Graphing Rational Functions using Transformations

Oct 13, 2025 by miodeya.com 90 views

Introduction

Graphing rational functions can be a complex task, but by using transformations of the graph of $y=\frac{1}{x}$ , we can simplify the process. In this article, we will explore how to graph three different rational functions using transformations and determine their domain and range.

Graphing Rational Functions using Transformations

The graph of $y=\frac{1}{x}$ is a simple rational function with a single asymptote at $x=0$ . To graph more complex rational functions, we can use transformations of this graph. There are four types of transformations: vertical stretch or compression, horizontal stretch or compression, reflection across the x-axis, and reflection across the y-axis.

Vertical Stretch or Compression

A vertical stretch or compression of the graph of $y=\frac{1}{x}$ can be achieved by multiplying the function by a constant. For example, if we have the function $r(x)=\frac{1}{x-1}$ , we can graph it by stretching the graph of $y=\frac{1}{x}$ vertically by a factor of 1.

Horizontal Stretch or Compression

A horizontal stretch or compression of the graph of $y=\frac{1}{x}$ can be achieved by replacing $x$ with $x-a$ or $x+a$ . For example, if we have the function $r(x)=\frac{-2}{x-2}$ , we can graph it by compressing the graph of $y=\frac{1}{x}$ horizontally by a factor of 2.

Reflection across the x-axis

A reflection across the x-axis of the graph of $y=\frac{1}{x}$ can be achieved by multiplying the function by -1. For example, if we have the function $r(x)=\frac{2x-9}{x-4}$ , we can graph it by reflecting the graph of $y=\frac{1}{x}$ across the x-axis.

Reflection across the y-axis

A reflection across the y-axis of the graph of $y=\frac{1}{x}$ can be achieved by replacing $x$ with $-x$ . However, this type of transformation is not applicable to the given rational functions.

Graphing the Rational Functions

Now that we have discussed the transformations, let's graph the three rational functions.

Graphing $r(x)=\frac{1}{x-1}$

To graph $r(x)=\frac{1}{x-1}$ , we can start by graphing the graph of $y=\frac{1}{x}$ and then apply a horizontal shift of 1 unit to the right. This will give us the graph of $r(x)=\frac{1}{x-1}$ .

Graphing $r(x)=\frac{-2}{x-2}$

To graph $r(x)=\frac{-2}{x-2}$ , we can start by graphing the graph of $y=\frac{1}{x}$ and then apply a horizontal compression of 2 units. This will give us the graph of $r(x)=\frac{-2}{x-2}$ .

Graphing $r(x)=\frac{2x-9}{x-4}$

To graph $r(x)=\frac{2x-9}{x-4}$ , we can start by graphing the graph of $y=\frac{1}{x}$ and then apply a horizontal shift of 4 units to the right and a vertical stretch of 2 units. This will give us the graph of $r(x)=\frac{2x-9}{x-4}$ .

Determining Domain and Range

Now that we have graphed the rational functions, let's determine their domain and range.

Domain and Range of $r(x)=\frac{1}{x-1}$

The domain of $r(x)=\frac{1}{x-1}$ is all real numbers except $x=1$ . The range is all real numbers except $y=0$ .

Domain and Range of $r(x)=\frac{-2}{x-2}$

The domain of $r(x)=\frac{-2}{x-2}$ is all real numbers except $x=2$ . The range is all real numbers except $y=0$ .

Domain and Range of $r(x)=\frac{2x-9}{x-4}$

The domain of $r(x)=\frac{2x-9}{x-4}$ is all real numbers except $x=4$ . The range is all real numbers except $y=0$ .

Conclusion

In this article, we have discussed how to graph rational functions using transformations of the graph of $y=\frac{1}{x}$ . We have also determined the domain and range of three different rational functions. By applying the transformations and understanding the properties of rational functions, we can graph and analyze complex rational functions.

References

[1] "Graphing Rational Functions" by Math Open Reference
[2] "Rational Functions" by Khan Academy
[3] "Transformations of Functions" by Math Is Fun
Q&A: Transformations of the Graph of $y=\frac{1}{x}$ : Graphing Rational Functions and Determining Domain and Range ====================================================================================

Introduction

In our previous article, we discussed how to graph rational functions using transformations of the graph of $y=\frac{1}{x}$ . We also determined the domain and range of three different rational functions. In this article, we will answer some frequently asked questions about transformations of the graph of $y=\frac{1}{x}$ and graphing rational functions.

Q&A

Q: What is the difference between a vertical stretch and a vertical compression?

A: A vertical stretch of the graph of $y=\frac{1}{x}$ is achieved by multiplying the function by a constant greater than 1. This will stretch the graph vertically. A vertical compression of the graph of $y=\frac{1}{x}$ is achieved by multiplying the function by a constant less than 1. This will compress the graph vertically.

Q: How do I determine the domain and range of a rational function?

A: To determine the domain and range of a rational function, you need to identify the values of $x$ that make the denominator equal to zero. These values are not in the domain of the function. The range is all real numbers except the value of $y$ that would make the denominator equal to zero.

Q: Can I use transformations to graph a rational function with a quadratic numerator?

A: Yes, you can use transformations to graph a rational function with a quadratic numerator. However, you need to factor the numerator and then apply the transformations.

Q: How do I graph a rational function with a horizontal asymptote?

A: To graph a rational function with a horizontal asymptote, you need to identify the horizontal asymptote and then graph the function using transformations.

Q: Can I use transformations to graph a rational function with a vertical asymptote?

A: Yes, you can use transformations to graph a rational function with a vertical asymptote. However, you need to identify the vertical asymptote and then apply the transformations.

Q: How do I determine the domain and range of a rational function with a vertical asymptote?

A: To determine the domain and range of a rational function with a vertical asymptote, you need to identify the values of $x$ that make the denominator equal to zero. These values are not in the domain of the function. The range is all real numbers except the value of $y$ that would make the denominator equal to zero.

Q: Can I use transformations to graph a rational function with a hole?

A: Yes, you can use transformations to graph a rational function with a hole. However, you need to identify the hole and then apply the transformations.

Q: How do I determine the domain and range of a rational function with a hole?

A: To determine the domain and range of a rational function with a hole, you need to identify the values of $x$ that make the denominator equal to zero. These values are not in the domain of the function. The range is all real numbers except the value of $y$ that would make the denominator equal to zero.

Conclusion

In this article, we have answered some frequently asked questions about transformations of the graph of $y=\frac{1}{x}$ and graphing rational functions. We hope that this article has been helpful in understanding the concepts of transformations and graphing rational functions.

References

[1] "Graphing Rational Functions" by Math Open Reference
[2] "Rational Functions" by Khan Academy
[3] "Transformations of Functions" by Math Is Fun