typically denoted as $x$ and $y$. The general form of an exponential function is $y = a \cdot b^x$, where $a$ and $b$ are constants. In our case, the equation $y = 8 \cdot 2^x$ represents an exponential function with a base of 2 and a coefficient of 8.

Completing the Table

To complete the table, we need to calculate the values of $y$ for each given value of $x$. We can do this by plugging in the values of $x$ into the equation $y = 8 \cdot 2^x$.

Plotting the Ordered Pairs

Now that we have completed the table, we can plot the ordered pairs on a graph. We will use a coordinate plane with $x$ on the horizontal axis and $y$ on the vertical axis.

Graph Analysis

As we can see from the graph, the function $y = 8 \cdot 2^x$ is an exponential function that grows rapidly as $x$ increases. The graph starts at the point (0, 8) and increases exponentially as $x$ increases. The graph also passes through the points (-3, 1), (-2, 2), (-1, 4), (1, 16), (2, 32), and (3, 64).

Key Features of the Graph

• The graph passes through the point (0, 8), which is the y-intercept.
• The graph increases exponentially as $x$ increases.
• The graph passes through the points (-3, 1), (-2, 2), (-1, 4), (1, 16), (2, 32), and (3, 64).
• The graph has a horizontal asymptote at $y = 0$.

Conclusion

In this article, we explored the graph of the equation $y = 8 \cdot 2^x$. We completed a table of ordered pairs and plotted them on a graph to visualize the behavior of this function. We analyzed the graph and identified its key features, including the y-intercept, exponential growth, and horizontal asymptote. This article provides a comprehensive understanding of the graph of an exponential function and its key features.

Exercises

1. Complete the table for the equation $y = 3 \cdot 4^x$.
2. Plot the ordered pairs on a graph for the equation $y = 2 \cdot 3^x$.
3. Analyze the graph of the equation $y = 5 \cdot 2^x$ and identify its key features.

References

• [1] "Exponential Functions." Khan Academy, Khan Academy, www.khanacademy.org/math/algebra/exponential-and-logarithmic-functions/exponential-functions/v/exponential-functions.

Discussion

Introduction

In our previous article, we explored the graph of the equation $y = 8 \cdot 2^x$. We completed a table of ordered pairs and plotted them on a graph to visualize the behavior of this function. In this article, we will answer some frequently asked questions about exponential functions and their graphs.

Q: What is an exponential function?

A: An exponential function is a type of mathematical function that describes a relationship between two variables, typically denoted as $x$ and $y$. The general form of an exponential function is $y = a \cdot b^x$, where $a$ and $b$ are constants.

Q: What is the base of an exponential function?

A: The base of an exponential function is the constant $b$ in the equation $y = a \cdot b^x$. In the equation $y = 8 \cdot 2^x$, the base is 2.

Q: What is the coefficient of an exponential function?

A: The coefficient of an exponential function is the constant $a$ in the equation $y = a \cdot b^x$. In the equation $y = 8 \cdot 2^x$, the coefficient is 8.

Q: How do exponential functions grow?

A: Exponential functions grow rapidly as $x$ increases. The graph of an exponential function will increase exponentially as $x$ increases.

Q: What is the y-intercept of an exponential function?

A: The y-intercept of an exponential function is the point where the graph intersects the y-axis. In the equation $y = 8 \cdot 2^x$, the y-intercept is (0, 8).

Q: What is the horizontal asymptote of an exponential function?

A: The horizontal asymptote of an exponential function is the horizontal line that the graph approaches as $x$ increases without bound. In the equation $y = 8 \cdot 2^x$, the horizontal asymptote is $y = 0$.

Q: How do you plot the graph of an exponential function?

A: To plot the graph of an exponential function, you need to complete a table of ordered pairs and plot them on a graph. You can use a coordinate plane with $x$ on the horizontal axis and $y$ on the vertical axis.

Q: What are some real-world applications of exponential functions?

A: Exponential functions have many real-world applications, including population growth, financial investments, and chemical reactions. For example, the population of a city may grow exponentially as the city expands, and the value of an investment may grow exponentially as the interest rate increases.

Conclusion

In this article, we answered some frequently asked questions about exponential functions and their graphs. We hope that this article has provided a comprehensive understanding of exponential functions and their key features.

Exercises

1. Complete the table for the equation $y = 3 \cdot 4^x$.
2. Plot the ordered pairs on a graph for the equation $y = 2 \cdot 3^x$.
3. Analyze the graph of the equation $y = 5 \cdot 2^x$ and identify its key features.

References

• [1] "Exponential Functions." Khan Academy, Khan Academy, www.khanacademy.org/math/algebra/exponential-and-logarithmic-functions/exponential-functions/v/exponential-functions.

Discussion

What are some other real-world applications of exponential functions? How do they relate to science, technology, engineering, and mathematics (STEM) fields? Share your thoughts and insights in the comments below.