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(a) Find the vertical and horizontal asymptotes.

(b) Find the intervals of increase or decrease.

(c) Find the local maximum and minimum values.

(d) Find the intervals of concavity and the inflection points.

(e) Use the information from parts $ (a) - (d) $ to sketch the graph of $ f $.

$ f(x) = \frac{x^2 - 4}{x^2 + 4} $

a) no vertical asymptote, horizontal asymptote $y=1$

b) decreasing: $(-\infty, 0) \quad$ increasing: $(0, \infty)$

c) local min: $(0,-1)$

d) Intervals of concavity: $x < -\frac{2 \sqrt{3}}{3}$ and $x > \frac{2 \sqrt{3}}{3},$ concave down, $-\frac{2 \sqrt{3}}{3} < $

$x < \frac{2 \sqrt{3}}{3},$ concave up. Inflection points at $x=\pm \frac{2 \sqrt{3}}{3}$

e) SEE GRAPH

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Campbell University

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All right. So first we're being asked to find the vertical and horizontal asking told of affect, So ah, the vertical and jerkers when the parliament equals zero. So in this case, X square plus four equals zero over a few Attempt to solve that You'll quickly realize that you'LL get an imaginary entrance. So there's actually no WeII this the economic chemicals, you know? So there is no class in tow, so no vertical tinto to find the horizontal axis. So we take the limit of the function to infinity and negative infinity. So we'll just tio we can do infinity and so ex women for over four plus four. So there's a couple of ways to convince us you can just look at the leading coefficient and realize that it's just a ratio of one to one. So be one, for if you want to be a bit more rigorous about it, you can actually pull out the factor of X where, so I'm not going to write the limit. I'm just gonna rewrite it, Write it, um, the better idea so I can show you that I'm actually applying limit loss so I pull out X square this will be one minded or over X squared and this will be all over X squared and I'll do one plus for Rex squared and then what will happen and I'll cancel the deck squares And if you actually applied the infinite if you plan to infinity you realized that this function we'll go to one minus four over infinity, which is zero and then four over infinity again one plus zero and I give you one of the one was just one. And if you do too negative, infinity doesn't make any difference because it's squared So you can also do plus or minus infinity I'll still give you one so we have a horizontal no at y equals one. So don't ask youto at Weyco sworn and that all we have for Athens yourself to find where the function is increasing the decreasing we have to apply the first derivative care So the prime of act and this is a question little problem, so you won't have to buy the question rules. So your final product should be sixteen x over X squared plus four and in the square the whole thing that the secret geo sixteen tactical gear only ex clinical zero and there is no ah, no undefined number for this causes squared on the bottom. So don't worry about that for interval evaluations are now we create a sign chart and we look at around zero for the sign of a crime. So if if it's lessened your your values that are negative, you get values are great and you get a crowd of number. And this tells us that it is decreasing from negative infinity to zero and it's increasing from zero to infinity. And there is also local men occurring because it is decreasing and then increasing. So it looks like this. So there is a loophole Men at X ical Tio Now Ah, we're going to look at the con cavity of this functions. So look at the second derivative cast. So we take the second do everyday again will kind of questionable. It's a bit maciver. It should come out to be sixty four minus forty eight x squared all over X square clothes for and and this is cute. They said the sequel is yours. So you do sixty four months excoriate square forty eight expert equal Tio, you were the X We're good. You Then you'LL get X square is equal to sixty four over forty eight and that's the same thing for third. So then this will give us access Decoy. So you take the square root of personal minus to over three because discredited forced to Now we're going to evaluate it around those two number So the sign chart. So this will be negative to over three and it will be a positive to over three. And then you're going to look at this time after Hubble Prime. The people in power is less than negative. Two or three. You get negative numbers between between these two get positive and greater than to get negative. This is Concord. Down, up, down. Oh, you have Kong kept down occurring from negative infinity to two negative to over three and from to over three to infinity And you have Khan caged up crying betweennegative two or three two positive two, three You're inflected Point occurs when the sign changes. So you have inflexion point at both negative two plus and minus two of the three. So you have class of money two over to me. Now you have enough information to draw the graph. Let's go ahead and draw the graph. So we have a horizontal ass enter at like a wild one. Something I didn't draw this red line, so reminder that it is decreasing zero. And it is. Khan gave down. So it's going to be kind of going down like this. So it has come down and then we have a local men and then it starts increasing at zero. It goes up and we have We have a slight you right here. And then it's going to go back to being a cop. So it's going to come give down and then climb up. And then back to be Kong came down. That's gonna look like this and that the graph of f of X, I believe that the bank Yes, exactly.