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When were doing rational functions and the one thing. I started off this instruction instruction with was how to find the vertical asymptotes and thats all i taught guys thats all we went over with this graph we did i did give you guys an equation to much. Im not sure if i left.

It up for this class y. Equals n. Of x over d of x.

Ok. But i didnt get to explain really anything else i just said ladies gentlemen rational expressions arent the reciprocal function theyre related. But basically youre gonna have a polynomial up top and a polynomial in the bottom ok.

Thats what it looks like so you can see this follows. Along whats nice about this one. Though its already factored right.

I already factored. It for you so that was nice of them then the other thing that i left in there before you guys before we ended class. As i said to find the vertical asymptotes you set the denominator equal to zero.

Then you solve so if you wanted to find the discontinuities might not understand what it is but if anything all i really expect you guys to do is to find the vertical asymptotes of every single graph. So or every single problem. So.

If you found the vertical asymptotes. You would have said. X plus.

3. Equals zero subtract. 3.

Subtract 3x equals negative. 3. And based on what you were taught last class period that would have been okay.

I would have it i would except you have to do that for every single problem because thats all i got thats all i had time to teach you guys. However. Now we have a little more time.

We can get into removable and non removable discontinuities. Because mathematically. This is not a part of the domain.

However its not a asymptote actually in this case. This is actually not an asymptote. So whatever you still the my definition is still the same to find the vertical asymptote set the denominator equal to zero.

However ladies and gentlemen what you should see is we can apply the division property right. If you applied the division property. The x plus 3s divide out leaving us with just x minus.

4. Right well. But still and thats a line.

So that looks something like this but still x. Cannot equal negative three well lines. Dont have asymptotes right you dont have asymptotes so how do we graph something a line but have x cannot equal negative three.

What that is called is a hole. So the graph is still a line. But originally you have a removable discontinuity.

So when you have a removable discontinuity. Youd still set the denominator equal to zero and find it. But its not gonna be an asymptote like my previous definition.

When its removable meaning you can divide it into one. Its called a hole and graphically it would look like this okay. So the difference of your discontinuities you either have an asymptote or a hole.

If its something that can be factored and simplified out its called removable because do you guys see how we remove the x plus 3s. So its removable when its removable. Its called a hole if its non removable.

Its an asymptote and you would create a nice little dotted line. Well do that next. But again.

The domain is still the same thing domain is all real numbers such that x cannot. Equal the discontinuity. Which is still the same x cant equal negative.

3. Right so that doesnt change the range doesnt change just think of it like an esco. Its all real numbers.

Except y cannot equal now we do need to figure out what that at x equals negative. 3. We need to figure out what that would be so if you plug in negative 3 4.

That negative. 3 minus 4 would be negative 7. So such that y cannot equal negative.

7. Because we need to figure out what would the y value be at negative. 3.

And then we need to find the x and the y intercepts. So x. Intercepts y.

Equals. Zero zero equals you can do it for either one. It doesnt really matter x.

Minus four. So you get x equals. Four.

And you could write that as a coordinate point as 4 comma 0. To find the y intercept x equals 0. Y equals negative 4 0.

Comma negative 4. And that was it and does that look like thats roughly about right yep. And thats all you guys had to do for that ok now .

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