Find the cubic function f(x) = ax³ + bx² + cx + d for which ƒ(...

Question

Find the cubic function f(x) = ax³ + bx² + cx + d for which ƒ( − 1) = 0, ƒ(1) = 2, ƒ(2) = 3, and ƒ(3) = 12.

Accepted Answer

Welcome back. I am so glad you're here. We are asked to identify this cubic function. We're told that it's a X cubed plus be X squared plus C. X plus D. And that's all equal to F of X. And so what are we being asked to do here? Well we are supposed to identify a B, C and D. Because we're actually given four different pairs for X and F of X. So that's what we've got here. This first pair, this F of one equals negative seven. Well that one is our X. Term and the negative seven is our F. Of X term. So we could fill that in to an equation A stays but then instead of X cubed we know our X is one. So it's a times one cubed plus B times X is still +11 squared plus C. Times one plus D equals. And our F of X that's equal to negative seven. And then we can do the same thing. We can write a second equation for this f of negative one. That's our acts equals negative 15. So we can fill that in. We have a again times now negative one. Instead of X cubed plus B times negative one squared plus C times negative one plus D equals. And now what it's equal to the F. Of X is negative 15. We can write a third equation with this F of two equals 02 is our X and zero is our F of X. So we have a times now it's two cubed plus B. Times two squared plus C, times two plus D equals and our F of X is zero. And then 1/4 equation here we've got half of negative two. There's our X equals negative 52. That's our F of X. So our fourth equation reads A times negative two cubed plus B, times negative two squared plus C times negative two plus D equals and our F of X again is negative 52. So we've got a system of equations and we recall from previous lessons there are ways to solve four systems of equations. So if you remember those ways, great, if you need a little bit more guidance stick with me we can simplify each of these equations. So looking at just the first one again one cubed. That's one. So one times a. Well that's just A and again one squared is one, so one times B is just B, C times one is just C and plus D equals negative seven, simplifying equation two negative one cubed is negative one. So this is a negative A plus negative one squared is just a one. So plus B and the negative one times C is minus C. Plus D equals negative 15 for equation +32 cubed is eight. So we've got eight A plus two squared is four plus four B plus two times CS two C plus D equals zero and the fourth equation negative two cubed is negative eight. A plus negative two squared is still positive for B minus two, C plus D equals negative 52. So there's our simplified version and we recall that we can turn this system of equations into a matrix. So using the coefficients and constant terms will create a matrix that has four rows to match our four equations. The coefficients of the first equation are +1111, And then the constant term is -7. And then going down to the second equation, we've got negative 11, negative 11 and negative 15. The third row will be 8421 and zero. And the fourth row will be negative 84 negative 21 negative 52. And we recall from previous lessons that we could create or we can work with this matrix and we can use Gauss jordans or Gaussian elimination to solve for A B, C and D. And I'm going to use Gauss Jordan elimination, which means I'm going to get this with a diagonal row of ones with zeros above and below them. So the first column will be one 000. The second column will be 0100. 3rd column will be 0010 in the fourth column will be 0001. Giving us our solutions for a B C and D. And then what we can do is go back up here and fill those in and we'll have our cubic function identified now if you have done some Gaussian and Gauss Jordan elimination, it's quite a few steps if you're confident in that great, go for it. You can check back with your answer by zooming to the end of this. If you need some practice with it, I'll go through the steps to solve it. Or if you miss something along the way, you can check and see where maybe you might have missed something. But this will be a journey. So stick with me. We can do it all right. So, what we want to start off doing is getting a one in the upper left hand corner. And yeah, we already have that. So one steps done. So now we want zeros in the rest of column one. So let's start with row to getting a zero in column one, row two. So we're going to replace row two in order to replace row to actually I'm going to need every inch of space here. So I'm going to write that over just a little more to the left. Okay, replacing row two, we're going to work with the row that has the one in the desired position already for this column, which is row one. And in this case we're just going to add row one to row two and that'll give us a zero in that position for row two because we're replacing road to we can copy over Rose 13 and four. So I'll do that 1111, negative seven 84, -84, -2 1. and -52. Okay. And now replacing road to so adding row one and row two together. One plus negative one is the zero that we want. One plus one is 21 plus negative one is 01 plus one is two. And negative seven plus negative 15 is negative 22. Okay. We have got that step done. Now working in column one we want a zero in row three will replace row three. And in order to do that, we'll