A Third Example of Graphing a Rational Function A Third Example of Graphing a Rational Function
A Third Example of Graphing a Rational Function
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- Let's graph another rational function, because you really
- can't get enough practice here.
- So let's say we have y is equal to x over x squared
- minus x minus 6.
- So the first thing we might want to do is just factor this
- denominator so we can identify our vertical asymptotes, if
- there are any.
- So what two numbers when I take their product I get
- negative 6 and if I add them up I get negative 1?
- So they have to be of different signs.
- So one's going to be a plus and one-- let me write my x's
- a little bit neater than that-- so one is going to be a
- positive and one is going to be a negative.
- A 2 and a 3 seem to be pretty close, because they're one
- apart, and I'm going to subtract the larger number
- because when I add them, I get a negative.
- So x minus 3 times x plus 2 seems to work.
- That gets negative 6.
- Negative 3x plus 2x.
- negative 3 times x plus 2 times x is
- negative x, so that works.
- So this is equal to x over x plus 2 times x minus 3.
- And like we saw in the last video, since these
- expressions, since the x plus 2 doesn't cancel out with
- anything in the numerator and the x minus 3 doesn't cancel
- out with anything in the numerator, we know that these
- can be used to find our vertical asymptotes.
- The vertical asymptotes are when either that term is equal
- to zero or when that term is equal to zero, because at
- those points, our equation is undefined.
- So this is equal to zero when x is equal to negative 2, and
- this is equal to zero when x is equal to positive 3.
- And you could try it out here.
- If x is equal to negative 2 or positive 3, you're going to
- get a zero in the
- denonminator, y will be undefined.
- So vertical asymptotes at x is equal to negative 2.
- So there's a vertical asymptote, a vertical
- asymptote right there.
- Another vertical asymptote is x is equal to 3.
- One, two, three.
- There is our other vertical asymptote.
- Now let's think about horizontal asymptotes, or if
- there are any.
- So what happens as x gets super-positive or
- And as we said before, you just have to look at the
- highest degree term on the numerator and the highest
- degree term on the denominator.
- Now, notice the highest degree term on the denominator is x
- squared, while the highest degree term on the numerator
- is only an x.
- So when x gets really large, what's going to happen?
- You could imagine, this is going to be like a million
- over a million squared, which is still one over a million.
- These terms over here don't matter much.
- But this term right here is going to grow faster than
- This is an x squared term.
- As x gets large, it's going to get way larger than
- everything, including this term on the top, so it's
- essentially going to go to zero.
- When the denominator just gets bigger, faster than the
- numerator as you're going to approach zero.
- So we have a horizontal asymptote at y is equal to 0.
- I could draw it as a dotted line over our x-axis.
- So that right there is the line, y is equal to 0.
- Once again, we identify that looking at the highest degree
- term there.
- The denominator has a higher-degree term, so it's
- going to grow faster than the numerator.
- You could try it out on your calculator.
- And that's true whether you go in the super-negative
- direction or the super-positive direction.
- This thing is going to overwhelm this thing up here,
- the denominator grows faster than the numerator, which
- essentially we're going to approach zero.
- You're going to get smaller and smaller fractions.
- Just remember, 1/10 and then-- let me actually just-- as x
- gets larger and larger and larger,
- what's going to happen?
- Let me just show you on my calculator.
- Let's say x is equal to 10.
- 10 divided by 10 squared minus 10, and normally you wouldn't
- have to do this.
- I just really want to show you the intuition.
- I'm not trying to graph.
- Let me exit from here.
- So if we have 10 over 10 squared minus 10, once again,
- you normally wouldn't have to do this.
- I just want to show you, give you the intuition.
- Let me put some parentheses there.
- Let me put some parentheses over here.
- So let me insert the parentheses there and put a
- parentheses over here.
- You get a small number.
- What happens if x gets even larger?
- Let me make, instead of a 10, let me make it all 100.
- Let me make these tens into hundreds, into 100.
- Insert 100 there, what do we get?
- We get even a smaller number.
- And if you try x is equal to 1,000, it's going to be be
- even smaller than that.
- That's because this term right here is growing faster than
- every single other term.
- That's why our horizontal asymptote is y is equal to 0.
- Now, the last thing we want to do, we've drawn all of our
- asymptotes, is just try out some points.
- So let's draw like a little table here.
