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Point) Do the following for the points 3,2) , (-1,1) , (0,-1) , (4,-1) , (5, 1): (If you are entering decimal approximations enter at least five decimal places(a) F...

Question

Point) Do the following for the points 3,2) , (-1,1) , (0,-1) , (4,-1) , (5, 1): (If you are entering decimal approximations enter at least five decimal places(a) Find the equation for the best-fitting parabola y ax2 + bx + for these points:(b) Find the equation for the best-fitting parabola with no constant term y a12 bx for these poinis:(c) Find the equation for the best-fitting parabola with no linear term y ax" + for these points:

point) Do the following for the points 3,2) , (-1,1) , (0,-1) , (4,-1) , (5, 1): (If you are entering decimal approximations enter at least five decimal places (a) Find the equation for the best-fitting parabola y ax2 + bx + for these points: (b) Find the equation for the best-fitting parabola with no constant term y a12 bx for these poinis: (c) Find the equation for the best-fitting parabola with no linear term y ax" + for these points:



Answers

Find the equation of the parabola $$y=a x^{2}+b x+c$$ that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. $$(-1,1),(0,-4),(1,-13)$$

Here we have three coordinate pairs and we're looking for the parable A that would contain all three pairs. So the equation for a parabola could take the form. Why is equal toe a X squared plus B X plus C, where A B and C are constants? And we confined those because of our three port repairs? We could make three equations and we'll have those three unknowns and we'll be ableto find are coefficients. So our first a question equation is a times X squared. In this case, negative one is our ex plus B times negative one plus c equals why, in this case, negative five. Our second equation we're going to use to for the X two x squared plus B times two plus c equals why which here is seven and our third equation. The X is five. So eight times five squared plus B times five plus c equals why, which is one. And we can clean these up a little bit. Negative one squared is one. So we have a minus. B plus C equals negative five on our second equation to swear this four. So for a plus to be plus C equals seven on our third equation five squared this 25 25 a plus five B plus C equals one. Now all three equations The sea has a coefficient of one. So we're gonna eliminate those first. That will be pretty straightforward. So it's called the top equation E one and our next equation e to on our 3rd 1 e three and we'll go ahead and take e one and multiply it by negative one. So we will have a negative A plus B minus C equals positive five and we'll take E to as is so that will be for a plus to the plus C equals positive. Seven. Negative and positive for a leaves us three a one plus to his threes. We have three be the seas got eliminated and five plus seven ist 12. We could divide every single term by three and we get a plus. B equals for now. Let's do the same thing over here using e three instead of e too. So well again, we'll take negative e one, which was negative. A positive be negative. C equals positive five. And when we add these two together 25 minus one is 24 so you get 24. A five plus one is six. So we get six. Be the Caesar eliminated and one plus five of six. If we divide every single term by six, we get that for a plus. Be equals one. And now, if we let's call this equation that we got to below a plus B equals for we'll call that a four and, well, take that and will multiply it by negative one so negative e for well, give us negative. A minus bi equals negative or so four minus one is 33 a. The bees eliminated equals negative three. And if we divide both sides by three, we get a is equal to negative one. So there's our first. Now it could come back over to eat for and say will be equals for minus a, which equals four minus negative one, which equals positive. Five. So B equals five. It's our second, and then we'll use a minus. B plus C will use this equation. You want to determine C so a is negative. One minus B. So minus five plus C equals negative five. So negative one and negative five is negative. Six. If we add six to both sides, we get that C equals positive one. So our solution set is negative. One five positive one. And that's that What we were asked to find. And if you actually wanted to know the equation for the proble, you would say that why equals negative X squared plus five x plus one. In fact, for fun. If you graphed that using a graphing utility and you plot all three points, you'll see that, indeed, they lie on the graph.

