Graphing Rational Functions: Definition, Examples

Do you know how graphing simple rational functions can be done with ease? You all must be acquainted with fractions. The top portion of a fraction is known as the numerator, and the bottom is known as the denominator. The numerator of a rational function reveals the x-intercept of the graph. On the other hand, the denominator reveals the vertical asymptotes of the graph. The factors of the numerator have integer powers greater than one in polynomials. 

Graphing Rational Functions – 1/x 

Can you guess which of the following rational functions is graphed below? The vertical asymptotes linked to the denominator’s factors will mirror one of the two reciprocal functions. You must know that when the degree of the factor in the denominator is odd, the graph heads positive infinity upwards on one side of the vertical asymptote. In contrast, the other side is towards negative infinity. 

Graphing Rational Functions – 1/x²

Here’s another example. Can you guess one more time which of the following rational functions is graphed below? You must know when the degree of the denominator is even; the characteristic graph either heads toward the positive infinity on both sides of the vertical asymptote or towards the negative infinity on both sides. 

Let us understand the above graphing rational function by an example. Suppose, the fraction is f (x) = (x + 1)² (x – 3)/(x + 3)² (x – 2), then, the graph of the function will be:

This graph is drawn by putting 1, 2,…, and so on to get the values for the function to plot a graph. Here a key points that can be drawn from the graph above: 

You must have noticed that at the x-intercept x = -1 corresponding to the (x + 1)² factor present in the numerator, the graph consistently bounces with the quadratic nature of the factor. 

Another thing to notice is that at the x-intercept x = 3 corresponding to the (x-3) factor of the numerator, the graph passes through the axis as in the case of a linear factor. 

Also, at the vertical asymptote, x = -3 corresponding to the (x + 3)² factor of the denominator, the graph is consistent with the behavior of the function f (x) = 1/x². Due to this, the graph heads up on both sides of the asymptote. 

Lastly, at the vertical asymptote x = 2, corresponding to the (x – 2) factor in the denominator, consistent behavior of the function f (x) = 1/x is followed. The graph heads towards positive infinity on the left side of the asymptote and towards negative infinity on the right side. 

How to Graph Rational Functions? – Key Tips to Follow 

If you have complexities making graphs of rational functions, then it’s time to follow these key tips listed below:

1. You must evaluate the function in the first place. Evaluate it at 0 to find the y-intercept. 
2. Next, you need to factor in the numerator and denominator.
3. After factoring, for factors in the numerator not common to the denominator, you need to determine where each factor is zero to find the x-intercepts. 
4. After performing step 3, find the multiplicities of the x-intercepts. You are doing this step to determine the behavior of the graph at those points. 
5. You need to note the multiplicities of the zeroes to determine the local behavior for the factors in the denominator. It would help to find the vertical asymptotes for the uncommon factors to the numerator. To do this, you need to set those factors equal to zero and then solve them. 
6. After getting zeroes in step 5, you need to find the removable discontinuities by setting those factors equal to 0. This must be done for the factors in the denominator common to the factors in the numerator.
7. Now, compare the numerator and denominator degrees to determine the slant or horizontal asymptotes. 
8. Finally, you can sketch the graph you wanted to. 

Let us learn graphing simple rational functions via an example. 

Example: Sketch a graph for the function, f (x) = (x + 2) (x – 3)/(x + 1)² (x -2) . 

Solution: You can follow the steps to sketch the graph for the following function:

Step 1: The first step to sketch the graph is to factor the function. Since the function is already factored, it saves your first step. 

Step 2: You need to find the intercepts. Evaluate the function at zero to get the y-intercept as shown,

f (0) = (0 + 2) (0 – 3)/(0 + 1)² (0 – 2)= 3.

Step 3: After getting the y-intercept, to find the x-intercept, determine the value when the numerator of the function is zero. Set each factor equal to zero, then find the x-intercepts at x = -2 and x = 3. You will see the behavior of the function is linear (multiplicity of 1) along with the graph through the intercept. 

Step 4: Now, you have x and y-intercepts as (-2, 0) and (3, 0), and (0, 3), respectively. 

