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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.

Watch this video lesson to learn how you can graph rational functions with linear polynomials in just a few steps. Also learn what kinds of functions these are and what you need to look for to graph them.

The phrase 'rational functions with linear polynomials' is a mouthful, that's for sure. But don't worry, it's not as bad as it sounds. If we break the phrase down into its parts, we will see that it's not too bad. First, **rational functions** are simply fractions of polynomial functions. A **linear polynomial** is a polynomial whose highest exponent is 1. So a rational function with linear polynomials is a fraction of two polynomials whose highest degree is 1. Polynomials are those expressions that combine terms consisting of variables and their coefficients with pluses and minuses.

An example of a rational function with linear polynomial is the function *f(x) = (x) / (x + 1)*. Look at the numerator and denominator, and we see that both are linear polynomials. Both the numerator and denominator consist of functions whose highest exponent is 1. A shortcut to identifying this type of function is just to look at the *x*. If both the numerator and denominator have an *x* with no exponent shown then you are looking at a rational function with linear polynomials. This is the type of function we will be graphing in this lesson.

The first step in graphing these functions is to look for asymptotes. We need to look for **vertical asymptotes**, the *x* values that make the function invalid, and **horizontal asymptotes**, the *y* value that the graph reaches for at the far left and far right of the graph.

To find our vertical asymptotes, we need to ask ourselves, what situation will make our function invalid? In other words, what situation causes the function to produce an error? Let's think about this. Our function is essentially a fraction. What do we know about fractions? What is the one number that we can never divide by? It's 0. So that tells us that if our denominator equals 0, then our function will produce an error.

So, at what *x* value does our denominator equal 0? How can we figure that out? We find this value by setting the denominator equal to 0. Our function is *f(x) = (x) / (x + 1)*, and our denominator is *x* + 1. Setting this equal to 0, we get *x* + 1 = 0. Solve for this by subtracting 1 from both sides, we get *x* = -1. So our vertical asymptote is the vertical line at *x* = -1. We can draw this on our graph with a dashed line at *x* = -1.

The next thing we look for is our horizontal asymptote. To find our horizontal asymptote, all we need to do is to look at our coefficients next to our variable *x*. In our case, the coefficients are 1 for the numerator and 1 for the denominator. So that means our horizontal asymptote is *y* = 1/1 = 1.

Yes, the horizontal asymptote is simply the fraction of the coefficients. How can you remember this? Think about the meaning of horizontal asymptote. It is the behavior of the graph at the far left and far right of the graph. This is where our *x* values are large. When our *x* is large, whatever we add or subtract won't make much difference, so we can ignore those numbers. Then since both the numerator and denominator have the same *x*, we can cancel them out. So what are we left with? We are left with the coefficients.

For our function *f(x) = (x) / (x + 1)*, to find the horizontal asymptote, we ignored the + 1 on the bottom, which leaves us with *y = x/x*. We have the same *x* on the top and bottom, which tells us we can cancel them out. This leaves us with just the coefficients *y* = 1/1. We can draw our horizontal asymptote *y* = 1 using a dashed line as well.

Now that we've drawn our asymptotes, we see that it separates our graph into four areas. One thing to note is that the graph will never touch or cross the vertical asymptote. So you will see the graph avoid this line like the plague! The horizontal asymptote, on the other hand, is like honey and attracts the graph. You will see the graph get closer and closer to this line.

To know exactly how our graph will look, we are going to plot some points on either side of the vertical asymptote as well as further away. To calculate these points, we plug in some *x* values and then just evaluate them to find our *y*. We will write these in table form so that when we are done, we can easily plot them on the graph.

The *x* values that I am going to use are -10, -5, -3, -2, -1.5, -0.5, 0, 1, 5, and 10. To find our *y* when *x* is -10, I plug in my -10 into my fraction and get *y* = -10 / (-10 + 1). Evaluating this, I get *y* = -10/-9 = 1.11. So our first point is (-10, 1.11). I continue this process for the rest of my points until my table is filled up.

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

-10 | 1.11 |

-5 | 1.25 |

-3 | 1.5 |

-2 | 2 |

-1.5 | 3 |

-0.5 | -1 |

0 | 0 |

1 | 0.5 |

5 | 0.83 |

10 | 0.9 |

Plotting these on the graph, I start to see the curves of my graph.

The last step is to connect the dots. I have to keep in mind the asymptotes, though. I see that the graph curves up as it approaches the vertical asymptote from the left and it curves down as it approaches the vertical asymptote from the right. I know that the graph cannot touch or cross this vertical asymptote, this 'plagued' line, so that means I have to draw my graph curving and up and down but never touching this line.

For my horizontal asymptote, I know that my graph gets closer and closer to this 'sweet as honey' line as it keeps going, so I am going to draw the line very close to the horizontal asymptote but not touching it either. Taking all this into consideration, I draw my graph, and I see that it consists of two curves.

I got my asymptotes, my points, and now my graph. So I am done!

What did we learn? We learned that a **rational function** is the fraction of two polynomial functions and that a **linear polynomial** is a polynomial whose highest degree is 1. We learned that a shortcut for identifying rational functions with linear polynomials is to just look for an *x* with no exponent in both the numerator and denominator of the function. The function can have only one *x* on top and one *x* on bottom.

To graph these functions, we look for the **vertical asymptote**, the *x* value that makes the function invalid, by setting the denominator equal to 0 and solving. We also look for the **horizontal asymptote**, the *y* value that the graph reaches towards as the *x* value gets very large and very small. To find the horizontal asymptote, we simplify our function to just the coefficients of our *x* variable. We graph both asymptotes using dashed lines. Then we plot some points near the vertical asymptote on both sides and some further away. We connect the dots, remembering that the graph cannot touch or cross the vertical asymptote, but it does reach for the horizontal asymptote.

Studying the details of this lesson could prepare you to:

- Determine whether an equation is a rational function with a linear polynomial
- Provide the definition of a horizontal asymptote
- Calculate a rational function's vertical and horizontal asymptotes
- Plot points on the graph

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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
- Analyzing the Graph of a Rational Function: Asymptotes, Domain, and Range 5:50
- Go to Rational Expressions and Functions

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