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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Graphing rational functions is not as hard or as scary as it sounds. Sure, the functions may be big, but watch this video lesson and you will see that graphing these functions can actually be easy.

**Rational functions** can be described as fractions made up of polynomial functions. A polynomial is an expression made up of variables with their coefficients separated by either addition or subtraction. If we had two polynomial functions that we have labeled *f(x)* and *g(x)*, then our rational function can be written as *f(x)* / *g(x)*. As an example, if my *f(x)* equaled 5*x* + 1 and my *g(x)* equaled *x*^2 + 3*x*, then my rational function will be (5*x* + 1) / (*x*^2 + 3*x*).

It looks scary, doesn't it? Don't worry, though, graphing these rational functions isn't all that bad.

There are only a few things that you need to find to graph them, which include the vertical asymptotes, the horizontal asymptotes and the points to plot. A vertical asymptote is an *x* value that makes our function invalid. In other words, it is the *x* value that produces an error in our function when we try to evaluate our function at that value. Our function and graph will never be this *x* value.

A horizontal asymptote is the *y* value that the graph approaches as the *x* values get very large and very small. In other words, the horizontal asymptote is the value that you see the graph approaching to the far left of the graph and the far right of the graph.

To find the vertical asymptotes of our rational function, we simply set our denominator equal to 0 and solve. Why do we do this? If our vertical asymptote is the *x* value that our graph cannot be, then our best way to find out these values is to set the denominator equal to 0 because it is at those values that the function is not valid. We cannot divide by 0, so by finding the *x* values when our denominator equals 0, we find those *x* values the function cannot be.

Let's use the rational function we began with and find its vertical asymptotes. We set the denominator equal to 0: *x*^2 + 3*x* = 0

We solve for *x* by using algebra. Our denominator is a quadratic, so we will use factoring to solve: *x*(*x* + 3) = 0

We set both factors equal to 0 and solve for each. *x* = 0 and *x* + 3 = 0

We have two *x* values here, actually. We have *x* = 0, and we have *x* = -3. So these are our two vertical asymptotes.

We can begin graphing our data by drawing dashed lines to represent our vertical asymptotes at *x* = 0 and *x* = -3.

Next, we need to find our horizontal asymptotes. To do this, we compare the first term of our numerator to the first term of our denominator. What we are looking for is the exponent of this term. There are three scenarios that we will look for in helping us find our horizontal asymptote.

The first scenario is when the exponent of the numerator is higher than the exponent of the denominator. When this happens, the *y* values of the graph will go towards either positive infinity or negative infinity as the *x* value gets very large and very small. Looking at our rational function, we see that this is not our case, so let's look at the second case.

The second case is when the exponent of the first term of the numerator is the same as the exponent of the first term in the denominator. When this is the case, we find the horizontal asymptote by dividing the coefficient of the first term of the numerator by the coefficient of the first term of the denominator. For example, if our rational function is (5*x* + 1) / (3*x* + 2), then our horizontal asymptote is *y* = 5/3. Is this our case? Looking at our function again, we see that this is also not our case. So we move on to look at the third scenario.

The third scenario is when the exponent of the first term in the numerator is smaller than the exponent of the first term in the denominator. When this happens, our horizontal asymptote is the line *y* = 0. Looking at our function, we can see that yes, this is our case. So that means we have a horizontal asymptote at *y* = 0. We can mark this with a dashed line as well. We recall that while it is never okay for our graph to cross a vertical asymptote, it is perfectly okay for the graph to cross a horizontal asymptote.

How can you remember these? Picture a mean fraction with big teeth that gets angry when you touch its ticklish spots, or its vertical asymptotes. When is the fraction ticklish? It's ticklish when the denominator is 0. Then picture the fraction monster running towards a piece of pie. This is the horizontal asymptote, or the behavior when the fraction monster is getting further and further away. The horizontal asymptote is determined by the first term of the numerator and denominator or what the monster is made of.

Great! Now that we have our asymptotes down, let's go ahead and plot some points. First, we find our *x* and *y* intercepts, or where the graph crosses the *x*-axis and *y*-axis. To find where the graph crosses the *x*-axis, we set *y* equal to 0 and solve for *x*. So we have 0 = (5*x* + 1) / (*x*^2 + 3*x*).

To get *x* by itself, we begin by multiplying both sides by the denominator to get 0 = 5*x* + 1. Then we subtract 1 from both sides to get -1 = 5*x*. Then we divide by 5 on both sides to find that our *x* intercept is *x* = -1/5. So we go ahead and plot a point there.

Next, we find our *y* intercept by setting *x* equal to 0 and solving for *y*. But wait, we can't have *x* = 0 because that is one of our vertical asymptotes. What does this mean? It means that we don't have a *y* intercept. Okay, we make a note of this; our graph will not touch the *x* = 0 line.

Now, let's plot a few points. I am choosing some *x* values between 0 and 3 as well as some numbers above 0 and below -3. I'm going to make a table with my points. What I am doing is plugging in my *x* and calculating the value for *y*.

x | y |
---|---|

-10 | -0.700 |

-7 | -1.214 |

-5 | -2.4 |

-4 | -4.750 |

-3.5 | -9.428 |

-2.5 | 9.200 |

-2 | 4.5 |

-1 | 2.000 |

-0.5 | 1.2 |

0.5 | 2.00 |

1 | 1.5 |

3 | 0.8889 |

5 | 0.65 |

10 | 0.3923 |

The points I picked were close to my vertical asymptotes and then further away to see what happens to the graph as it gets farther away. I can plot all these points on my graph to see what kind of behavior I get.

Hmm, these points look interesting. I see the graph curving as it gets close to each asymptote. I see that the graph has three parts. One on the left side of *x* = -3, one between *x* = -3 and *x* = 0, and one to the right of *x* = 0. I remember that the graph will not touch either of the asymptotes, meaning that the graph curves up or down depending on the direction of the graph as it approaches the asymptote. I connect the dots, keeping all of this in mind.

We are done! We have graphed our rational function. We see that our graph can have more than one part depending on how many vertical asymptotes it has. We see that each vertical asymptote splits our graph up. So overall, graphing our rational function wasn't bad.

We learned that **rational functions** can be described as fractions made up of polynomial functions. To graph them, we first find the vertical asymptotes by setting our denominator equal to 0 and solving for *x*. We then find our horizontal asymptotes by setting *y* equal to 0 and solving for *x*.

We know that the graph cannot ever touch a vertical asymptote, but the graph can touch a horizontal asymptote.

After finding the asymptotes, we find some points. We see if we can find the *x* and *y* intercepts by setting *y* and then *x* equal to 0, respectively.

We then calculate some points as close to the asymptotes and then further away from them to see what kind of behavior the graph has. Then we connect the dots to finish graphing the rational function.

After studying this lesson, you should have the capabilities necessary to:

- Graph a rational function by identifying the vertical and horizontal asymptotes
- Remember the process of finding
*x*and*y*intercepts - Plot points

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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

- How to Multiply and Divide Rational Expressions 8:07
- Multiplying and Dividing Rational Expressions: Practice Problems 4:40
- How to Add and Subtract Rational Expressions 8:02
- Practice Adding and Subtracting Rational Expressions 9:12
- How to Solve a Rational Equation 7:58
- Horizontal and Vertical Asymptotes 7:47
- Graphing Rational Functions That Have Linear Polynomials: Steps & Examples 7:55
- Graphing Rational Functions That Have Polynomials of Various Degrees: Steps & Examples 8:59
- Go to Rational Expressions and Functions

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