How Would You Describe The Difference Between The Graphs Of F ( X ) = 2 3 X 3 + 1 F(x)=\frac{2}{3} X^3+1 F ( X ) = 3 2 X 3 + 1 And G ( X ) = 2 3 ( − X ) 3 + 1 G(x)=\frac{2}{3}(-x)^3+1 G ( X ) = 3 2 ( − X ) 3 + 1 ?A. G ( X G(x G ( X ] Is A Reflection Of F ( X F(x F ( X ] Over The X X X -axis.B. G ( X G(x G ( X ] Is A Reflection Of

Mar 11, 2025 by ADMIN 336 views

**Understanding the Difference Between Two Cubic Functions**

Introduction

When dealing with functions, it's essential to understand the properties and characteristics of each function. In this article, we will explore the difference between two cubic functions, $f(x)=\frac{2}{3} x^3+1$ and $g(x)=\frac{2}{3}(-x)^3+1$ . We will analyze the graphs of these functions and discuss the key differences between them.

Graphing the Functions

To visualize the graphs of $f(x)$ and $g(x)$ , we can start by plotting a few points on each graph. Let's begin with $f(x)=\frac{2}{3} x^3+1$ . We can choose some values of $x$ and calculate the corresponding values of $f(x)$ .

$x$	$f(x)$
-2	$\frac{2}{3}(-2)^3+1 = -\frac{16}{3}+1 = -\frac{11}{3}$
-1	$\frac{2}{3}(-1)^3+1 = -\frac{2}{3}+1 = \frac{1}{3}$
0	$\frac{2}{3}(0)^3+1 = 1$
1	$\frac{2}{3}(1)^3+1 = \frac{2}{3}+1 = \frac{5}{3}$
2	$\frac{2}{3}(2)^3+1 = \frac{16}{3}+1 = \frac{19}{3}$

Now, let's plot these points on a coordinate plane. We can see that the graph of $f(x)$ is a cubic curve that opens upward.

Next, let's analyze the graph of $g(x)=\frac{2}{3}(-x)^3+1$ . We can choose some values of $x$ and calculate the corresponding values of $g(x)$ .

$x$	$g(x)$
-2	$\frac{2}{3}(-(-2))^3+1 = \frac{2}{3}(2)^3+1 = \frac{16}{3}+1 = \frac{19}{3}$
-1	$\frac{2}{3}(-(-1))^3+1 = \frac{2}{3}(1)^3+1 = \frac{2}{3}+1 = \frac{5}{3}$
0	$\frac{2}{3}(-0)^3+1 = 1$
1	$\frac{2}{3}(-1)^3+1 = -\frac{2}{3}+1 = \frac{1}{3}$
2	$\frac{2}{3}(-2)^3+1 = -\frac{16}{3}+1 = -\frac{11}{3}$

Now, let's plot these points on a coordinate plane. We can see that the graph of $g(x)$ is also a cubic curve, but it opens downward.

Analyzing the Graphs

From the graphs of $f(x)$ and $g(x)$ , we can observe that both functions have the same shape, but they are reflected over the $x$ -axis. This means that the graph of $g(x)$ is a reflection of the graph of $f(x)$ over the $x$ -axis.

Conclusion

In conclusion, the graph of $g(x)=\frac{2}{3}(-x)^3+1$ is a reflection of the graph of $f(x)=\frac{2}{3} x^3+1$ over the $x$ -axis. This is because the function $g(x)$ is obtained by replacing $x$ with $-x$ in the function $f(x)$ , which results in a reflection of the graph over the $x$ -axis.

Key Takeaways

The graph of $g(x)=\frac{2}{3}(-x)^3+1$ is a reflection of the graph of $f(x)=\frac{2}{3} x^3+1$ over the $x$ -axis.
The function $g(x)$ is obtained by replacing $x$ with $-x$ in the function $f(x)$ .
The graph of $g(x)$ opens downward, while the graph of $f(x)$ opens upward.

Final Thoughts

In this article, we have analyzed the difference between two cubic functions, $f(x)=\frac{2}{3} x^3+1$ and $g(x)=\frac{2}{3}(-x)^3+1$ . We have discussed the key differences between the graphs of these functions and have concluded that the graph of $g(x)$ is a reflection of the graph of $f(x)$ over the $x$ -axis. This is an essential concept in mathematics, and it has many applications in various fields, such as physics, engineering, and economics.

Q: What is the main difference between the graphs of $f(x)=\frac{2}{3} x^3+1$ and $g(x)=\frac{2}{3}(-x)^3+1$ ?

A: The main difference between the graphs of $f(x)$ and $g(x)$ is that the graph of $g(x)$ is a reflection of the graph of $f(x)$ over the $x$ -axis.

Q: Why is the graph of $g(x)$ a reflection of the graph of $f(x)$ over the $x$ -axis?

A: The graph of $g(x)$ is a reflection of the graph of $f(x)$ over the $x$ -axis because the function $g(x)$ is obtained by replacing $x$ with $-x$ in the function $f(x)$ .

Q: What is the effect of replacing $x$ with $-x$ in the function $f(x)$ ?

A: Replacing $x$ with $-x$ in the function $f(x)$ results in a reflection of the graph of $f(x)$ over the $x$ -axis.

Q: How can we determine if a function is a reflection of another function over the $x$ -axis?

A: We can determine if a function is a reflection of another function over the $x$ -axis by checking if the function is obtained by replacing $x$ with $-x$ in the original function.

Q: What is the significance of the graph of $g(x)$ opening downward, while the graph of $f(x)$ opens upward?

A: The graph of $g(x)$ opening downward, while the graph of $f(x)$ opens upward, is a result of the reflection of the graph of $f(x)$ over the $x$ -axis.

Q: Can we conclude that the graph of $g(x)$ is a reflection of the graph of $f(x)$ over the $x$ -axis?

A: Yes, we can conclude that the graph of $g(x)$ is a reflection of the graph of $f(x)$ over the $x$ -axis, based on the analysis of the graphs and the effect of replacing $x$ with $-x$ in the function $f(x)$ .

Q: What are some real-world applications of understanding the difference between the graphs of two cubic functions?

A: Understanding the difference between the graphs of two cubic functions has many real-world applications, such as in physics, engineering, and economics. For example, it can be used to model the motion of objects, the growth of populations, and the behavior of financial markets.

Q: How can we use the knowledge of the difference between the graphs of two cubic functions to solve problems?

A: We can use the knowledge of the difference between the graphs of two cubic functions to solve problems by analyzing the graphs and using the properties of the functions to make predictions and draw conclusions.

Q: What are some common mistakes to avoid when analyzing the difference between the graphs of two cubic functions?

A: Some common mistakes to avoid when analyzing the difference between the graphs of two cubic functions include:

Not checking if the function is a reflection of the original function over the $x$ -axis
Not analyzing the effect of replacing $x$ with $-x$ in the function
Not considering the properties of the functions, such as the direction of the opening of the graph

Q: How can we ensure that we are accurately analyzing the difference between the graphs of two cubic functions?

A: We can ensure that we are accurately analyzing the difference between the graphs of two cubic functions by:

Double-checking our calculations and analysis
Considering multiple perspectives and viewpoints
Using visual aids, such as graphs and charts, to help with the analysis

Q: What are some additional resources that can help us learn more about the difference between the graphs of two cubic functions?

A: Some additional resources that can help us learn more about the difference between the graphs of two cubic functions include:

Textbooks and online resources on algebra and calculus
Graphing calculators and computer software
Online tutorials and video lectures
Practice problems and exercises