When it comes to rational functions, it’s important to understand how the graph of a rational function looks like and how to identify various rational functions based on their graphs. In this article, we will explore the graphing of rational functions and how to determine which rational function is graphed below 10.

## Understanding Rational Functions

Rational functions are functions that can be expressed as the ratio of two polynomials. They are typically written in the form:

*f(x) = p(x)/q(x)*

where *p(x)* and *q(x)* are polynomials and *q(x) ≠ 0*.

The graph of a rational function can have various characteristics such as asymptotes, intercepts, holes, and end behavior. Understanding these characteristics can help in identifying the specific rational function from its graph.

## Identifying Rational Functions from Graphs

Graphs of rational functions often exhibit specific features that can be used to determine the type of rational function being graphed. Some of the key features to look for include:

– Vertical Asymptotes

– Horizontal or Slant Asymptotes

– x-intercepts

– y-intercepts

– Holes in the graph

Let’s take a look at an example of a graph and identify which of the following rational functions is graphed below 10:

1. *f(x) = (x-2)/(x+1)*

2. *g(x) = (2x+1)/(x-3)*

3. *h(x) = (x+2)/(x^2-4)*

## Graph Analysis

Let’s analyze the graph of a rational function and determine which of the given rational functions it represents.

x | y |
---|---|

-5 | -0.8 |

-3 | 1.5 |

-1 | 3 |

0 | 5 |

1 | 6 |

2 | 8 |

Based on the given graph, we can observe the following characteristics:

– There is a vertical asymptote at x = -1

– The graph passes through the point (-2, 1)

– The graph is increasing as x approaches positive infinity

– The graph is decreasing as x approaches negative infinity

From the given characteristics, we can eliminate options 2 and 3 as neither have a vertical asymptote at x = -1. We can now focus on analyzing the remaining option.

## Analyzing Option 1: *f(x) = (x-2)/(x+1)*

Let’s analyze the rational function *f(x) = (x-2)/(x+1)* and determine if it matches the given graph.

Vertical Asymptote:

– The rational function *f(x) = (x-2)/(x+1)* has a vertical asymptote at x = -1, matching the characteristics of the given graph.

Intercept:

– The rational function *f(x) = (x-2)/(x+1)* passes through the point (-2, 1), matching the characteristics of the given graph.

End Behavior:

– As x approaches positive infinity, the rational function is increasing, matching the characteristics of the given graph.

– As x approaches negative infinity, the rational function is decreasing, matching the characteristics of the given graph.

Based on the analysis, we can conclude that the given graph matches the rational function ** f(x) = (x-2)/(x+1)**.

## Conclusion

In conclusion, understanding the characteristics of rational functions and how they manifest in their graphs is essential for identifying the specific rational function from its graph. By analyzing the given graph, we were able to determine that the rational function *f(x) = (x-2)/(x+1)* is graphed below 10.

It’s important to practice graphing and identifying rational functions to build a strong understanding of their properties and how they are represented in their graphs. This knowledge can be valuable in various mathematical and real-world applications.

As mathematics continues to evolve, it’s crucial to stay updated with the latest knowledge and techniques for analyzing and identifying mathematical functions to expand one’s problem-solving skills and analytical thinking.

By mastering the art of interpreting graphs of rational functions, individuals can enhance their mathematical proficiency and approach problem-solving with confidence and precision.