The Difference Between the Graphs

Oct 13, 2025 by magicalceylontours.com 80 views

Introduction

In mathematics, functions are used to describe the relationship between variables. When dealing with functions, it's essential to understand the properties and behavior of these functions, especially when they are transformed or modified. In this article, we will explore the difference between two cubic functions, $f(x)=\frac{2}{3} x^3$ and $g(x)=\frac{2}{3}(-x)^3$ , and discuss how they are related.

The Functions

Let's start by examining the two functions:

$f(x)=\frac{2}{3} x^3$
$g(x)=\frac{2}{3}(-x)^3$

Both functions are cubic functions, meaning they have a cubic term in their expression. The only difference between the two functions is the sign of the variable $x$ in the function $g(x)$ .

Graphing the Functions

To understand the difference between the two functions, let's graph them. We can use a graphing tool or software to visualize the functions.

When we graph $f(x)=\frac{2}{3} x^3$ , we get a cubic function that opens upwards. The graph is symmetric about the origin, and it has a positive leading coefficient.

When we graph $g(x)=\frac{2}{3}(-x)^3$ , we get a cubic function that also opens upwards. However, the graph is symmetric about the origin, and it has a negative leading coefficient.

The Difference Between the Graphs

Now that we have graphed both functions, let's discuss the difference between them.

The graph of $g(x)=\frac{2}{3}(-x)^3$ is a reflection of the graph of $f(x)=\frac{2}{3} x^3$ over the $x$ -axis. This means that if we were to reflect the graph of $f(x)$ over the $x$ -axis, we would get the graph of $g(x)$ .

Why is this the case?

The reason why the graph of $g(x)$ is a reflection of the graph of $f(x)$ over the $x$ -axis is because of the negative sign in the function $g(x)$ . When we raise a negative number to an odd power, the result is negative. In this case, the negative sign in the function $g(x)$ causes the graph to be reflected over the $x$ -axis.

Conclusion

In conclusion, the graph of $g(x)=\frac{2}{3}(-x)^3$ is a reflection of the graph of $f(x)=\frac{2}{3} x^3$ over the $x$ -axis. This is because of the negative sign in the function $g(x)$ , which causes the graph to be reflected over the $x$ -axis.

Key Takeaways

The graph of $g(x)=\frac{2}{3}(-x)^3$ is a reflection of the graph of $f(x)=\frac{2}{3} x^3$ over the $x$ -axis.
The negative sign in the function $g(x)$ causes the graph to be reflected over the $x$ -axis.
When we raise a negative number to an odd power, the result is negative.

Final Thoughts

$In this article, we explored the difference between two cubic functions, $f(x)=\frac{2}{3} x^3$ and $g(x)=\frac{2}{3}(-x)^3$. We discussed how the graph of $g(x)$ is a reflection of the graph of $f(x)$ over the $x$-axis, and why this is the case. We also highlighted the key takeaways from this article, including the fact that the negative sign in the function $g(x)$ causes the graph to be reflected over the $x$-axis.<br/> **Q&A: Understanding the Difference Between Two Cubic Functions**$

Introduction

In our previous article, we explored the difference between two cubic functions, $f(x)=\frac{2}{3} x^3$ and $g(x)=\frac{2}{3}(-x)^3$ . We discussed how the graph of $g(x)$ is a reflection of the graph of $f(x)$ over the $x$ -axis, and why this is the case. In this article, we will answer some frequently asked questions about the difference between these two functions.

Q: What is the main difference between the two functions?

A: The main difference between the two functions is the sign of the variable $x$ in the function $g(x)$ . In the function $f(x)$ , the variable $x$ is positive, while in the function $g(x)$ , the variable $x$ is negative.

Q: Why does the graph of $g(x)$ reflect the graph of $f(x)$ over the $x$ -axis?

A: The graph of $g(x)$ reflects the graph of $f(x)$ over the $x$ -axis because of the negative sign in the function $g(x)$ . When we raise a negative number to an odd power, the result is negative. In this case, the negative sign in the function $g(x)$ causes the graph to be reflected over the $x$ -axis.

Q: What happens when we raise a negative number to an even power?

A: When we raise a negative number to an even power, the result is positive. For example, $(-2)^2 = 4$ , which is a positive number.

Q: Can we generalize the result for any cubic function?

A: Yes, we can generalize the result for any cubic function. If we have a cubic function of the form $f(x) = ax^3 + bx^2 + cx + d$ , where $a$ , $b$ , $c$ , and $d$ are constants, and we replace $x$ with $-x$ , we will get a new function of the form $g(x) = a(-x)^3 + b(-x)^2 + c(-x) + d$ . If $a$ is positive, the graph of $g(x)$ will be a reflection of the graph of $f(x)$ over the $x$ -axis.

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

A: Cubic functions have many real-world applications, including:

Modeling population growth: Cubic functions can be used to model population growth, where the population grows at a rate that is proportional to the cube of the current population.
Modeling chemical reactions: Cubic functions can be used to model chemical reactions, where the rate of reaction is proportional to the cube of the concentration of the reactants.
Modeling physical systems: Cubic functions can be used to model physical systems, such as the motion of a pendulum or the vibration of a spring.

Q: How can we use technology to visualize cubic functions?

A: We can use technology, such as graphing calculators or computer software, to visualize cubic functions. These tools allow us to easily graph and analyze cubic functions, and to explore their properties and behavior.

Conclusion

In this article, we answered some frequently asked questions about the difference between two cubic functions, $f(x)=\frac{2}{3} x^3$ and $g(x)=\frac{2}{3}(-x)^3$ . We discussed how the graph of $g(x)$ reflects the graph of $f(x)$ over the $x$ -axis, and why this is the case. We also highlighted some real-world applications of cubic functions and discussed how technology can be used to visualize these functions.

Key Takeaways

The graph of $g(x)=\frac{2}{3}(-x)^3$ reflects the graph of $f(x)=\frac{2}{3} x^3$ over the $x$ -axis.
The negative sign in the function $g(x)$ causes the graph to be reflected over the $x$ -axis.
Cubic functions have many real-world applications, including modeling population growth, chemical reactions, and physical systems.
Technology can be used to visualize cubic functions and explore their properties and behavior.