Question

What is the cubic function with these four points: (2,5)(3,7.2)(6.3,7.8)(9,10.7)?

Answer 1

You can represent a cubic equation in the form \(y = ax^3 + bx^2 + cx + d\)
By plugging the four sets of coordinates you're given into x and y, you can get four linear equations, which you can solve to find a, b, c, and d. Do you follow?

Answer 2

yeah ive gotten that far i dont know how to find a b and c

Answer 3

and d

Answer 4

The same way you solve any other system of linear equations.

Answer 5

What are your equations?

Answer 6

you use the template of \(\bf y=ax^3+bx^2+cx+d\) and use each point (x, y) values to obtain 1 equation

you have 4 points, thus 4 equations, and you'd end up with a system of equations of 4 variables, then you'd eliminate to find the values

Answer 7

5=8a+4b+2c+d
7.2=18a+9b+3c+d
7.8=250.05a+39.69b+6.3c+d
10.7=729a+81b+9c+d

Answer 8

Ok, so I wouldn't waste time messing around and just use matrices, unless you're supposed to crunch this out manually for some reason.

Answer 9

i dont know how to use matrices

Answer 10

\[\left[\begin{matrix}8 & 4 & 2 & 1 \\ 18 & 9 & 3 & 1 \\ 250.05 & 39.69 & 6.3 & 1 \\ 729 & 81 & 9 & 1\end{matrix}\right]\left[\begin{matrix}a \\ b \\ c \\ d\end{matrix}\right] = \left[\begin{matrix}5 \\ 7.2 \\ 7.8 \\ 10.7 \end{matrix}\right]\]

Answer 11

Which means
\[\left[\begin{matrix}a \\ b \\ c \\ d\end{matrix}\right] = \left[\begin{matrix}5 \\ 7.2 \\ 7.8 \\ 10.7 \end{matrix}\right]\left[\begin{matrix}8 & 4 & 2 & 1 \\ 18 & 9 & 3 & 1 \\ 250.05 & 39.69 & 6.3 & 1 \\ 729 & 81 & 9 & 1\end{matrix}\right]^{\huge-1}\]

Answer 12

Don't ask me how to take the inverse of a 4x4 matrix. Calculators are good at it.

Answer 13

okay