Follow The Steps For Graphing A Rational Function To Graph The Function:$ R(x) = \frac{x^2 + 2x - 24}{x^2 - 2x - 8} }$1. Domain Of { R(x) $}$ What Is The Domain Of { R(x) $ ? \[ ? \[ ? \[ (- \infty, -2) \cup (-2, 4)

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Understanding Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. It is a fundamental concept in algebra and is used to model various real-world phenomena. In this article, we will focus on graphing rational functions, which involves finding the domain, identifying the x-intercepts, and sketching the graph.

Step 1: Finding 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 other words, it is the set of all x-values that make the function non-zero. To find the domain of a rational function, we need to identify the values of x that make the denominator zero.

Finding the Domain of R(x)

The given rational function is:

R(x)=x2+2x−24x2−2x−8{ R(x) = \frac{x^2 + 2x - 24}{x^2 - 2x - 8} }

To find the domain of R(x), we need to find the values of x that make the denominator zero. We can do this by factoring the denominator:

x2−2x−8=(x−4)(x+2){ x^2 - 2x - 8 = (x - 4)(x + 2) }

Now, we can set each factor equal to zero and solve for x:

x−4=0  ⟹  x=4{ x - 4 = 0 \implies x = 4 }

x+2=0  ⟹  x=−2{ x + 2 = 0 \implies x = -2 }

Therefore, the domain of R(x) is:

(−∞,−2)∪(−2,4){ (- \infty, -2) \cup (-2, 4) }

Step 2: Identifying the X-Intercepts

The x-intercepts of a rational function are the values of x that make the function equal to zero. To find the x-intercepts, we need to set the numerator equal to zero and solve for x.

Finding the X-Intercepts of R(x)

The given rational function is:

R(x)=x2+2x−24x2−2x−8{ R(x) = \frac{x^2 + 2x - 24}{x^2 - 2x - 8} }

To find the x-intercepts, we need to set the numerator equal to zero:

x2+2x−24=0{ x^2 + 2x - 24 = 0 }

We can factor the numerator:

(x+6)(x−4)=0{ (x + 6)(x - 4) = 0 }

Now, we can set each factor equal to zero and solve for x:

x+6=0  ⟹  x=−6{ x + 6 = 0 \implies x = -6 }

x−4=0  ⟹  x=4{ x - 4 = 0 \implies x = 4 }

Therefore, the x-intercepts of R(x) are x = -6 and x = 4.

Step 3: Sketching the Graph

Now that we have found the domain and x-intercepts, we can sketch the graph of R(x). To do this, we need to identify the asymptotes and the behavior of the function as x approaches positive and negative infinity.

Sketching the Graph of R(x)

The given rational function is:

R(x)=x2+2x−24x2−2x−8{ R(x) = \frac{x^2 + 2x - 24}{x^2 - 2x - 8} }

To sketch the graph, we need to identify the asymptotes. The vertical asymptotes are the values of x that make the denominator zero, which are x = -2 and x = 4. The horizontal asymptote is the value that the function approaches as x approaches positive and negative infinity.

As x approaches positive and negative infinity, the function approaches the value of the ratio of the leading coefficients of the numerator and denominator:

lim⁡x→∞x2+2x−24x2−2x−8=11=1{ \lim_{x \to \infty} \frac{x^2 + 2x - 24}{x^2 - 2x - 8} = \frac{1}{1} = 1 }

Therefore, the horizontal asymptote is y = 1.

Graphing the Rational Function

Now that we have identified the asymptotes and the behavior of the function as x approaches positive and negative infinity, we can sketch the graph of R(x). The graph will have vertical asymptotes at x = -2 and x = 4, and a horizontal asymptote at y = 1.

Conclusion

In this article, we have followed the steps for graphing a rational function. We have found the domain, identified the x-intercepts, and sketched the graph. The domain of the rational function is (-∞, -2) ∪ (-2, 4), and the x-intercepts are x = -6 and x = 4. The graph has vertical asymptotes at x = -2 and x = 4, and a horizontal asymptote at y = 1.

Graphing Rational Functions: A Step-by-Step Guide

By following these steps, you can graph any rational function. Remember to find the domain, identify the x-intercepts, and sketch the graph. With practice, you will become proficient in graphing rational functions and be able to model various real-world phenomena.

References

  • [1] "Rational Functions" by Math Open Reference
  • [2] "Graphing Rational Functions" by Khan Academy
  • [3] "Rational Functions" by Wolfram MathWorld

Further Reading

  • "Algebra and Trigonometry" by Michael Sullivan
  • "Calculus" by Michael Spivak
  • "Mathematics for the Nonmathematician" by Morris Kline
    Graphing Rational Functions: A Q&A Guide =============================================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about graphing rational functions.

Q: What is a rational function?

A: A rational function is a function that can be expressed as the ratio of two polynomials. It is a fundamental concept in algebra and is used to model various real-world phenomena.

Q: How do I find the domain of a rational function?

A: To find the domain of a rational function, you need to identify the values of x that make the denominator zero. You can do this by factoring the denominator and setting each factor equal to zero.

Q: What are the x-intercepts of a rational function?

A: The x-intercepts of a rational function are the values of x that make the function equal to zero. To find the x-intercepts, you need to set the numerator equal to zero and solve for x.

Q: How do I sketch the graph of a rational function?

A: To sketch the graph of a rational function, you need to identify the asymptotes and the behavior of the function as x approaches positive and negative infinity. You can do this by finding the vertical and horizontal asymptotes and using them to sketch the graph.

Q: What are the vertical and horizontal asymptotes of a rational function?

A: The vertical asymptotes of a rational function are the values of x that make the denominator zero. The horizontal asymptote is the value that the function approaches as x approaches positive and negative infinity.

Q: How do I find the horizontal asymptote of a rational function?

A: To find the horizontal asymptote of a rational function, you need to find the ratio of the leading coefficients of the numerator and denominator. This will give you the value that the function approaches as x approaches positive and negative infinity.

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

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

  • Not factoring the denominator correctly
  • Not identifying the x-intercepts correctly
  • Not sketching the graph correctly
  • Not using the asymptotes to guide the graph

Q: How can I practice graphing rational functions?

A: You can practice graphing rational functions by working through examples and exercises in a textbook or online resource. You can also try graphing rational functions on a graphing calculator or online graphing tool.

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

A: Graphing rational functions has many real-world applications, including:

  • Modeling population growth and decline
  • Modeling the spread of disease
  • Modeling the behavior of electrical circuits
  • Modeling the behavior of mechanical systems

Conclusion

In this article, we have answered some of the most frequently asked questions about graphing rational functions. We have covered topics such as finding the domain, identifying the x-intercepts, and sketching the graph. We have also discussed common mistakes to avoid and provided some real-world applications of graphing rational functions.

Graphing Rational Functions: A Q&A Guide

By following the steps outlined in this article, you can become proficient in graphing rational functions and be able to model various real-world phenomena.

References

  • [1] "Rational Functions" by Math Open Reference
  • [2] "Graphing Rational Functions" by Khan Academy
  • [3] "Rational Functions" by Wolfram MathWorld

Further Reading

  • "Algebra and Trigonometry" by Michael Sullivan
  • "Calculus" by Michael Spivak
  • "Mathematics for the Nonmathematician" by Morris Kline