Step 5: Next, to get the vertical asymptotes, you need to keep the value of the denominator equal to zero. That will occur when x + 1 = 0 and x – 2 = 0, giving the value for vertical asymptotes at x = -1 and x = 2. Since there are no common factors in the numerator and denominator, there will be no removable discontinuities. 

Step 6: At last, the degree of the denominator is larger than the degree of the numerator, which tells that the graph has a horizontal asymptote at y = 0. 

Step 7: To sketch the graph, you need to plot the three intercepts. Since you know that the y-intercept is positive and there is no x-intercept between the vertical asymptotes, you get to know that the function must remain positive between the asymptotes. It will let you fill the middle portion of the graph. 

Step 8: The factor associated with the vertical asymptote at x = -1 is squared, so you know the behavior of such a function will be the same on both sides of the asymptote. The graph directs towards positive infinity as the inputs approach the asymptote on the right. This way, the graph will also head towards the positive infinity on the left. 

Step 9: The factor is not squared in the vertical asymptote at x = 2. Hence, the graph will have opposite behavior on either side of the asymptote. After the graph passes the x-intercepts, you will see that the graph will then level off toward an output of zero. This was, however, indicated by the horizontal asymptote. 

Try it yourself! 

Example 2: Can you Sketch the Graph of the Function: f (x) = 4x + 12x + 1?

Answer: 

Example 3: Graph the Function: 2x + 5x – 1.

Answer: 

After Graphing Rational Functions, How to Write Them?

If you are given a graph of a rational function and need to write the function, you must check the intercepts in the first place. If a rational function has x-intercepts at x = x1, x2, x3…, xn, vertical asymptotes at x = v1, v2, …vm, and no xi = any vj, then the function can be written in the form: 

f (x) = a(x – x1)p1 (x – x2)p2… (x – xn)pn(x – v1)q1(x – v2)q2… (x – vm)qn

Where the behavior of the graph can find the powers of pi or qi on each factor at the corresponding intercept or asymptote, you can also find the stretch factor ‘a’ from the given value of the function other than the x-intercept or by the horizontal asymptote if it is nonzero. 

After Graphing Simple Rational Functions, Write the Function

The steps to write the function after graphing simple rational functions are illustrated below:

Step 1: You need to determine the factors of the numerator. Examine the behavior of the graph at the x-intercepts to find the zeros and their multiplicities. This step is easy to find the simplest function with small multiplicities, such as 1 or 3. It gets complicated when larger numbers, such as 8 or 9, come into the picture. 

Step 2: Next, you need to determine the factors of the denominator. Examine the behavior of both sides of each vertical asymptote to determine the factors and their powers. 

Step 3: After performing steps 1 and 2, the last step is to use any clear point on the graph to find the stretch factor. 

It is crucial to learn graphing rational functions as they can be useful tools for representing real-life situations where you need to find the solutions to complex problems. The equations representing inverse, direct or joint variation are rational formulas that can model many real-life situations. Hence, it would help if you practiced graphing rational functions.

Frequently Asked Questions

1. What is the Definition of a Rational Function?

Ans. A rational function is a function of the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomials. Rational functions are important because they can be expressed as the quotient of two polynomials, which makes them easy to factor.

2. How Do You Know If a Function is Rational?

Ans. If you want to know if a function is rational, you can either look at its graph or at its equation. If the graph of a function contains only vertical and horizontal lines, then it is rational. If the equation contains only integers, then it is rational.

3. How to Graph a Rational Function?

Ans. You can graph rational functions by using a vertical line test. The vertical line test is a way to determine whether a rational function is defined at a given point. It’s based on the fact that if you graph two different points on the function, then draw a vertical line connecting them, you’ll get a straight line if the function is defined at those points.

4. How to Find Asymptotes of Rational Functions?

Ans. Asymptotes of rational functions are the roots of the denominator, and they are always an integer.To find the asymptotes of a rational function, first, isolate the denominator by taking it outside of brackets. Then test all the roots of that denominator for being an integer. If you find an integer root, then you have found your asymptote.

5. What are the Applications of Rational Function?

Ans. Rational functions are the most common type of polynomial functions, and they can be applied in many different ways. One application of rational functions is in solving systems of linear equations. Rational functions can also be used to represent the relationship between two variables on a graph, known as a phase line.