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12. Determine the equation of the polynomial function shown in the graph below: The curve passes through the point (3, 12/. Write the equation in its factorized for...

Question

12. Determine the equation of the polynomial function shown in the graph below: The curve passes through the point (3, 12/. Write the equation in its factorized form:(x) = ax} bx? 2x + 12 The curve defined by the function intersects the X-axis at three points_ axis at three points: Two are known, and -3 Determine the equation of the function_

12. Determine the equation of the polynomial function shown in the graph below: The curve passes through the point (3, 12/. Write the equation in its factorized form: (x) = ax} bx? 2x + 12 The curve defined by the function intersects the X-axis at three points_ axis at three points: Two are known, and -3 Determine the equation of the function_



Answers

Find the equation and sketch the graph for each function. A quadratic function that passes through $(1,2)$ and has $x$ -intercepts $(-1,0)$ and $(3,0)$

But this problem. We are using the guidelines for graphing polynomial functions in order to graph this equation. The first step is to identify end behavior, and for that we need to identify the degree. The degree is simply the exponents of the X term that's raised to the highest power here. It's three. So we have two great three now, the way we identify in behavior from that, as we say, OK, if the degree is even in the end, behaviors are in the same direction as each other. If the degree is odd, they're in opposite directions from each other. Okay, so here we have already identified degree three. It's not. They are in opposite directions of each other, but we still need to know whether another graph goes up on the left or dental left her up on the right or down on the right. That's where the leading coefficient comes in. Here and here. We don't see anything, Meaning it's the implied positive one. All right, So what that tells us is that if the leading coefficient calling that egg, if it's positive, then it's left, goes down and right it goes up on the other hand. If a is negative left will go up and right will go down. All right, So we identified The leading profession is positive. One mean and behavior is left, goes down and right. Goes up. Okay, so then the next step is to identify our Why intercept, which is Says, what happens to why, when x equals zero. So I'm just gonna substitute zero for each extra term that simplifies to positive 12. I'm sorry. So then are weiners up to zero comma 12. Next up is identifying are zeros. So when we can do this is looking at the coefficients. And if all the coefficients add up to zero, then one is one of our roots. All right, so are coefficients one minus 13 plus 12. That indeed equals zero. So X equals one. It's one of our roots. Next step, we're gonna use, um, synthetic division to identify the rest of the roots. Okay, so we already identified one. We're gonna use that number zero, cause we know don't have an ex term that raised to the second power. The one just drops down. One most part by one is one at this together one negative. 12 in syrup. It's the resulting equation. Here, then, is X squared plus X minus 12. So then we need to numbers that multiply to get negative 12 and add together to get positive one. That means we need X plus four multiplied by X minus three. Remember, standard form is X minus C to the power of him. So subtraction is the operation Sea is zero and M is the multiple city. With that in mind are zeros. Then our X equals negative. Four index equals positive. Three All of these air raised to the first power They all have odd multiplicity, ease meaning at each of the zeros The graph who crossed the X axis at that point. Now we congrats, thes So why intercept zero positive 12. That's not really on my graph, but this will give us a rough idea. And then our zeros positive one negative four and positive three. Okay, so that gives us some information. We still need to know what's happening on the mid intervals or the areas between the zeros, the way we're gonna soul that he's we're gonna look at, OK, what happens when X equals two. So, like we did before. Substitute to in for each of our X terms and we get negative. Six. Graphing that to negative six. All right, so now we need to know. OK, well, what happens? It between our weiners up to a negative four. Okay, thanks. Equals negative one like that in and we get Who positive? 24. It's a bit of a game changer. Great. Made it clear that out, in a way, just fixing this so it shows a little bit more of how this is gonna happen. That's so that's 24 right there. That's 12. Okay, so then what happens at negative two and negative three? Right? So I'm not gonna write that out. Could we know the procedure? But negative two ends up being positive. 30 on the negative. Three results in positive 24. Okay, so let's reader this top part to reflect that. So let's say this is our 12 this or 24. That could be our 30. And that's our 24 then. Okay, Now we have sufficient points, remember? And behaviors on the left is down. Right is up. The graph crosses the X axis at each of our zeros, so working from negative to positive, we can begin. So after this zero and negative for the graph is just gonna continue down because it won't touch the x axis again and then continues up. So now we've connected those points. We're going back down to our zero at positive one and then down to the next point at negative six AM back up toward 03 and then our end behavior is ups, we continue.

