Plot All Of The Existing Features Of The Following Rational Function. If You Get A Fraction Or Decimal, Plot As Close To The True Location As Possible. F ( X ) = 3 X − 12 X 2 − 4 X F(x) = \frac{3x - 12}{x^2 - 4x} F ( X ) = X 2 − 4 X 3 X − 12 ​ Features To Identify:- Vertical Asymptote- Horizontal

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Introduction

Rational functions are a type of function that can be expressed as the ratio of two polynomials. They are commonly used in mathematics to model real-world phenomena, such as the growth of populations, the decay of radioactive materials, and the behavior of electrical circuits. In this article, we will explore the features of a rational function, specifically the vertical and horizontal asymptotes, and learn how to plot them.

The Rational Function

The rational function we will be working with is:

f(x)=3x−12x2−4xf(x) = \frac{3x - 12}{x^2 - 4x}

This function has two polynomials in the numerator and denominator, which makes it a rational function.

Vertical Asymptotes

Vertical asymptotes are values of x that make the denominator of the rational function equal to zero. In other words, they are the values of x that make the function undefined. To find the vertical asymptotes, we need to set the denominator equal to zero and solve for x.

x2−4x=0x^2 - 4x = 0

We can factor the quadratic expression as:

x(x−4)=0x(x - 4) = 0

This gives us two possible values for x:

x=0orx=4x = 0 \quad \text{or} \quad x = 4

These values of x make the denominator equal to zero, so they are the vertical asymptotes of the function.

Plotting the Vertical Asymptotes

To plot the vertical asymptotes, we need to draw vertical lines at the values of x that we found. In this case, we need to draw vertical lines at x = 0 and x = 4.

Horizontal Asymptotes

Horizontal asymptotes are values of y that the function approaches as x approaches infinity or negative infinity. To find the horizontal asymptotes, we need to look at the degrees of the polynomials in the numerator and denominator.

The degree of the numerator is 1, and the degree of the denominator is 2. Since the degree of the denominator is higher than the degree of the numerator, the horizontal asymptote is y = 0.

Plotting the Horizontal Asymptote

To plot the horizontal asymptote, we need to draw a horizontal line at y = 0.

Plotting the Features

Now that we have found the vertical and horizontal asymptotes, we can plot them on a graph. We will use a coordinate plane with x on the horizontal axis and y on the vertical axis.

To plot the vertical asymptotes, we will draw vertical lines at x = 0 and x = 4. To plot the horizontal asymptote, we will draw a horizontal line at y = 0.

Plotting the Function

To plot the function, we need to find the values of y that correspond to different values of x. We can do this by plugging in different values of x into the function and solving for y.

For example, if we plug in x = 1, we get:

f(1)=3(1)−12(1)2−4(1)=−9−3=3f(1) = \frac{3(1) - 12}{(1)^2 - 4(1)} = \frac{-9}{-3} = 3

So, the point (1, 3) is on the graph of the function.

We can repeat this process for different values of x to get more points on the graph.

Plotting the Features of the Function

Now that we have plotted the function, we can see the vertical and horizontal asymptotes. The vertical asymptotes are the vertical lines at x = 0 and x = 4, and the horizontal asymptote is the horizontal line at y = 0.

Conclusion

In this article, we learned how to plot the features of a rational function, specifically the vertical and horizontal asymptotes. We found the vertical asymptotes by setting the denominator equal to zero and solving for x, and we found the horizontal asymptote by looking at the degrees of the polynomials in the numerator and denominator. We then plotted the vertical and horizontal asymptotes on a graph and plotted the function itself.

References

  • [1] "Rational Functions" by Math Open Reference
  • [2] "Asymptotes" by Math Is Fun
  • [3] "Graphing Rational Functions" by Purplemath

Glossary

  • Rational Function: A type of function that can be expressed as the ratio of two polynomials.
  • Vertical Asymptote: A value of x that makes the denominator of the rational function equal to zero.
  • Horizontal Asymptote: A value of y that the function approaches as x approaches infinity or negative infinity.
  • Degree of a Polynomial: The highest power of x in a polynomial.

Further Reading

  • "Rational Functions" by Khan Academy
  • "Asymptotes" by Wolfram MathWorld
  • "Graphing Rational Functions" by Mathway
    Q&A: Plotting the Features of a Rational Function =====================================================

Introduction

In our previous article, we explored the features of a rational function, specifically the vertical and horizontal asymptotes. We learned how to find and plot these features on a graph. In this article, we will answer some common questions about plotting the features of a rational function.

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 commonly used in mathematics to model real-world phenomena, such as the growth of populations, the decay of radioactive materials, and the behavior of electrical circuits.

Q: What are vertical asymptotes?

Vertical asymptotes are values of x that make the denominator of the rational function equal to zero. In other words, they are the values of x that make the function undefined.

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

To find the vertical asymptotes, you need to set the denominator equal to zero and solve for x. This will give you the values of x that make the function undefined.

Q: What are horizontal asymptotes?

Horizontal asymptotes are values of y that the function approaches as x approaches infinity or negative infinity. They are an important feature of rational functions, as they can help us understand the behavior of the function as x gets very large or very small.

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

To find the horizontal asymptote, you need to look at the degrees of the polynomials in the numerator and denominator. 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: How do I plot the features of a rational function?

To plot the features of a rational function, you need to draw vertical lines at the values of x that make the denominator equal to zero (the vertical asymptotes), and a horizontal line at the value of y that the function approaches as x approaches infinity or negative infinity (the horizontal asymptote).

Q: What is the difference between a vertical asymptote and a hole in a graph?

A vertical asymptote is a value of x that makes the denominator of the rational function equal to zero, while a hole in a graph is a value of x that makes the numerator and denominator equal to zero. In other words, a vertical asymptote is a value of x that makes the function undefined, while a hole in a graph is a value of x that makes the function equal to zero.

Q: How do I determine if a graph has a hole or a vertical asymptote?

To determine if a graph has a hole or a vertical asymptote, you need to look at the factors of the numerator and denominator. If the factor is common to both the numerator and denominator, it is a hole. If the factor is only in the denominator, it is a vertical asymptote.

Q: What is the significance of the degree of a polynomial in a rational function?

The degree of a polynomial in a rational function determines the behavior of the function as x approaches infinity or negative infinity. If the degree of the numerator is less than the degree of the denominator, the function approaches zero as x approaches infinity or negative infinity. If the degree of the numerator is equal to the degree of the denominator, the function approaches the ratio of the leading coefficients of the numerator and denominator as x approaches infinity or negative infinity.

Conclusion

In this article, we answered some common questions about plotting the features of a rational function. We learned how to find and plot the vertical and horizontal asymptotes, and how to determine if a graph has a hole or a vertical asymptote. We also discussed the significance of the degree of a polynomial in a rational function.

References

  • [1] "Rational Functions" by Math Open Reference
  • [2] "Asymptotes" by Math Is Fun
  • [3] "Graphing Rational Functions" by Purplemath

Glossary

  • Rational Function: A type of function that can be expressed as the ratio of two polynomials.
  • Vertical Asymptote: A value of x that makes the denominator of the rational function equal to zero.
  • Horizontal Asymptote: A value of y that the function approaches as x approaches infinity or negative infinity.
  • Degree of a Polynomial: The highest power of x in a polynomial.

Further Reading

  • "Rational Functions" by Khan Academy
  • "Asymptotes" by Wolfram MathWorld
  • "Graphing Rational Functions" by Mathway