- There's our table.
- When x is equal to 0, what is y?
- x is 0, we have 0 over all of this.
- 0 minus 6, 0 over negative 6 is just 0.
- When x is equal to-- I don't know, let's just try when x is
- equal to 1, what do we have?
- We have 1 over-- I'll write it here.
- 1 over 1 squared minus 1.
- Now that's just 0, so we have negative 6.
- When x is equal to negative 1, what do we have?
- When x is equal to negative 1, we have negative 1 over
- negative 1 squared, which is 1 minus negative 1.
- So that's plus 1-- right, minus negative 1-- minus 6.
- So what is this right here?
- This is negative 1, so this is going to be negative 1 over 2
- minus 6 over negative 4.
- This is going to be equal to 1/4, so we're going to get a
- positive value.
- So we have-- let me draw this-- negative 1, we're at
- 1/4 right here.
- That's about right there.
- I'll do it in a darker color.
- We had the point 0, 0, and then at x is equal to 1, we
- had negative 1/6.
- So you could keep graphing more and more points, but it
- looks like as we approach this vertical asymptote from the
- right, we go to positive infinity.
- And that should make sense.
- Let's see, if we were to put in-- we're approaching
- negative 2 from the right.
- So if you were to put in negative 1.9999999, this term
- is going to be a very small positive number.
- This term's going to be a negative number.
- This term's going to be negative number.
- The negatives cancel out.
- You have a very small positive number in the denominator.
- 1 over that gives you a very positive number.
- Now, as we approach the other vertical asymptote from the
- left, we're going to go super-negative.
- My gut tells me that because when I tried x is equal to 1,
- I already went to a negative value.
- But you could imagine if you did 2.99999, right?
- Let me draw that a little bit better.
- You get the idea.
- If x is equal to 2.999, so we get really close to the
- asymptote, this is going to be positive, this is going to be
- negative, that's going to be positive, and this is going to
- be a small number.
- So you're going to have 1 over a very small negative number,
- which is a very, I guess, negative number.
- It's a negative of 1 over a very small number, so you're
- going to approach negative infinity.
- Now, let's try some point out here to see what happens.
- So what happens when x is equal to 4?
- When x is equal to 4, you have 4 over 16 minus 4 minus 6.
- What is that?
- That's 16 minus 10.
- That is 6.
- So this is equal to 4/6, which is equal to 2/3.
- So the point 4, 2/3 is here, so one, two, three, four.
- 2/3, just like that.
- So that gives me the sense, look, I have to approach this
- horizontal asymptote as we go further and further out.
- We're going to probably approach positive
- infinity like this.
- Let me draw it a little neater than that.
- Like that.
- You get the idea.
- Then over here, we're going to get closer and closer to our
- horizontal asymptote as we approach infinity.
- This should be a smoother-looking curve right
- around there.
- I'm making a mess here.
- This should be a smoother-looking curve.
- You get the idea, I think.
- Now, let's see what happens when x is equal to negative 3.
- So when x is equal to negative 3, we have negative 3 over
- negative 3 squared, which is 9 minus negative 3, so that's
- plus 3 minus 6.
- So what is this equal to?
- This is equal to negative 3 over-- this is 12 minus 6 over
- 6, right, which is equal to negative 1/2, So negative 3,
- negative 1/2.
- Negative 1/2 is right there.
- So we're going to approach this asymptote as we get
- really negative.
- And we're probably going to go straight down like that as we
- approach this vertical asymptote right there.
- And you could try more points if you don't believe me.
- But let us graph it just to verify it for ourselves.
- So our equation is x divided by x squared minus x minus 6.
- And let's graph it.
- And there you go.
- There you go.
- All right, looks pretty good.
- Our asymptote is zero, we go down.
- Vertical asymptote, bam!
- Go up there, then we go back down here, then we go just
- like that again.
- So once again, this looks just about exactly what we got.
- Obviously, the graphing calculator, it kind of a
- pitters out as you get close to these values and it does
- weird things, but it has the same general shape.
- We could actually close the range a little bit if we want,
- if we want to graph it.
- Let's make our x minimum value, let's make it 5.
- And let's take our x maximum value, let's
- make that 5 as well.
- We're kind of zooming in a little bit.
- So let's graph it now.
- There you go!
- It's the same shape as we graphed right here.
- Hopefully, you found that satisfying.
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