# How to Compare Graphs of Rational Functions: A Case Study of mc020-1.jpg and mc020-2.jpg

Rational functions are functions that can be written as the ratio of two polynomial functions. They often have interesting features such as asymptotes, holes, and discontinuities. In this article, we will learn how to compare the graph of a rational function to its parent function, using the example of mc020-1.jpg and mc020-2.jpg.

## What is the Parent Function of a Rational Function?

The parent function of a rational function is the simplest rational function that has the same general shape as the given function. For example, the parent function of mc020-1.jpg is mc020-2.jpg, which is also written as f(x) = 1/x.

The graph of the parent function f(x) = 1/x is shown below:

![Graph of f(x) = 1/x](https://www.mathsisfun.com/algebra/images/graph-1-over-x.svg)

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The graph has two important features:

• A vertical asymptote at x = 0, which means that the function approaches infinity or negative infinity as x approaches 0 from either side.
• A horizontal asymptote at y = 0, which means that the function approaches 0 as x approaches infinity or negative infinity.

## How to Compare the Graph of a Rational Function to its Parent Function?

To compare the graph of a rational function to its parent function, we need to look at how the function is transformed from the parent function. There are four types of transformations that can affect the graph of a rational function:

• Vertical shift: This means adding or subtracting a constant to the function, which moves the graph up or down by that amount. For example, f(x) = 1/x + 2 is a vertical shift of f(x) = 1/x by 2 units up.
• Horizontal shift: This means adding or subtracting a constant to the input variable x, which moves the graph left or right by that amount. For example, f(x) = 1/(x – 3) is a horizontal shift of f(x) = 1/x by 3 units right.
• Vertical stretch or compression: This means multiplying or dividing the function by a constant greater than 1 or less than 1, respectively, which changes the steepness of the graph. For example, f(x) = 2/x is a vertical stretch of f(x) = 1/x by a factor of 2.
• Horizontal stretch or compression: This means multiplying or dividing the input variable x by a constant greater than 1 or less than 1, respectively, which changes the width of the graph. For example, f(x) = 1/(2x) is a horizontal compression of f(x) = 1/x by a factor of 2.
• Reflection: This means changing the sign of the function or the input variable x, which flips the graph over the x-axis or y-axis, respectively. For example, f(x) = -1/x is a reflection of f(x) = 1/x over the x-axis.

These transformations can affect the location and direction of the asymptotes, as well as the shape and position of the graph.

## A Case Study of mc020-1.jpg and mc020-2.jpg

Now that we know how to compare rational functions to their parent functions, let’s apply it to our example of mc020-1.jpg and mc020-2.jpg.

First, we need to identify the parent function of mc020-1.jpg. We can do this by simplifying the expression as much as possible:

mc020-3.jpg

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We can see that the parent function is mc020-4.jpg, which is equivalent to f(x) = -1/x.

Next, we need to identify how mc020-1.jpg is transformed from its parent function. We can do this by comparing the expressions:

mc020-5.jpg

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We can see that there are three transformations:

• A horizontal shift by -4 units left
• A vertical shift by -3 units down
• A reflection over the y-axis

These transformations will affect the graph of mc020-1.jpg in the following ways:

• The vertical asymptote will shift from x = 0 to x = -4
• The horizontal asymptote will shift from y = 0 to y = -3
• The graph will be flipped over the y-axis

The graph of mc020-1.jpg is shown below:

![Graph of mc020-1.jpg](https://www.desmos.com/calculator/8xqjwzqy0b?embed)

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We can compare it to the graph of the parent function mc020-2.jpg, which is shown below:

![Graph of mc020-2.jpg](https://www.desmos.com/calculator/8xqjwzqy0b?embed)

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We can see that the graphs have the same general shape, but they are shifted, reflected, and stretched in different ways.

## Conclusion

In this article, we learned how to compare the graph of a rational function to its parent function, using the example of mc020-1.jpg and mc020-2.jpg. We learned that rational functions can be transformed by vertical and horizontal shifts, vertical and horizontal stretches or compressions, and reflections. These transformations can affect the location and direction of the asymptotes, as well as the shape and position of the graph. We also learned how to identify the parent function of a rational function by simplifying the expression as much as possible. 