Rational Parent Function

Introduction Of Rational Parent Function

As you start working with graphs and functions, you realize that graphs and the related functions are most common in their looks and follow similar patterns. These functions share the same degree of angles and have the same rational parent functions.

A rational parent function is nothing but a family of functions represented in its simplest form. We use parent functions to get some direction in graphing tasks that belong to the same family of functions.

Some of the most common parent functions are linear, constant, cubic, quadratic, absolute value, exponential, radical, reciprocal, and logarithmic functions. To find rational parent functions, you have to follow the following functions.

• The highest degree of the functions.
• Whether it has cube roots and square roots or not
• Is the function found in a denominator or exponent?
• The function's graph can increase and decrease depending on which character of the rational parent function will be determined.

Rational Function Equation 

Parent functions are the members of different functions within the same family represented in the simplest manner. When two or more functions share similar shapes for their graphs and the same highest degree, it is called a rational parent function.

Here are four graphs which are U shaped and popularly called parabolas. All these graphs have similar highest degrees, and their shapes are similar. Therefore, they can be grouped into one family of functions. But it is for you to guess which family of functions would fall.

These four functions are specimens of quadratic functions. In their simplest form, you have to write x=y^2. So, the rational parent function for this entire family would be "y=x² “.

These functions are represented as the simplest form of parts and give you an immediate idea of what a single function from the given family would look like.

Rational Parent Function X and Y Intercepts

In a graph, there are two main axes. X and Y. These two axes represent the basis of two-dimensional space. Now assume the X and Y axes are drawn in a graph. When you draw small lines crossing the X axis, it intersects the axis. These lines are vertical.

Similarly, when you draw vertical lines crossing the Y axis, it is called Y-axis intercepts. As you go further towards the eastern side of the graph, it will be denoted as X=Y+1. Similarly, as it moves towards the western or left side of the graph, it is represented as X=Y-1.

Rational Parent Function End Behavior

We have diversified the end behaviour of rational parent functions in the following steps.

Step 1-

You have to look at the degrees of denominator and numerator to adjudge the end function of the rational parent function. When the denominator's degree exceeds the numerator's degree, a "horizontal asymptote" of "Y=0" is called a " horizontal asymptote". This is called the end behaviour of the function.

Step 2-

When both the degrees of denominator and numerator are equal, it is called a horizontal asymptote. This horizontal asymptote is attributed to Y=a/b. Here the leading coefficient of the numerator is "a", and the leading coefficient of the denominator is "b". This can be termed the end behaviour of the coefficient.

Step 3-

The graph forms a " slant or oblique asymptote " when the numerator's degree is greater than the denominator's degree; then, the graph forms a "slant or oblique asymptote". This happens when the numerator's degree exceeds by 1 degree from the denominator. This function is also asymptomatic of a polynomial. To find out the quotient, you have to make long divisions to the quotient. If it goes on like this, then "y=q(x)". Here, the end behaviour is represented through the q(x) quotient.

Graphing Rational Functions

You better understand the rational graphing function of an end behaviour through the following steps.

A. Asymptotes of Rational Functions

Asymptotes represent rational functions. Asymptotes are nothing but values of x and y represented through various dotted lines. Some of the most used asymptotes are vertical asymptotes, slant asymptotes and horizontal asymptotes. The following section discusses the variety of asymptotes.

B. Horizontal Asymptotes

Horizontal asymptotes are also called slant asymptotes. It describes how the graph will head as X's value keeps increasing. You can cross or touch a slant or horizontal asymptote, which differs from a vertical asymptote.

You can bring in different comparisons of the degree of denominators and numerators to find out the slant or horizontal asymptotes.

C. Vertical Asymptotes

When you enter "holes" in a graph, it is called vertical asymptotes. Here the function does not have a value. Instead, they represent the places where the value of X is not allowed. Hence, you cannot provide a "zero" value at the denominator of the rational function. Here, any value of X that would turn the denominator to zero is called a vertical asymptote.

To represent a vertical asymptote, you put dotted lines to denote that those places cannot include the functions. Vertical asymptotes inform the domain of the particular graph.

D. Oblique Asymptotes

Oblique asymptotes are when the denominator’s degree of a rational parent function is one less than the numerator. It is also called the slant asymptote. It is represented through a line denoted by "y=mx+b", where the value of "m" can never be zero.

Here is a function represented by "f(x)=x+1/x. Now, y=x is an oblique asymptote in the same function. Here the value of the line or X will always remain 0.

So, here are the main features of the rational parent function. Of course, you have to be more diligent to make it work accordingly. Through this article, we learned the fundamental structures of rational parent functions, their role in geometry and the ways these functions are represented.

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