Introduction

In the article, Graphing rational function I, we discussed what are rational functions and how to graph them using intercepts and asymptotes. In this article, we will solve some more examples related to graphing rational functions. But before proceeding to solve examples step by step, let us go through a quick introduction of rational functions.

The rational functions are obtained after taking the ratio between two polynomials. In other words, we can say that when we divide two polynomials, we get a new function known as a rational function. Division by zero in a rational function is not possible like normal division. Before graphing rational functions, you should know the asymptotes and intercepts of the function. Use dashed line to sketch the asymptote of the function. The graph should not cross vertical asymptote of the function.

 

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Examples of Rational Functions

1. \frac{x + 5}{x^3 - 3x + 8}

2. \frac{x^4 + 7x^3 - 2}{x + 1}

3. \frac{8x + 9}{x - 8}

The above three functions are rational because they are obtained after dividing polynomials.

Let us now proceed to solve some more examples related to graphing the rational functions.

Example 1

Graph the following function:

f (x) = \frac{x^2}{3x - 6}

Solution

We will graph the above rational function by following the steps below:

Step 1 - Find the asymptotes of the function

To compute the vertical asymptote of the function set the denominator of the function equal to zero and solve the equation for x. Remember that the vertical asymptote of the function will be the line that should not be crossed by the graph. The reason behind this is that when the denominator of the function is zero, the function becomes undefined.

3x - 6 = 0

x = 2

Hence, the vertical asymptote of the function in this example is 2.

For the horizontal asymptotes of the function, we need to look at the numerator and assess its degree. If the degree of the numerator is greater than the degree of the denominator, then we can assume that the function has no horizontal asymptote. The degree of the numerator of the function in this example is greater than the degree of the denominator, hence, we can say that there is no asymptote of this function.

 

Step 2 - Compute the vertical intercept

By substituting x = 0, calculate the vertical intercept of the function.

f (x) = \frac{x^2}{3x - 6}

y = 0

This shows that the vertical intercept of this rational function is (0, 0).

 

Step 3 - Calculate the horizontal intercept

Set the function equal to zero to calculate the horizontal intercept.

f (x) = \frac{x^2}{3x - 6}

\frac{x^2}{3x - 6} = 0

The horizontal intercept of the function is also (0,0).

 

Step 4 - Find y-values by substituting different x-values

In this step, we will plug in different x-values to get the y-values.

y = \frac{x^2}{3x - 6}

At x = 0, y = 0

At x = 1, y = -\frac{1}{3}

At x = 3, y = 3

At x = 4, y = \frac{8}{3}

At x = 5, y = \frac{25}{9}

At x = -1, y = -\frac{1}{9}

These are only few points, you can substitute more values to get points for the graph.

 

Step 5 - Plot the points in an xy-plane and join the points for a smooth curve

After many points from the step 3 will be plotted, we will get the graph below:

Graph of the example 1

 

Example 2

Graph the following function:

f (x) = \frac{x^2}{x^3 - 1}

Solution

We will graph the above rational function by following the steps below:

Step 1 - Find the asymptotes of the function

To compute the vertical asymptote of the function, we will set the denominator equal to 0. We will solve the equation for x to get the vertical asymptote.

x^3 - 1 = 0

x = 1

Hence, the vertical asymptote of the function in this example is 1.

For the horizontal asymptote of the function, we need to look at the polynomial in the numerator and assess its degree. If the degree of the numerator is greater than the degree of the denominator, then we can assume that the function has no horizontal asymptote. On the other hand, if the degree of the numerator is equal to the degree of the denominator, then we take the ratio of leading coefficients. In the third case, if the degree of numerator is less than the degree of denominator, then the horizontal asymptote is zero. In this example, the horizontal asymptote of the function is zero.

 

Step 2 - Compute the vertical intercept

By substituting x = 0, calculate the vertical intercept of the function.

f (x) = \frac{x^2}{x^3 - 1}

y = 0

This shows that the vertical intercept of this rational function is (0, 0).

 

Step 3 - Calculate the horizontal intercept

Set the function equal to zero to calculate the horizontal intercept.

f (x) = \frac{x^2}{x^3 - 1}

\frac{x^2}{x^3 - 1} = 0

The horizontal intercept of the function is also (0,0).

 

Step 4 - Find y-values by substituting different x-values

In this step, substitute different x-values to get the y-values.

f (x) = \frac{x^2}{x^3 - 1}

At x = 0, y = 0

At x = -1, y = -\frac{1}{2}

At x = -2, y = -\frac{4}{9}

At x = 2, y = \frac{4}{7}

At x = 3, y = \frac{9}{26}

At x = 4, y = \frac{16}{63}

At x = 5, y = \frac{25}{124}

These are only few points, you can substitute more values to get points for the graph.

 

Step 5 - Plot the points in an xy-plane and join the points for a smooth curve

The graphical representation of the function is given below:

Graph of the example 2

 

Example 3

Graph the following function:

f (x) = \frac{x^4 + x}{x^2 - 1}

Solution

We will graph the above rational function by following the steps below:

Step 1 - Find the asymptotes of the function

To compute the vertical asymptote of the function the denominator of the function will be equal to zero. Remember that the vertical asymptote of the function will be the line that should not be crossed by the graph. The reason behind this is that when the denominator of the function is zero, the resulting function we get is undefined.

x^2 - 1 = 0

x^2 = 1

x = \pm 1

Hence, the vertical asymptotes of the function in this example are 1 and -1.

For the horizontal asymptote of the function, we need to look at the numerator and assess its degree. If the degree of the numerator is greater than the degree of the denominator, then we can assume that the function has no horizontal asymptote. In this example, the degree of the numerator is greater than the denominator, hence there is no horizontal asymptote of this function.

Step 2 - Compute the vertical intercept

By substituting x = 0, calculate the vertical intercept of the function.

f (x) = \frac{x^4 + x}{x^2 - 1}

y = 0

This shows that the y-intercept of this rational function is (0, 0).

 

Step 3 - Calculate the horizontal intercept

Set the function equal to zero to calculate the horizontal intercept.

f (x) = \frac{x^4 + x}{x^2 - 1}

\frac{x^2}{x^3 - 1} = 0

The horizontal intercept of the function is also (0,0).

 

Step 4 - Find y-values by substituting different x-values

In this step, substitute different x-values to get the y-values.

f (x) = \frac{x^4 + x}{x^2 - 1}

At x = 0, y = 0

At x = 2, y = 6

At x = 3, y = \frac{21}{2}

At x = 4, y = \frac{52}{3}

At x = -2, y = \frac{14}{3}

At x = -3, y = \frac{39}{4}

At x = -4, y = \frac{84}{5}

These are only few points, you can substitute more values to get points for the graph.

 

Step 5 - Plot the points in an xy-plane and join the points for a smooth curve

Sketch the graph after plotting all the points:

Graph of the example 3

 

 

 

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Rafia Shabbir