❖ Graphing a Rational Function ❖

Graphing a Rational Function - Example 1

In this video, I walk through the complete process of graphing a rational function step-by-step. Learn how to find and plot key features of the function such as the x- and y-intercepts, vertical and horizontal asymptotes, and multiplicities of zeros. We also go through plotting additional points to understand the behavior of the rational function in all its regions. This example will help you master the art of graphing rational functions, a crucial concept in algebra and precalculus.

By the end of this video, you'll be able to:

Identify and calculate the intercepts of a rational function.

Find the vertical and horizontal asymptotes and understand how they affect the graph.

Determine the multiplicity of zeros and their impact on the function's shape.

Plot key points and sketch the complete graph of the rational function.

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[patrickjmt]

Yes, you can pass through a horizontal asymptote!

[Missy]

i'm pretty sure it took our class a whole month to learn all of what you just did in 7 minutes...

[ceprodarisystem]

"NOTE: A common mistake that students make is to think that a graph cannot cross a slant or horizontal asymptote. This is not the case! A graph CAN cross slant and horizontal asymptotes (sometimes more than once). It's those vertical asymptote critters that a graph cannot cross"

[jetstream]

I understand how to find the asymptotes and intercepts...BUT HOW DO YOU DRAW THOSE CURVES!!! This class is murder T-T

[gegasmeef]

Had to replay this coz I wasn't listening the first time...force of habit

[KingKromah]

I might be going to Harvard this Fall because of you. I just wanted you to know how thankful I am.

[domoboy]

he just explained this flawlessly, which that my professor couldn't lol...

[patrickjmt]

spread to word to your friends! i would appreciate it! :)

[AtomicMushs]

To those who have trouble understanding why the middle section crosses the asymptote, the occurrence is called a cross-over point and is an unique exception to when a asymptote is crossed. You can find them by subbing in the H.A  value into h(x) [HA is three in this case] and solving for x after supping in 3. If the answer  is impossible e.g 0 = 16, then there is no crossover. But if there is a value for x. then that is when you have your crossover on your asymptote

P.S the point is this question is x = 3/2

[cpseazay]

I'm studying for my college algebra final right now and I can't even begin to explain how much your videos have helped me. Thank you!!!

[patrickjmt]

glad i could help :) thanks for spreading the word!

[adisadawes]

This saved my last test grade, which saved my total grade, saving my GPA, saving my life haha. THANK YOU

[patrickjmt]

you do the same thing. the only thing that changes are the horizontal asymptotes. if you are unsure about that part, i have a video called ' a shortcut to find horizontal asymptotes of rational functions ' on my channel. just do a search for it.

[holaeskayla]

How is the middle line (between vertical asymptotes x= -5 and x=1) on the graph going through the horizontal asymptote? I thought that lines can't go through asymptotes.

[patrickjmt]

cause you are mistaken in what asymptotes are :) a rational function will never cross the vertical asymptotes, but it can 100% for sure cross a horizontal aymptote (as shown). you should look at the definition and noticed that it nowhere says anything about not being able to cross a horizontal asymptote (when in doubt, reread the definition)

[melariemantanona]

Can someone explain to me why he passed through the horizontal asymptote?

[patrickjmt]

@InsaneHands2 nope, no bs. i figure people just want to get their homework done. ; )

[patrickjmt]

@anotherhappyguy a rational function never cross vertical, correct. but they for sure can cross horizontal.

[subhajitghosh]

Well I have a confusion, in this example for  -5 < x < 1 the graph intersect y=2. shouldn't be there a hole where it crosses the  line?? please

[patrickjmt]

rational functions can cross horizontal asymptotes. they do not cross vertical asymptotes.