WEBVTT
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we want to sketch the girl. Why is he
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doing excellently minus violets? So in this chapter,
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they give us this laundry list of steps we should
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follow if we want to Sketch, graph, have
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some function. So just go on and follow the
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steps. So the first thing they tell us to
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find is our domain of the function. Well,
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this is a polynomial, and we know Paula.
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No meals have a domain of all real numbers.
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Although numbers or negative infinity to infinity. The next
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thing they suggest refined is our intercepts. So let's
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go ahead on back. So off. Why Intercept
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? Actually, go ahead. And that low down
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here. So are why intercept will be when X
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is equal to zero. So why is going to
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be zero to the minus by time zero, which
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is just d'oh and then find out ex intercepts,
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Remember, we're going to set why you could deserve
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get zero is equal to X to my ass.
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Bye. Let's go ahead and factor this and people
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get zero physical too. That's excellent work minus five
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and using the zero product park to get X is
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equal zero or ex physical to plus or minus the
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route. Uh, five. And this year's approximately
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one point. I will just need to know that
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approximation for later. The last thing they want us
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to check is for symmetry, So this won't be
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a periodic function, but we can check to see
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if it's going to be even more odd. So
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we won't check effort. Negative X and doing that
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, we will get well, negative X to the
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power Negative minus five. Mine effects. Well,
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we could go ahead and pull that negative out to
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power. That would be negative. One to the
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fifth rush. Really Negative X. And then those
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two natives there cancel out. And now notice if
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the factor that negative out we get excellent with minus
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five x. And this here is negative. Yeah
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, the banks. So with this implies is that
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we have a odd function or symmetry about origin.
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Oh, or origin. What next? They suggest
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we find any acid is what we know. That
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since this is a polynomial, there should be no
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acid trips. But we can use similar logic.
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50 are in behavior of this craft. Should be
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so. I'm gonna go ahead and call this looks
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So as X goes to infinity. Well, this
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is a odd function with a positive coefficient. So
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we know that should go to in 30 on DDE
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. But they're being a odd degree polynomial than it
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should have the opposite in behavior for negative infinity.
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So it would go to negative infinity as X goes
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to negative. So the next two steps, I'm
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gonna combine Minto one. And this was supposed to
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be looking for intervals where it increases decreases, and
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we're going to look for local maxes and meds.
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So that means we want to figure out what wide
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primates have said it better than zero. So taking
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the derivative Oh, exit. We're gonna use power
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rule to take the girl with the Chinese. So
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we're going to get bye, thanks to the fourth
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by this. Now we want to figure out where
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there are actually some really do that. Um,
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let's go ahead and set up a secret. You
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zero. So we confined our critical points so saying
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to seek with zero, we would divide by five
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and we get thanks to the fourth minus. One
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is equal to zero. And then that would tell
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us Exit is gonna be plus or minus one once
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we go ahead and move the one over and take
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the fourth of one, All right. And actually
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, let me go ahead and get out of this
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. Exited with a negative one and ex physical one
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. Now what we want to do is figure out
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where it is increasing with decreasing. So why crime
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would be larger than zero. So just for remedy
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of problem, I went ahead. And Artie soul
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this so this inequality will hold on negative infinity to
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negative one and one to infinity. So remember this
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here tells us it is increasing. Next, we
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want to find where the function is decreasing, which
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were being warned. Why pride? Strictly less than
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zero decreasing and again already one had solved this.
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And it should just be negative. One too one
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. All right, so on negative one toe one
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. We know our function is gonna be decreasing.
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So over here, I'm just gonna go ahead and
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write this. So it's decreasing from negative 1 to
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1. And it would be increasing before negative one
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, an increasing after one. So that's telling us
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that except the one is a minimum and exited with
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a negative one as a maximum. Remember both of
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these being local. All right. Next it says
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we want to determine the comm parity implants of inflection
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. Oh, come on, cavity on inflection points
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. So this means we're gonna want to look a
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second derivative. So why double prime while this one
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, the derivative respect X o our first derivative,
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which is five x before my life. So once
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again to take the dribble of X reports, it's
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our rule. So would get five times 4 20
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x to the third power, and then the driven
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five is just zero. And let's go ahead and
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submissive zero insult that really quickly. So zero.
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Well, that's just gonna be X is equal to
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zero. Now we can go ahead and do the
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same thing and look for where. Why double kind
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of Florida's zero, which would tell us our functions
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can keep up there and then look for worldwide about
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time. It's less than zero, which would be
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con cave down. So live double prime zero or
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X cubed is gonna be larger than zero on zero
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Betty 20 excuse, officials say, and then it
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will be strictly less than zero on negative entity to
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zero. So to the left of zero, this
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is calm down. And to the right, it's
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Kong cable. So that tells us X zero should
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be a inflection point of our ground. So we
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followed all the steps that they suggested we do before
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we start actually trying to graft this. So let's
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first go ahead and just brought down our intercepts.
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So we have the intercept 00 and we have.
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And actually, before we do that, we know
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that this should be an odd function from step three
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. So all I'm gonna do is I'm going to
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only graft the left side of this and by only
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graphical left side, I would know I would just
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reflect whatever I did on the left, over to
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the right. All right, so we will are
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not reflected. All right? But reflected across the
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origin or the line y z x. So on
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the left side, we would just have 10 just
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going to be negative for root of I And what
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other things on the left we have So we know
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from step or that are in behavior should be going
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too negative insanity and we know that we have a
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local Max at X equals negative one. So negative
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one will be right here. And, you know
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, this would be eight Max. All right,
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so that should be everything for the left side of
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this. Now, let's just go ahead and start
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cropping. So our first point of importance is going
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to be our zero here at negative work. Beautified
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. And then after that, we know we're going
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to have a maximum around here somewhere. So what
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, They actually do that in a different color?
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So we're gonna have a maximum, maybe around here
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, and then it keeps on going. But then
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the next thing were of interest is our point of
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infection at exit zero. So at this point,
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you need to make sure those boys from Concord up
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on paper, it should be content down, really
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stressed that here and then on the other side,
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it should start to be con cave. And now
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that we have this symmetry, let's just go ahead
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and put in the line. Why is it ex
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and that we're just going to reflect everything? Um
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, across are not wise, and we're just gonna
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look everything across the board, you know, So
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instead of being a Max moment, one is going
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to be amenable. So we just call it here
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, put our maximum, and then this is going
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to just go up until it crosses the X access
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begin like that. So you could have went ahead
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and plotted in all the other points. But for
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this side here, I'm just using the fact from
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number three that we know this is and odd function
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with symmetry about the