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Let's go over the answers to the assessment you just completed. So, here was the first
problem. Based on your knowledge of vertical asymptotes and limits, which equation matches
this graph? Let's zoom in on the graph for a second and look at some of its main features.
So, what stands out to us as important about this graph? Well that first thing, that's
pretty important, is that the graph has a vertical asymptote at X equals 2. And then,
also, on either side of that asymptote, the graph's going to positive infinity. So, going
to positive infinity on both sides of the asymptote is pretty important. So, let's see
if we can knock out any choices based on those two pieces of information, and then we'll
dig in a little deeper if we have to. So, first of all, we know there's an asymptote
at X equals 2. That means that when I plug in X equals 2 to the correct equation, I should
be getting 0 in the denominator. So, that automatically knocks out one choice for us
-- choice number 4. Because if I plug in 2 to number 4, I don't get 0 in the denominator.
So, we've knocked out one choice. Now, let's use the fact that the graph goes to positive
infinity on both sides of the asymptote to knock out another choice. So, the fact that
it goes to positive infinity on both sides of the asymptote tells me that if I plug in
values that are a little bit greater than 2 and a little bit less than 2, my function
should be positive for both of those scenarios. So, look at choice number 3. If I plug in
values that are slightly greater than 2, I get a positive in that case. But, what if
I plug in values that are slightly less than 2? Like 1.9. If I plugged in 1.9 here, I would
get a negative value overall, but we see that's not what happens with the function. We have
positive values as we approach the asymptote. So, I know it can't be number 3 because of
that reason. So now we're left with two options, and these two options are pretty similar.
And actually distinguishing between them is pretty subtle. So, what do we have? Both of
these options -- number 1 and number 2 -- have X minus 2 quantity squared in the denominator.
But option number 1 has X in the numerator while option number 2 has 1 in the numerator.
Let me show you this graph one more time. I'm going to point out a fact here that's
going to be key to distinguishing between them. Notice here on the left hand side of
the graph that our function dips into negative y values. We have some negative y values on
this graph. But, look at choice number 2. Choice number 2 is 1 over X minus 2 quantity
squared. So, in choice number 2 the numerator is always positive because it's 1, and the
denominator is also always positive because it's squared. So, choice number 2 could never
be negative. So because of that reason, I know that it's not the choice I'm looking
for. The correct choice is choice number 1. It has an asymptote at X equals 2, which we
want. On either side of X equals 2, it's positive, which we want. But it does allow for negative
values, which we see that our graph has. So that's why choice number 1 is correct in this
case. So, let's continue and look at the other problems. This is the second assessment you
completed. And same directions, just a different graph. Let me zoom in on this graph for us.
What do we see here? Well, the main thing that stand out to me is that in this case
we have two vertical asymptotes. We have one at X equals negative 2 and one at X equals
positive 2. That's a pretty unique thing, so let's see if that fact in itself will help
us pick the right answer. So, I need to pick an answer that allows for two vertical asymptotes.
So, whether I plug in 2 or negative 2 to the denominator, I should get zero. So, that automatically
knocks out a lot of choices. Choice number 1 only has one vertical asymptote, at X equals
negative 2. Choice number 2 also has only one vertical asymptote, at X equals positive
2. And same thing for choice number 4. Choice 4 only has 1 vertical asymptote, at X equals
4. So that was pretty nice, right? I was eliminating three choices right off the bat. So, it looks
like choice 3 is the correct answer. Let's just make sure. So, out to the side here I'm
just going to show you that if I factor that denominator into X plus 2 and X minus 2, that's
a difference of squares and I'm just factoring it, then I can clearly see that there are
two vertical asymptotes. There are two X values that make this denominator zero. And they're
the 2 and the negative 2 that I was looking for. So, 3 is the correct answer for this
function. Alright, one more. So, this one's different, right? This function is unique
out of all the ones we've looked at so far in that it doesn't have any vertical asymptotes.
So, let me zoom in on it for a second. This is what it looks like up close. No vertical
asymptotes. Just a friendly, continuous function. So, out of these answer choices, we need to
pick a choice that would not have a vertical asymptote. So, let's see if we can eliminate
any, based on that fact. Well, I can eliminate choice number 1 because it would have a vertical
asymptote at X equals negative 2. Ok, number 2, I'm not so sure about that one. Let's skip
over that one. Number 3 -- well this is the one I just looked at at the previous problem.
We know that this function, 1 over X squared minus 4, has vertical asymptotes at X equals
negative 2 and X equals positive 2. So, it's not what I'm looking for. And number 4 -- I
know that that would have a vertical asymptote at X equals 4. So, it looks like number 2
is the winner in this case. And that's because there's no X value that I can plug into the
denominator to make the denominator equal zero, which means there are no vertical asymptotes
and that's exactly what I was looking for in this case.