And this problem we are asked to determine the equation for a parabola that goes through three points. So remember the general form for probable, it's Y equals a. X squared plus bx plus C. So we're giving these points, we have the 0.-2 -3. So that's why is -3 in that point X is minus two. Well minus two squared, that's positive four. That's four. A X is -2 sets minus to B. Let's see. Were given the 0.-10. So why is zero X is minus one minus one squared is positive one. So that's a X -1. So that's -B. Let's see. And the .1/2 -3. So why is -3 X. Is a half a half squared and be 1/4. So that's 1/4 a. Was one half B. Let's see. Okay, So this is equation one. Call this equation too and we'll rewrite this last equation. I'm multiplying by four so that all the fractions go away from me. The denominators go away So I multiplied by four, I get -12 equals a plus. Well four times a half. That's too. So that's to be was foreseen Mcauliffe's equation three. So this was four times equation Well, let me let me do this. So this be an equation three. We'll call this equation three and call this equation four. So four times equation three went in there. Okay now this means I have three equations, three unknowns. Right? I focus on this equation. This equation in this equation. Okay, so now I want to combine these so that one of the variables drops out. Well let's take equation one minus equation too, So -3 0, that's -3. For a man to say that's three A minus to b minus of minus. So that's plus, so it's minus to B plus B. So that's minus B. And the seas drop out. I'm gonna call this equation five. Okay now I need to do something with equation for we'll get rid of the sea here So let's go -4 times. It's four times equation two So I'll have -4 times zero. That's zero is minus four A. My surmise that's plus for b minus four C. Okay then I'm gonna add Croatian forward to it. That's minus 12 is A. Plus two b. Plus four C. And of course the seas cancel out by design that's what we're trying to do. So when I add these together that's -12 -4 people say that's -3 a. Four B. Plus twos be that's plus six B. We call this equation six then. all right so between between equation five and 6 I want to get rid of one of the variables A. Or B. Well I have three a.m. minus three. A. So all I do is take equation 5-plus equations six. So that's -3 Plus and -12. That's A -15. The A's drop out minus B Plus six B. That's five B Divide by three or 75. So that's -3 is B. So I now have my B. And then go back into equation five or 6 to get A. Now let's go into equation five here that means I have minus three equals three A minus of minus that's plus three. So subtract three From both scientists -6 equals three A. five x 3. So -2 is A. And now I have A. And I have B. And so now I just need to go up period equation 12 or three you get. See equation two looks pretty simple, doesn't it? So I'm going to take my A. In my being into equation too eight months to be S -3. So that means I have zero equals Age -2 -2. B. is a -3. So minus of -3. So that's plus three. Uh Plus see so that's zero equals one. Plus. See because my eyes two plus three is one So -1 is C. And so now I have all three of my coefficients. And so my general form of my equation why equals or my equation by preval sorry A. X. Squared. So it's minus two X. Squared Plus B -3 X. Plus. See why this one. And then we're asked to graph this to verify that these three points are all on this equation of this Pamela. Okay so let's go to our graphing calculator here minus two. Let's see minus to a seat. I need to do. Why equals minus two X. Weird -3 X -1. Okay minus three x minus one. Okay. And we had the points -2, 3. Which is there? Because you can see We have the points -10, which is right there. Sorry. Right There. There we go -10. And we had the point a half minus three. Half minus three. Right there. So all three points are on this problem.

We have given why equals two a x squared plus B x plus c So this is the general equation off parabola, and we have given the points here. So the first pointed the first point here, given 00 and the second point, age three Tzeitel. And the third point is 44 And now we have to find the equation off parabola. So for this, finding the equation off parabola see decision. If this point lies on the but Ebola, so this must satisfy the equation. So for this, here's why that is little so zero x zero also so see, close to zero that when the question number one that they see equals 20 and now see here So why equal for little? And here, this one with a nine e and then see this one of the nine end And here three b c equals zero, so not to write it. And now for the car park. So that is why equals to four. So see why close to four under through 16 it and for be that is here also cuz it'll So we have to solve this for we have to find the value of it and be so for this. So now they're simply nine. A plus three b equals little that wins nine A equals toe. This is minus off three B, or I can say that B equals toe. This is minus three a. Now put the value. This is for 16 and weak was toe minus three. That the minus 12 will give four a equals to food that is equals toe. And because toe that this year minus city so because two minus three and equals toe one So that will give inquisitiveness and C equals 20 will you Why equals two X squared minus three. So this is the question and also we have to trace the car. So the car with here so this could represent y equals two x squared minus three x Also see the point. This is 00 Yes, and now 30 that is extremely and y zero. And also 44 for here is a 440.44 So all points like here, this is 30 and this is the zero. All the points under graph next means are in question. It under 40

So in this problem Were asked to determine the formula for Parabola that goes through three points. Remember the general form of a problem Y equals a X squared plus bx plus C. And were given that it goes to the .20. So why is zero excess too? So that's four A. Because two squared is four plus to be let's see, Goes to the .3 -1. So why is -1 X. Is three. So three square deaths nine A plus X. Is three. That's 3 B. Let's see. And then through the .40 zero equals XS four. That's 16 a. Plus for me. Let's see. Try on the line here. Just to keep our work kind of neat. It's called this equation one. This equation too. This equation three. So now I need to combine these so that I end up with two equations two unknowns. To be to continue solving with. All right, well, let's sing. Okay, I know that I can get rid of the seas pretty easily can't I? So let's go -1 times equation one. Actually instead of doing that let's just do this. Let's take equation one minus equation to how about So 0- of -1. That's a positive one four A -98. That's -5 a. And to beam on his three B. That's minus B. And of course the seas cancel out. I'm gonna call that equation for now. Let's take equation to minus equation three So -1 monastery. That's -1 9. -16 A. That's -7 a. Three be months four B. That's minus B. And the seas dropout by design Only call that equation five. Okay now let's do equation four minus equation five because then the B will drop out for us so four minus five so that's one minus of minus once. That's one plus one. That's two -5- of -7- and- has missed plus. So that's minus five plus seven. So that's two A. And the bees cancel out. Bye Bye to. So one is a. So one equals A. All right now Take a equal to one. Put that back in either equation for or equation five doesn't matter which one I am I going to equation for. So one equals a. Is once that's -5. Let us be had 5 to both sides. So six equals -B. And so -6 equals b. And there is my B. Now I want to take both of these in the equation 12 or three doesn't matter which one and determine the value for C. So let's just choose equation one. So that means zero equals four. A. Well A. Is once that's four plus To be be is -6. So that's -12 which means that's actually going to be subtracting as to the adding now for us plus C. So zero equals will four minus 12. That's minus eight plus C. moved it to the side by adding it so eight equals c. and so now I have All three of my coefficients. So my equation is why equals X squared -6 x eight. Okay. And then we're asked to graph this to verify goes through those three points. So in our graphing calculator we go why equals X squared minus six X. What's eight? Okay. And so we went through the point 20 which is right there, 3 -1 is right there And 40 which is right there. So there we go went through all three points.


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