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Fit a cubic equation t0 the following data:12 4.61,63.64,43.42.22.83.8...

Question

Fit a cubic equation t0 the following data:12 4.61,63.64,43.42.22.83.8

Fit a cubic equation t0 the following data: 12 4.6 1,6 3.6 4,4 3.4 2.2 2.8 3.8



Answers

Find a cubic model for each set of values.
$$
(0,-12),(1,10),(2,4),(3,42)
$$

All right, So we're given four different points. We need to figure out the cubic function that they all lie on. Remember when we're given a point that each of these has an ex con? Oh, why So what we can do is we can take those X coordinates and plug them in for the X variable, and we can take the Y value, plug it in for the white. So the first cubic function I could write would be negative. Six equals and the A, B, C and D now turn into the variables and the negative four will go in for the X. OK, so wherever there's an ex, I'm putting in a negative four and I'm leaving a B c and d him for my next equation. The zero will go in for the why and the exes will be negative one for the next point. X is one. And why is negative 16 So negative 16 equals a times one cubed plus B times one squared plus C times one plus D For the last equation, eight equals a times three cubed plus B times three squared plus C times three plus deep. Can we can write a new. So now that we have my four equations with the four unknowns, I can write my matrix equation and my matrix equation would have the constants with coefficients first, then the variables. And then that would all equal the Constance. So I'd have negative four cubed negative for squared negative four and then one the negative one cubed negative one squared negative 11 Then I would have one cubed one squared one and one and then three cubed three squared three and one. And the reason why I'm not squaring them or cubing them is just because the calculator can do that for me. And it's really easy to miss a negative sign or mess up some simple math and throw your answer off at the end. So instead, well, just that the calculator do that lifting for us, we're gonna multiplied by the variable matrix and then we'll have the constants. Negative 60 negative. 16 and eight. Okay, so the coefficient matrix is a The variables is X, and the constants are being so a X equals B. So what I can do to solve for the variable matrix is to take a inverse and multiplied by day, Multiply by B. So on my calculator, I'll have a new matrix. That's four by four. And I'm gonna type in those numbers, making sure that if I type those in, I'm using parentheses. Okay? And we're gonna do this for the entire matrix. Okay? And then one cube, which for one's one cubed one squared one and one. Those aren't so bad. So we can go ahead and type close in as just what they are. Okay? And then we have a new matrix, that is four rows, One column, and that one looks like negative. 60 negative. 16 and eight. I'm gonna take a inverse and multiply by by beef came. So I found my coefficients. They are 12 negative. Nine and negative. 10. So my cubic function would look like one x cubed. Plus two X squared, minus nine X minus 10. Okay,

Hello, everybody. Welcome. Uh, we want to find a cubic model for each set of values. So pretty much what we have here is five, um, different coordinate points, each of each of which followed the same rule. So we just want to figure out what that rule is and create a equation, but it's off it. So, uh, we want to see how X right. So this X is being manipulated to get FX, or why? So, um and we know it's also going to be a cubic model, so we know it's going to be X Cube. Somewhere in there, we're gonna get F X is equal to X cube, and this is going to be a shifter drawing a bit, but we know there's going to be an excuse because the question tells us is going to be a cubic model. So, uh, let's just, uh, tube negative to for a moment. So we get negative to Cube is equal to negative eight. Right. So now what happens to negative eight to make it negative? Seven. You have to add one. Correct. Be negative. Eight plus one. I'm sorry. I mean, just rewrite this elsewhere. We get negative eight plus one is able to negative seven. And, uh, that is actually also the why value in the corner point. So when we add one to our equation and see if this works for the other points So this is our equation. So let's choose another point. Not negative to God. My negative seven. Let's just say, um, to common nine. So why don't we find what f off, too, will be if we use our equation? Well, in lackeys, you have effort to is able to cube plus one right to cubed plus one f of two is equal to nine. And what's the recording of your two common nine right to common? Nine. And so this just shows us that this was a good way of checking to make sure that our equation actually works. So our final equation our final cubic function is X cubed. FX is able to x cubed plus one

So this is a calculator exercise, and we're trying to find ah cubic function that goes through these points. So in your calculator, most likely most of my students have a t I product a t i 83 t I 84 you want to go to stay at at it and you want to put in your X coordinates Negative three negative to negative one and zero. And then you wanna put in list to the corresponding y coordinates 91 84 93 and 100. And then after you enter those you could go to stat and calculate and you want cubic wreg mhm. And then you hit enter and it should spit out to you. Why equals a X cubed plus B X squared plus c x plus d and then it gives you the coefficients. It should tell you that a is negative. Three. That be is negative. 10. That c is zero and that d is 100. But then it also should print out that an R squared is equal to one. And that's important that that's one. If this is something lower than one, that means that all the data does not lie on an exact cubic function. But for our case, it does show one which means all of these points are exactly on this particular cubic function, and that cubic function can be written as negative three X cubed minus 10 X squared. We don't have to write plus zero c R plus zero acts you may and then plus 100. So there is our cubic function that contains all of these four points.

Were given a graph of this cubic polynomial and we're gonna make a cubic polynomial actually fit to this graph. And so how we're gonna do that is we're gonna have to use the known information from this graph, which is just the intercepts. So we know that there's an X intercept at negative 21 and five. So that's when are y is going to be equal to zero at negative 21 and five. So we since we know that we can set up a cubic polynomial already where we have this A. Which since we don't know exactly what um this constant is, we're gonna say it's a but we know that we're gonna have zeros at negative two. So we have X plus two, 01 so X minus one and a zero at five. So we have X minus five. And the reason that we know this is because if we plug in negative two, we're gonna have zero here and zero times anything is zero. So we'll have y Is equal to zero and the same will happen if we plug in one or five. So we have this part of our equation already set up now we just need to figure out what it is. And we're going to do that by using this y intercept at two. So at two we know that X. Is equal to zero. So we can just plug in this point into our equation and solve for A. So we have to it's equal to a. Of zero plus two is just too zero minus one is negative one and zero minus five is negative five. So we have two is equal to two times negative one times negative five is 10. So we have two is equal to 10 A. So a. If we divide by 10 A. is equal to 1/5. So now we just have to plug this back into our equation, which is why equals 1 5th X plus two x minus one X -5. And we can just leave our equation like this and now we have found a cubic polynomial that fits our graph.


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