Which Statement Describes The Graph Of $f(x) = 4x^7 + 40x^6 + 100x^5$?A. The Graph Crosses The $x$-axis At $x = 0$ And Touches The $x$-axis At $x = 5$.B. The Graph Touches The $x$-axis At $x =

Introduction

In mathematics, a polynomial function is a function that can be expressed as a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power. The graph of a polynomial function can be analyzed to determine its behavior and characteristics. In this article, we will examine the graph of the function $f(x) = 4x^7 + 40x^6 + 100x^5$ and determine which statement describes its behavior.

The Function $f(x) = 4x^7 + 40x^6 + 100x^5$

The given function is a polynomial function of degree 7, which means it has 7 terms. The leading term is $4x^7$, which indicates that the function has a positive leading coefficient. This means that the function will have a positive value for large positive values of $x$ and a negative value for large negative values of $x$.

Analyzing the Graph

To analyze the graph of the function, we need to examine its behavior at the origin and its behavior as $x$ approaches infinity.

Behavior at the Origin

The function $f(x) = 4x^7 + 40x^6 + 100x^5$ is a polynomial function, and as such, it is continuous at the origin. However, we need to determine whether the function crosses or touches the $x$-axis at the origin.

To determine this, we can evaluate the function at $x = 0$. We have:

$$f(0) = 4(0)^7 + 40(0)^6 + 100(0)^5 = 0$$

This means that the function passes through the origin, but it does not necessarily mean that the function crosses the $x$-axis at the origin.

Behavior as $x$ Approaches Infinity

As $x$ approaches infinity, the leading term $4x^7$ dominates the function. Since the leading coefficient is positive, the function will have a positive value for large positive values of $x$.

Behavior as $x$ Approaches Negative Infinity

As $x$ approaches negative infinity, the leading term $4x^7$ dominates the function. Since the leading coefficient is positive, the function will have a negative value for large negative values of $x$.

Conclusion

Based on the analysis of the function $f(x) = 4x^7 + 40x^6 + 100x^5$, we can conclude that the graph of the function does not cross the $x$-axis at $x = 0$. The function passes through the origin, but it does not touch the $x$-axis at $x = 0$.

Which Statement Describes the Graph of $f(x) = 4x^7 + 40x^6 + 100x^5$?

Based on the analysis of the function, we can conclude that the correct statement is:

The graph passes through the origin, but it does not touch the $x$-axis at $x = 0$.

This statement is consistent with the behavior of the function at the origin and its behavior as $x$ approaches infinity and negative infinity.

Discussion

The graph of a polynomial function can be analyzed to determine its behavior and characteristics. In this article, we examined the graph of the function $f(x) = 4x^7 + 40x^6 + 100x^5$ and determined which statement describes its behavior.

The analysis of the function showed that the graph passes through the origin, but it does not touch the $x$-axis at $x = 0$. This behavior is consistent with the behavior of the function as $x$ approaches infinity and negative infinity.

The graph of a polynomial function can be analyzed using various techniques, including the use of derivatives and limits. In this article, we used the analysis of the function at the origin and its behavior as $x$ approaches infinity and negative infinity to determine which statement describes its behavior.

References

• [1] Thomas, G. B. (2010). Calculus and Analytic Geometry. Addison-Wesley.
• [2] Larson, R. E. (2013). Calculus: Early Transcendentals. Brooks Cole.
• [3] Rogawski, J. (2011). Calculus: Early Transcendentals. W.H. Freeman and Company.

Keywords

• Polynomial function
• Graph of a polynomial function
• Behavior of a polynomial function
• Analysis of a polynomial function
• Derivatives and limits
• Calculus and analytic geometry
  Q&A: Understanding the Graph of a Polynomial Function =====================================================

Introduction

In our previous article, we analyzed the graph of the function $f(x) = 4x^7 + 40x^6 + 100x^5$ and determined which statement describes its behavior. In this article, we will answer some frequently asked questions about the graph of a polynomial function.

Q: What is a polynomial function?

A: A polynomial function is a function that can be expressed as a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power.

Q: What is the degree of a polynomial function?

A: The degree of a polynomial function is the highest power of the variable in the function. For example, the degree of the function $f(x) = 4x^7 + 40x^6 + 100x^5$ is 7.

Q: How do I determine the behavior of a polynomial function at the origin?

A: To determine the behavior of a polynomial function at the origin, you can evaluate the function at $x = 0$. If the function is equal to 0 at $x = 0$, then the function passes through the origin. If the function is not equal to 0 at $x = 0$, then the function does not pass through the origin.

Q: How do I determine the behavior of a polynomial function as $x$ approaches infinity?

A: To determine the behavior of a polynomial function as $x$ approaches infinity, you can examine the leading term of the function. If the leading term is positive, then the function will have a positive value for large positive values of $x$. If the leading term is negative, then the function will have a negative value for large positive values of $x$.

Q: How do I determine the behavior of a polynomial function as $x$ approaches negative infinity?

A: To determine the behavior of a polynomial function as $x$ approaches negative infinity, you can examine the leading term of the function. If the leading term is positive, then the function will have a negative value for large negative values of $x$. If the leading term is negative, then the function will have a positive value for large negative values of $x$.

Q: What is the difference between a polynomial function and a rational function?

A: A polynomial function is a function that can be expressed as a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power. A rational function is a function that can be expressed as the ratio of two polynomial functions.

Q: How do I graph a polynomial function?

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

Q: What are some common types of polynomial functions?

A: Some common types of polynomial functions include:

• Linear functions: $f(x) = ax + b$
• Quadratic functions: $f(x) = ax^2 + bx + c$
• Cubic functions: $f(x) = ax^3 + bx^2 + cx + d$
• Quartic functions: $f(x) = ax^4 + bx^3 + cx^2 + dx + e$

Conclusion

In this article, we answered some frequently asked questions about the graph of a polynomial function. We hope that this information is helpful to you in your studies.

References

• [1] Thomas, G. B. (2010). Calculus and Analytic Geometry. Addison-Wesley.
• [2] Larson, R. E. (2013). Calculus: Early Transcendentals. Brooks Cole.
• [3] Rogawski, J. (2011). Calculus: Early Transcendentals. W.H. Freeman and Company.

Keywords

• Polynomial function
• Graph of a polynomial function
• Behavior of a polynomial function
• Analysis of a polynomial function
• Derivatives and limits
• Calculus and analytic geometry