Analyzing the Values in the Table

Oct 13, 2025 by windailys.com 71 views

The Table of Exponential Function: Understanding the Vertically Shifted Parent Function

In mathematics, exponential functions are a fundamental concept that plays a crucial role in various mathematical operations and applications. The parent function, ${$ f(x) = 2^x $}$, is a basic exponential function that serves as a building block for more complex functions. When this parent function is vertically shifted, it results in a new function, ${$ g(x) $}$, which is represented in the table below.

x	0	1	2	3	4
g(x)	-11	-10	-9	-8	-7

The table represents the vertically shifted parent function, ${$ g(x) $}$, which has been transformed from the original parent function, ${$ f(x) = 2^x $}$. To understand the transformation, let's analyze the values in the table.

Analyzing the Values in the Table

x	g(x)
0	-11
1	-10
2	-9
3	-8
4	-7

From the table, we can observe that the values of ${$ g(x) $}$ are decreasing as the value of ${$ x $}$ increases. This indicates that the function is decreasing exponentially. To understand the relationship between the parent function and the vertically shifted function, let's compare the values of ${$ f(x) $}$ and ${$ g(x) $}$.

x	f(x)	g(x)
0	1	-11
1	2	-10
2	4	-9
3	8	-8
4	16	-7

From the table, we can observe that the values of ${$ f(x) $}$ are increasing exponentially, while the values of ${$ g(x) $}$ are decreasing exponentially. This indicates that the vertically shifted function, ${$ g(x) $}$, is a reflection of the parent function, ${$ f(x) $}$, about the x-axis.

To determine the vertical shift, let's analyze the difference between the values of ${$ f(x) $}$ and ${$ g(x) $}$.

x	f(x)	g(x)	Difference
0	1	-11	-12
1	2	-10	-12
2	4	-9	-13
3	8	-8	-16
4	16	-7	-23

From the table, we can observe that the difference between the values of ${$ f(x) $}$ and ${$ g(x) $}$ is constant, which is -12. This indicates that the vertically shifted function, ${$ g(x) $}$, has been shifted down by 12 units.

In conclusion, the table represents the vertically shifted parent function, ${$ g(x) $}$, which has been transformed from the original parent function, ${$ f(x) = 2^x $}$. By analyzing the values in the table, we can understand the relationship between the parent function and the vertically shifted function. The vertically shifted function, ${$ g(x) $}$, is a reflection of the parent function, ${$ f(x) $}$, about the x-axis, and has been shifted down by 12 units.

Exponential functions have numerous applications in various fields, including:

Finance: Exponential functions are used to calculate compound interest and depreciation.
Biology: Exponential functions are used to model population growth and decay.
Physics: Exponential functions are used to model radioactive decay and exponential growth.
Computer Science: Exponential functions are used to model the growth of algorithms and data structures.

Exponential functions can be seen in various real-world examples, including:

Compound Interest: When money is invested in a savings account, the interest earned is calculated using an exponential function.
Population Growth: The population of a city or country can be modeled using an exponential function.
Radioactive Decay: The decay of radioactive materials can be modeled using an exponential function.
Algorithmic Complexity: The time complexity of algorithms can be modeled using an exponential function.

In conclusion, exponential functions are a fundamental concept in mathematics that has numerous applications in various fields. The vertically shifted parent function, ${$ g(x) $}$, is a reflection of the parent function, ${$ f(x) = 2^x $}$, about the x-axis, and has been shifted down by 12 units. Exponential functions can be seen in various real-world examples, including compound interest, population growth, radioactive decay, and algorithmic complexity.
Exponential Function Q&A: Understanding the Vertically Shifted Parent Function

In our previous article, we explored the concept of exponential functions and the vertically shifted parent function, ${$ g(x) $}$. In this article, we will answer some frequently asked questions about exponential functions and the vertically shifted parent function.

Q: What is an exponential function?

A: An exponential function is a mathematical function that describes a relationship between two variables, where one variable is raised to a power that is proportional to the other variable.

Q: What is the parent function, ${$ f(x) = 2^x $}$?

A: The parent function, ${$ f(x) = 2^x $}$, is a basic exponential function that serves as a building block for more complex functions.

Q: What is the vertically shifted parent function, ${$ g(x) $}$?

A: The vertically shifted parent function, ${$ g(x) $}$, is a reflection of the parent function, ${$ f(x) = 2^x $}$, about the x-axis, and has been shifted down by 12 units.

Q: How do I determine the vertical shift of the function?

A: To determine the vertical shift, you can analyze the difference between the values of ${$ f(x) $}$ and ${$ g(x) $}$. If the difference is constant, it indicates that the function has been shifted down by that amount.

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

A: Exponential functions can be seen in various real-world examples, including:

Compound Interest: When money is invested in a savings account, the interest earned is calculated using an exponential function.
Population Growth: The population of a city or country can be modeled using an exponential function.
Radioactive Decay: The decay of radioactive materials can be modeled using an exponential function.
Algorithmic Complexity: The time complexity of algorithms can be modeled using an exponential function.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer program. You can also use a table of values to plot the function.

Q: What are some common mistakes to avoid when working with exponential functions?

A: Some common mistakes to avoid when working with exponential functions include:

Not checking the domain and range of the function: Make sure to check the domain and range of the function to ensure that it is defined and valid.
Not using the correct notation: Use the correct notation for exponential functions, such as ${$ f(x) = 2^x $}$.
Not considering the vertical shift: Make sure to consider the vertical shift of the function when graphing or working with it.

In conclusion, exponential functions are a fundamental concept in mathematics that has numerous applications in various fields. The vertically shifted parent function, ${$ g(x) $}$, is a reflection of the parent function, ${$ f(x) = 2^x $}$, about the x-axis, and has been shifted down by 12 units. By understanding the concept of exponential functions and the vertically shifted parent function, you can better analyze and solve problems involving exponential functions.

For more information on exponential functions and the vertically shifted parent function, you can consult the following resources:

Math textbooks: Consult a math textbook for a comprehensive explanation of exponential functions and the vertically shifted parent function.
Online resources: Visit online resources, such as Khan Academy or Mathway, for interactive lessons and practice problems.
Graphing calculators: Use a graphing calculator to visualize and explore exponential functions.

To practice working with exponential functions and the vertically shifted parent function, try the following problems:

Graph the function ${$ f(x) = 2^x $}$: Use a graphing calculator or a computer program to graph the function.
Determine the vertical shift of the function ${$ g(x) $}$: Analyze the difference between the values of ${$ f(x) $}$ and ${$ g(x) $}$ to determine the vertical shift.
Solve the equation ${$ 2^x = 16 $}$: Use the properties of exponential functions to solve the equation.

By practicing and working with exponential functions and the vertically shifted parent function, you can develop a deeper understanding of the concept and improve your problem-solving skills.