Follow The Steps For Graphing A Rational Function To Graph The Function $R(x)=\frac{x 2+2x-24}{x 2-2x-8}$1. If Needed, First Write The Given Function As A Single Rational Expression. Then, Factor The Numerator And Denominator Of

Mar 12, 2025 by ADMIN 229 views

**Graphing Rational Functions: A Step-by-Step Guide**

Understanding Rational Functions

Rational functions are a type of function that can be expressed as the ratio of two polynomials. They are an essential concept in algebra and are used to model various real-world phenomena. In this article, we will focus on graphing rational functions, specifically the function $R(x)=\frac{x^2+2x-24}{x^2-2x-8}$ .

Step 1: Factor the Numerator and Denominator

To graph the rational function, we first need to factor the numerator and denominator. Factoring the numerator and denominator will help us identify any common factors and simplify the expression.

Factoring the Numerator

The numerator of the function is $x^2+2x-24$ . We can factor this expression by finding two numbers whose product is $-24$ and whose sum is $2$ . These numbers are $6$ and $-4$ , so we can write the numerator as:

$x^2+2x-24 = (x+6)(x-4)$

Factoring the Denominator

The denominator of the function is $x^2-2x-8$ . We can factor this expression by finding two numbers whose product is $-8$ and whose sum is $-2$ . These numbers are $-4$ and $2$ , so we can write the denominator as:

$x^2-2x-8 = (x-4)(x+2)$

Step 2: Identify Common Factors

Now that we have factored the numerator and denominator, we can identify any common factors. In this case, we can see that both the numerator and denominator have a common factor of $(x-4)$ .

Canceling Common Factors

Since we have a common factor of $(x-4)$ in both the numerator and denominator, we can cancel it out. This will simplify the expression and make it easier to graph.

$R(x)=\frac{(x+6)(x-4)}{(x-4)(x+2)}$

Canceling the common factor $(x-4)$ , we get:

$R(x)=\frac{x+6}{x+2}$

Step 3: Find the Domain

The domain of a rational function is the set of all possible input values (x-values) for which the function is defined. In the case of a rational function, the domain is all real numbers except for the values that make the denominator equal to zero.

Finding the Domain

To find the domain of the function, we need to find the values of x that make the denominator $(x+2)$ equal to zero.

$x+2=0$

Solving for x, we get:

$x=-2$

So, the domain of the function is all real numbers except for $x=-2$ .

Step 4: Find the Vertical Asymptote

A vertical asymptote is a vertical line that the graph of a function approaches but never touches. In the case of a rational function, the vertical asymptote occurs at the value of x that makes the denominator equal to zero.

Finding the Vertical Asymptote

We have already found the value of x that makes the denominator equal to zero, which is $x=-2$ . Therefore, the vertical asymptote is at $x=-2$ .

Step 5: Find the Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches but never touches. In the case of a rational function, the horizontal asymptote occurs when the degree of the numerator is equal to the degree of the denominator.

Finding the Horizontal Asymptote

In this case, the degree of the numerator is 2, and the degree of the denominator is also 2. Therefore, the horizontal asymptote is at $y=1$ .

Step 6: Graph the Function

Now that we have found the domain, vertical asymptote, and horizontal asymptote, we can graph the function.

Graphing the Function

To graph the function, we can start by plotting the vertical asymptote at $x=-2$ . Then, we can plot the horizontal asymptote at $y=1$ . Finally, we can plot the points on the graph that satisfy the equation $y=\frac{x+6}{x+2}$ .

Conclusion

Frequently Asked Questions

Graphing rational functions can be a challenging task, but with the right guidance, it can be made easier. In this article, we will answer some of the most frequently asked questions about graphing rational functions.

Q: What is a rational function?

A rational function is a type of function that can be expressed as the ratio of two polynomials. It is a function that has a numerator and a denominator, and the denominator cannot be zero.

A: A rational function is a function that can be expressed as the ratio of two polynomials. It is a function that has a numerator and a denominator, and the denominator cannot be zero.

Q: How do I graph a rational function?

To graph a rational function, you need to follow these steps:

Factor the numerator and denominator.
Identify common factors and cancel them out.
Find the domain of the function.
Find the vertical asymptote.
Find the horizontal asymptote.
Graph the function.

A: To graph a rational function, you need to follow these steps:

Factor the numerator and denominator.
Identify common factors and cancel them out.
Find the domain of the function.
Find the vertical asymptote.
Find the horizontal asymptote.
Graph the function.

Q: What is the domain of a rational function?

A: The domain of a rational function is the set of all possible input values (x-values) for which the function is defined. In the case of a rational function, the domain is all real numbers except for the values that make the denominator equal to zero.

Q: What is a vertical asymptote?

A: A vertical asymptote is a vertical line that the graph of a function approaches but never touches. In the case of a rational function, the vertical asymptote occurs at the value of x that makes the denominator equal to zero.

Q: What is a horizontal asymptote?

A: A horizontal asymptote is a horizontal line that the graph of a function approaches but never touches. In the case of a rational function, the horizontal asymptote occurs when the degree of the numerator is equal to the degree of the denominator.

Q: How do I find the vertical and horizontal asymptotes of a rational function?

To find the vertical and horizontal asymptotes of a rational function, you need to follow these steps:

Factor the numerator and denominator.
Identify common factors and cancel them out.
Find the value of x that makes the denominator equal to zero (vertical asymptote).
Find the degree of the numerator and denominator (horizontal asymptote).

A: To find the vertical and horizontal asymptotes of a rational function, you need to follow these steps:

Factor the numerator and denominator.
Identify common factors and cancel them out.
Find the value of x that makes the denominator equal to zero (vertical asymptote).
Find the degree of the numerator and denominator (horizontal asymptote).

Q: What are some common mistakes to avoid when graphing rational functions?

Some common mistakes to avoid when graphing rational functions include:

Not factoring the numerator and denominator.
Not identifying common factors and canceling them out.
Not finding the domain of the function.
Not finding the vertical and horizontal asymptotes.
Not graphing the function correctly.

A: Some common mistakes to avoid when graphing rational functions include:

Not factoring the numerator and denominator.
Not identifying common factors and canceling them out.
Not finding the domain of the function.
Not finding the vertical and horizontal asymptotes.
Not graphing the function correctly.

Conclusion

Graphing rational functions can be a challenging task, but with the right guidance, it can be made easier. By following the steps outlined in this article, you can create a clear and accurate graph of a rational function. Remember to factor the numerator and denominator, identify common factors and cancel them out, find the domain of the function, find the vertical and horizontal asymptotes, and graph the function correctly.