Identify The Graph Of The Function:${ F(x) = \frac{10 - 10x 2}{x 2} }$

Introduction

In mathematics, rational functions are a type of function that can be expressed as the ratio of two polynomials. These functions are commonly used in various fields, including physics, engineering, and economics. In this article, we will focus on identifying the graph of a specific rational function, $f(x) = \frac{10 - 10x^2}{x^2}$.

Understanding Rational Functions

Rational functions are defined as the ratio of two polynomials, where the numerator and denominator are both polynomials. The general form of a rational function is:

$$f(x) = \frac{p(x)}{q(x)}$$

where $p(x)$ and $q(x)$ are polynomials. Rational functions can be classified into different types based on their behavior, such as asymptotes, holes, and vertical tangents.

Asymptotes and Holes

Asymptotes are lines that the graph of a rational function approaches as $x$ approaches a certain value. There are two types of asymptotes: vertical and horizontal. Vertical asymptotes occur when the denominator of the rational function is equal to zero, while horizontal asymptotes occur when the degree of the numerator is less than the degree of the denominator.

Holes, on the other hand, are points on the graph of a rational function where the function is not defined. Holes occur when there is a common factor between the numerator and denominator of the rational function.

**Identifying the Graph of $f(x) = \frac{10 - 10x^2}{x^2}$

To identify the graph of $f(x) = \frac{10 - 10x^2}{x^2}$, we need to analyze its behavior. The numerator of the function is $10 - 10x^2$, which is a quadratic polynomial. The denominator of the function is $x^2$, which is also a quadratic polynomial.

Degree of the Numerator and Denominator

The degree of the numerator is 2, while the degree of the denominator is also 2. Since the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote of the function is $y = 1$.

Vertical Asymptote

The denominator of the function is $x^2$, which is equal to zero when $x = 0$. Therefore, the vertical asymptote of the function is $x = 0$.

Holes

There are no common factors between the numerator and denominator of the function, so there are no holes on the graph.

Graph of the Function

Based on the analysis above, the graph of the function $f(x) = \frac{10 - 10x^2}{x^2}$ has a horizontal asymptote at $y = 1$ and a vertical asymptote at $x = 0$. The graph also has a hole at $x = 0$.

Conclusion

In this article, we identified the graph of the rational function $f(x) = \frac{10 - 10x^2}{x^2}$. We analyzed the behavior of the function, including its asymptotes and holes. The graph of the function has a horizontal asymptote at $y = 1$ and a vertical asymptote at $x = 0$. The graph also has a hole at $x = 0$.

Key Takeaways

• Rational functions can be classified into different types based on their behavior.
• Asymptotes and holes are important features of rational functions.
• The degree of the numerator and denominator of a rational function determines its horizontal asymptote.
• The denominator of a rational function determines its vertical asymptote.

Further Reading

For more information on rational functions and their graphs, we recommend the following resources:

• Wikipedia: Rational Function
• Math Open Reference: Rational Functions
• Khan Academy: Rational Functions

References

• [1] Thomas, G. B. (2014). Calculus and Analytic Geometry. Addison-Wesley.
• [2] Larson, R. E., & Edwards, B. I. (2013). Calculus: Early Transcendentals. Cengage Learning.
• [3] Anton, H. (2014). Calculus: A New Horizon. John Wiley & Sons.
  Q&A: Identifying the Graph of a Rational Function =====================================================

Introduction

In our previous article, we identified the graph of the rational function $f(x) = \frac{10 - 10x^2}{x^2}$. In this article, we will answer some common questions related to identifying the graph of a rational function.

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

A: A rational function is a function that can be expressed as the ratio of two polynomials, while 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: How do I determine the degree of the numerator and denominator of a rational function?

A: To determine the degree of the numerator and denominator of a rational function, you need to count the highest power of the variable in each polynomial. For example, in the rational function $f(x) = \frac{2x^3 + 3x^2 - 4x + 1}{x^2 + 2x - 3}$, the degree of the numerator is 3 and the degree of the denominator is 2.

Q: What is the horizontal asymptote of a rational function?

A: The horizontal asymptote of a rational function is the horizontal line that the graph of the function approaches as $x$ approaches infinity. To find the horizontal asymptote, you need to compare the degrees of the numerator and denominator of the rational function. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator.

Q: What is the vertical asymptote of a rational function?

A: The vertical asymptote of a rational function is the vertical line that the graph of the function approaches as $x$ approaches a certain value. To find the vertical asymptote, you need to set the denominator of the rational function equal to zero and solve for $x$.

Q: How do I identify holes in a rational function?

A: To identify holes in a rational function, you need to factor the numerator and denominator of the function and look for common factors. If there are common factors, you need to cancel them out to find the holes.

Q: Can a rational function have more than one horizontal asymptote?

A: No, a rational function can only have one horizontal asymptote. However, a rational function can have multiple vertical asymptotes.

Q: Can a rational function have a hole and a vertical asymptote at the same point?

A: Yes, a rational function can have a hole and a vertical asymptote at the same point. This occurs when there is a common factor between the numerator and denominator of the function that is equal to zero.

Q: How do I graph a rational function?

A: To graph a rational function, you need to identify the horizontal and vertical asymptotes, holes, and any other features of the function. You can use a graphing calculator or software to graph the function.

Conclusion

In this article, we answered some common questions related to identifying the graph of a rational function. We hope that this article has been helpful in understanding the concepts of rational functions and their graphs.

Key Takeaways

• Rational functions can be classified into different types based on their behavior.
• Asymptotes and holes are important features of rational functions.
• The degree of the numerator and denominator of a rational function determines its horizontal asymptote.
• The denominator of a rational function determines its vertical asymptote.

Further Reading

For more information on rational functions and their graphs, we recommend the following resources:

• Wikipedia: Rational Function
• Math Open Reference: Rational Functions
• Khan Academy: Rational Functions

References

• [1] Thomas, G. B. (2014). Calculus and Analytic Geometry. Addison-Wesley.
• [2] Larson, R. E., & Edwards, B. I. (2013). Calculus: Early Transcendentals. Cengage Learning.
• [3] Anton, H. (2014). Calculus: A New Horizon. John Wiley & Sons.