work with the row that has the zero or the one in the desired position, which is row one will take row one and multiply it by negative eight and then add that to row three. Because we're replacing row three, we can copy over rows 12 and four. So I'll do that 1111, negative 70 to to negative 22. And negative 84, negative 21, negative 52. Now replacing Row three one times negative eight is negative eight plus eight is the zero that we want. One times negative eight is negative eight plus four is negative four, one times negative eight is negative eight plus two equals a negative 61 times negative eight is negative eight plus one is negative seven, negative seven times negative eight is a positive 56 plus zero is positive 56. Okay, continuing with column one. We want a zero in this position. In row four will replace row four. So to do that, we work with the row that has the one in the desired position which is row one will multiply row one times eight this time and add that to row four. Because we're replacing row four, we can copy over rows one through 31111, negative 70 to 0. To negative 22 zero, negative four, negative six, negative seven and 56. And now replacing row 48 times one is eight plus negative eight is the zero that we want. Column one is done eight times one is eight plus four is 12. 8 times one is eight plus negative two is times one is eight plus one is 98 times negative seven is negative 56 plus negative. 52 is negative. 108. Okay, now we want to get the one in the desired position for column two. So that's where this too is we're replacing row two. In order to get a one there will multiply everything in row two times its reciprocal. So times one half one half times row two. Because we're replacing road to we can copy over rose 13 and 41111 negative seven, skipping row 20, negative four negative six negative 7, 56 and 0 69. 100 negative 108. Okay. And now road two times one half, zero times one half is 02 times one half is one and again zero and one a negative 22 times one half is negative 11. Okay, now we want zeros in the rest of the positions in column two. So we can start with row one getting row one replaced, getting a zero in that position. We work with the row that has the one in the desired position which is row two. And in this case we can multiply row two times a negative one and add that to row one to get a zero in the desired position. Because we're replacing row one we can copy over rose to three and four. I'll do that. 0101, negative 11, zero negative four negative six, negative 7 56. 0 12 69, negative 108. Now replacing row 10 times negative one is zero plus one is still 11 times negative one is negative one plus zero is the 01 times negative one plus one is the zero that we want. Ye zero times negative one is zero plus one is still 11 times negative one is negative one plus one is zero and negative 11 times negative one is positive 11 plus negative seven gives us positive four. Okay, still getting zeros in column two. We've got this negative four in row three. So we'll replace row three. We'll do that by working with the row that has the one in the desired position which is road to will multiply row two times four And then add that to row three. Because we're replacing row three, we can copy over rows 12 and 4101040101, negative 11 and 0 12 69, negative 108. Now replacing row three, zero times four is zero plus zero is 01 times four is four plus negative four. Is the zero that we want zero times 40 plus negative six is still negative 61 times four is four plus negative seven is negative three, Four times negative 11 is negative 44 plus gives us a 12. Alright. And one last position in column two where we need a zero, that's row four To get a zero in that position will work with the road that has the one in the desired position which is still row two will take row two times a negative 12. And add that to row four. Because we're replacing row four, we can copy over rows 12 and 3101040101, negative 11, 00 negative six, negative three and 12. Now replacing row 40 times negative 12 is zero plus 00, one, times negative 12 is negative 12 plus 12 is the zero that we want ye zero times negative 12 is zero plus six is +61 times negative 12 is negative, 12 plus nine is negative three negative 11 times negative 12 is positive 132 plus negative 108 equals 24. Now we are looking at column three and we want to have our one in our desired position in that diagonal. So that's the third row will be replacing the third row in order to get a one there. We're going to multiply row three times the negative reciprocal of that negative six so that it will become a positive one. So multiplying row three times negative 1/6. Because we're replacing row three we can copy over rows 12 and 4101040101, negative 11 006, negative 3 24. And replacing row 30 times negative 1/6 is zero both times negative six times negative 1/6 is the one that we want negative three times negative. 