polynomial equation help plzzz?

Question

Determine the equation of the polynomial function that matches the following data set.
x y
1 1
2 -3
3 5
4 37
5 105

Answer 1

you can find the equation without a calculator this way.

First start by charting the successive differences of f(1), f(2), .., f(5). To do this, draw the 'triangle' below:

1 -3 5 37 105 [these are the f(x) values]
. . .-4 8 32 68 [differences between them]
. . . . ..12 24 36 [2nd differences]
. . . . . . . . 12 12 [3rd differences]

The triangle's rows calculate differences between terms from the row above (e.g. 32 - 8 = 24). Once we reach the 3rd difference row, all the terms are the same (i.e. 12). Thus, the original function can be fit with an order 3 polynomial (a cubic).

A general cubic has equation f(x) = Ax^3 + Bx^2 + Cx + D.

To find the specific values of A, B, C and D you will need to solve a system of 4 equations (because there are four variables). The system to solve comes from the data points given in the problem:

f(1) = A(1)^3 + B(1)^2 + C(1) + D = 1
f(2) = A(2)^3 + B(2)^2 + C(2) + D = -3
f(3) = A(3)^3 + B(3)^2 + C(3) + D = 5
f(4) = A(4)^3 + B(4)^2 + C(4) + D = 37

or,

A + B + C + D = 1
8A + 4B + 2C + D = -3
27A + 9B + 3C + D = 5
64A + 16B + 4C + D = 37

--->
7A + 3B + C = -4
19A + 5B + C = 8
37A + 7B + C = 32

--->
12A + 2B = 12
18A + 2B = 24

--->
6A = 12

So, A=2 and back filling we find that B=-6, C=0, and D=5.

the equation is f(x) = 2x^3 - 6x^2 +5.

Answer 2

This is the answer if you don't have a graphing calculator, or aren't supposed to use one.

The First logical thing to do would be to do a very fast sketch to see the profile of the graph. You would notice that there are 2 turning points, which can mean this is a cubic equation, where highest power of x is 3.

So, the answer is in the form:
y = Ax^3+Bx^2+Cx+D

There are 4 unknowns, so we form 4 equations.
When x=1,y=1, 1 = A+B+C+D -------------------(1)
When x=2,y=-3, -3 = 8A+4B+2C+D -----------(2)
When x=3,y=5, 5 = 27A + 9B + 3C+ D -------(3)
When x=4, y=37, 37 = 64A + 16B + 4C + D --(4)

Then, we solve them.
To start off, rearrange (1), D = 1-A-B-C
Substitute this expression into (2), (3), (4).

Yes it's tedious I think.

Answer 3

The following procedure (messy and tedious) will solve it. Suppose that the polynomial function is ax^4 + bx^3 + cx^2 + dx + e. Then, using the numbers given, one can formulate equations such as these:
a + b + c + d + e = 1
16a + 8b + 4c + 2d + e = -3
81a + 27b + 9c + 3d + e = 5
256a + 64b + 16c + 4d + e = 37
and so on. These can then be solved as a linear system by the usual means.

Answer 4

Hi,

Grab your TI83. Go to STAT and enter your numbers in L1 and L2. Then go to STAT, Calc, and choose QuartReg. Hit ENTER and the calculator gives you y = 2x^3 - 6x^2 + 5. If you enter that and check the table, it's perfect!!

I hope that helps!! :-)

Answer 5

2x^3-6x^2+5

Answer 6

2x^3-6x^2+5
Please give me best answer thanks!