Consider the following four ordered pairs: (1, 6) (2, 11) (4, 39) (7, -54). Each of these ordered pairs represents an (x, y) pair. Let y = f(x). We can also think of each pair as an (x, f(x)) pair.
Let’s say we wanted to find a third-degree polynomial that fits these ordered pairs. You could substitute the values of the ordered pairs into this equation:
f(x) = A + B(x – xo) + C(x – xo)(x – x1) + D(x – xo)(x – x1)(x – x2), where A, B, C, and D are constants.
In the equation above, xo, x1, and x2 represent the x-coordinates of three of the ordered pairs that are given. In this case we will let xo=1, x1 =2, and x2=4.
By substituting the known values from the ordered pairs into the equation, we can find the constants A, B, C, and D.
Substituting (1, 6) in the equation by letting x=1 and f(x)=6, after doing the math and simplifying, we obtain 6 = A.
Substituting (2,11) in the equation, 11 = A + (2 – 1)B; so 11 = A + B.
Substituting (4, 39) in the equation, 39 = A + (4 – 1)B + (4 -1)(4 – 2)C; so 39 = A + 3B + 6C .
Substituting (7, -54) in the equation, we get -54 = A+ (7 – 1)B + (7 – 1)(7 – 2)C + (7 -1)(7 -2)(7 -4)D; so -54 =A + 6B + 30C + 90D.
The value of A is ____.
A)5
B)7
C)3
D)6
The value of B is ____.
A)7
B)6
C)4
D)5
The value of C is _____.
A)5
B)6
C)7
D)3
The value of D is _____.
A)-2
B)-3
C)3
D)2
Which polynomial function best represents the four ordered pairs given in the example?
Reminder: The ordered pairs are (1, 6); (2, 11); (4, 39); and (7, -54).
A)x3 + 8x2 + 10x - 1 = f(x)
B)x3 - 4x2 + 10x - 1 = f(x)
C)-2x3 + 17x2 - 32x + 23 = f(x)
D)-2x3 - 17x2 - 23x + 32 = f(x)
Janine, a science student, measured the volume of carbon dioxide gas that can be absorbed by one cubic centimeter of charcoal at pressure x. She made the following four sets of measurements:
(120, 3.1)
(340, 5.5)
(534, 7.1)
(698, 8.3)
Janine finds that the volume, f(x), to be a function of pressure, x, and decides to use the formula discussed earlier in this project to fit her data. Recall that the formula is:
f(x) = A + B(x – xo) + C(x – xo)(x – x1) + D(x – xo)(x – x1)(x – x2)
where A, B, C, and D are constants, while xo, x1, x2 are the x-coordinates of three of the data points given.
Following the method similar to first example, when we use Janine’s data, what would the value of A be?
A)3.1
B)120
C)5.5
D)340
Using Janine's data, the value of B is ____.
A)3/275
B)220
C)12/110
D)3/550
Select the polynomial function that includes these four ordered pairs.
(8, -207); (12, 169); (15, 976); (20, 3801)
D) f(x) = x3 - 10x2 - 10x + 1