But this problem you're using guidelines but crapping polynomial functions in order to grow up. What first piece of information that we need here is the end behavior in the degrees going up, assault the greens, the highest power of the exponents of the highest powered here That's three. So when the degree is an odd number, that indicates that the invaders are in opposite directions from each other on the left, in the light. But that still doesn't tell us who left his upper. Left it down, right? It's up. Raised him. That's where we need the leading coefficient, the politicians off the highest power bets, more fortune. That is a year. So if a it's positive but greater than zero and left me down right will be up. On the other hand, if a is less than zero, I'm sorry this was supposed to be greeted with If they has negatively lessons here, we love to be light will be down going back to our equation. We see that there is nothing in front of the execute implying the positive one. Therefore, we haven't end behavior of left down, right? Next piece is why intercept what happens to Why? When x people's syrup Cynthia Cleese equal to why? And substitute zero for each Exton zero multiplied by anything is zero see ever. Why interesting? Uh, zero. Well, I go ahead and grab that. It's gonna run out a little bit of my graph. That's okay. I'm gonna try and keep it to scale them. Okay, Next is our Ciro's. So the first thing you do to test my zero sc a positive one is a zero. The function the way I do that is by adding together the coefficients of the equation to have one minus 13 plus 12 which does, in fact equals zero. Therefore, X minus one is aware of the bunch, except this one. This is your what's out on the graph. And then I'm gonna utilize that to continue solving for more zeros using thin synthetic division positive one. No input our whole efficiency of one. And then a place wanted Brexit with Don't forget about your sign to you. Correct? All right. First term, drop straight down. 11 multiplied by one is 101 There's you continue that pattern, but t negative 13 plus one is negative. 12 and then zero means that one is in vector route. In this division leaves us with the new polynomial X squared plus six minus 12. Okay, we need to introduce that multiply to get negative 12 and add together to equal positive one. What comes to mind? Three is negative for were actually positive. Foreign. Negative three now remember? Okay, here we go. Positive for warmed Negative three. Intervene. Remember standing form years X minus C to power in operation subtraction. Therefore, this shows us, but the zeros are negative. Four and positive three isn't that I can grab those negative four and positive Now I need a few intermediary points. Like, let's see, what happens at X equals negative too. And what about X equals negative three and for that matter, negative one. And another one that we need is positive too. We follow the same procedure that we did over here to solve for wires. And then you show all these, but I will tell them in. So when negative two is our X value. Why is 30 but make it three or while value is 24 minutes One again, it's 24. Which makes sense once we put these on the ground and the white for positive. Two is negative. Six. Okay, so now we need to grab all of these. Positive. Too. Negative. Six negative. Serene will be positive. 24. So this is not just feel anymore, but you need a place with It's the negative. Two is gonna be even hired left. The negative one is back to our your medical 24. Then I remember anything you love to be down bright toe up and I'm ready to sketch or grab beginning at negative four. My work my way up to 24 and then back down to my wife. Is it down to positive one? Make it six up to a final zero. Can I continue on my end behavior? And I want to go back to negative for Continue back down for that year and then I'm just gonna go back over this one more time and smooth it out. So we have a nice grab. It's not perfect, but it's there. And then don't forget your arrows. This does continue

Again. We can use the formula of X equals minus B. Move it to a which is equal to 12 over six, which is equal to two. Now. We know that why is equal toe f of to which is equal to 12 minus 24 plus one, which is minus 11. So we have the coordinates off of the attacks being two comma minus 11.

Don't we have a problem? Number 14 invents. Look, we need to find the work. Did so better. Well, what was your question? It is a X squared minus 12 x. Listen, Mr we didn't do, you know, wanting that everything quantity question off for me X squared plus B explicitly or any form people into Paraguana. It was work picks, uh, exported. That is the parabola were prevented by this. Who's you support? Text one sick where things is always a minus bi bi the way and value of this at minus beal work the way. So if you compare these two in question the street vision being get the value of a equal to well, if people three value being acquittal minus trail better see, it was toe. So the xcor metal verdicts will be minus B over. Doing that is minus minus Tread who were cruelly to cream? It really drove over six. That is toe. So why cordon it off work? It must be after that. We need to play in the well. You have to in given equation, which is three x squared, minus treatment plus one. So three with good minus 12. Do do Glassman it because to a square four into three. Trail minus 24. Listen, that is 13 minus 24. It is minus 11. So no, we have word eggs. He will do do and minus We live in. Thank you very much.


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