1/6 is going to be a positive to know it's not it's a positive one half. Excuse me and 12 times negative 16. That will be a negative two. Okay, now we need to get zeroes in the rest of column three. So let's start with row one. We need to have a zero where that one is to get a zero there we work with the row that has the one in the desired position which is row three. And in this case we'll multiply row three times a negative one and then add that to row one. And because we're replacing row one, we can copy over rose to three and 40101, negative 11 half negative two, 006, negative 3 24. And now replacing row 10 times negative one is zero plus one is 10 times negative one is zero plus zero is 01 times negative one is negative one plus one is the zero that we want one half times negative one is negative one half plus zero is negative one half and negative two times negative one is positive two plus four is six. Okay, one more position in column three where we need a zero, that's going to be in row four. So replacing row four, We'll get a zero there by working with the row that has the one in the desired position, which is row three will multiply row three times negative six And then add that to row four. Because we're replacing row four, we can copy over. Call rows 12 and negative one half 60101, negative 11 and 0011 half negative two. Now replacing row four, zero times negative six is zero plus zero is zero, both times one times negative six is negative six plus six is the zero that we want one half times negative six is negative three plus negative three is negative six negative six times negative two is positive 12 plus 24 is positive, 36. Okay, now looking at column four, we want our one in our desired position, which is where this negative six is in row four. So again, we're replacing row four In order to get a one there, we're going to multiply it by its negative reciprocal. So this will be times a negative 1/6. Because we're replacing row four, we can copy over rows 12 and three. I'll do that 100 negative one half 60101 negative 11 0011 half negative two. And now replacing row four, zero times negative 1/6 is zero, all three times negative six times negative 16 is the one that we want and 36 times negative 1/6 is negative six. Okay, we're getting there now, column four, we just need to get zeros in the rest of the spots. So let's work on row one, replacing row one to get a zero. There will work with the road that has the one in the desired position which is row four will multiply row four times one half. And then we will add that to row one because we are replacing row one. We can copy over rows 23 and 40101 negative 11, 0011 half negative two and negative six. Now replacing Rule one zero times one half is zero plus one is 10 times one half is zero plus zero is zero, both times one times one half is one half plus negative one half is the zero that we want. And negative six times one half is negative three plus six is positive three. Okay. Just two more. Just two more. Okay, we want a zero in this second row of column four. So we'll be replacing row two In order to do that. We work with the road that has the one in the desired position which is still row four. And real quick. I'm just going to draw a line here because these are almost kind of bleeding together. Okay and taking row four to get a zero in row two. Let's multiply row four times negative one. So negative row four plus row two will give us a zero in the desired position. Because we're replacing road to. We can copy over rose 13 and four. So 100030101. Oops. That's the role that we're replacing. I apologize, row 30011 half negative two and 0001 negative six. We are so close. Okay. Doing the work. Zero times negative one is zero plus zero is 00, times negative one is zero plus one is 10, times negative 10 plus 001, times negative one is negative one plus one is the zero that we want. Ye a negative six times negative one is negative one is positive six plus negative 11 is negative five. one More. We can do this third row of column four to get a zero in that position. We're going to work with the row that has the one in the desired position which is still row four. We're going to take row four and we'll multiply it by a negative one half And then add that to row three. And because we're replacing row three we can copy over rows 12 and four. So negative five and negative six. Now replacing Row three zero times one negative one half is zero plus zero is zero. Both of these times zero times negative one half is still zero. But plus one is +11 times negative one half is negative one half plus one half is the zero that we want. A negative six times negative one half is a positive three plus negative two is a positive one. Look at this this gives us our diagonal ones surrounded by zeros. And we recall from the beginning of this that this gives us our A R. B. R. C. And our D. So remember 3000 years ago when we started we said that we had a x cubed. So that's going to be three X. Cubed and then plus be X squared. Well that B. Is a minus five so it's minus five X. Squared. And then it's going to be a. Plus C. X. So that's going to be a plus one X. Or plus X. And then plus D. That's minus six. All equals R. F. Of X. That's our answer. Now let's bring this back up to the top and actually I'm going to take away that because that'll probably accidentally cross something out up there. Okay getting all the way back to the top we'll get there it will be a mess for a second. Okay now looking at our answer choices this matches with answer choice. A answer choice. A well done. You've done it we'll catch you on the next one.

Find the cubic function f(x) = ax³ + bx² + cx + d for which ƒ( − 1) = 0, ƒ(1) = 2, ƒ(2) = 3, and ƒ(3) = 12.

Key Concepts

Cubic Functions

Using Function Values to Form Equations

Solving Systems of Linear Equations

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