Question 121168: What function represents the data in the following tables?
a) x-0,1,2,3
y-24,12,6,3
b) x-0,3,6,9
y-5,10,20,40
Answer by Edwin McCravy(20055)   (Show Source):
You can put this solution on YOUR website!
a) x- 0 | 1 | 2 | 3
y- 24 |12 | 6 | 3
That means it contains the 4 points
(0,24), (1,12), (2,6) and (3,3)
The polynomial of least degree which has 4
coefficients is a third degree (cubic) polynomial:
y = ax³ + bx² + cx + d
Subnstitute each of those points in:
24 = a(0)³ + b(0)² + c(0) + d
24 = d
12 = a(1)³ + b(1)² + c(1) + d
12 = a + b + c + d
6 = a(2)³ + b(2)² + c(2) + d
6 = 8a + 4b + 2c + d
3 = a(3)³ + b(3)² + c(3) + d
3 = 27a + 9b + 3c + d
So you have the system of 4 equations
in 4 unknowns:
24 = d
12 = a + b + c + d
6 = 8a + 4b + 2c + d
3 = 27a + 9b + 3c + d
Can you solve that? If not post again.
a = -1/2, b = 9/2, c = -16, d = 24.
So the formula
y = ax³ + bx² + cx + d
becomes
y =  x³ +  x² - 16x + 24
or letting y = f(x)
f(x) = x³ + x² - 16x + 24
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This is done the exact same way:
b) x- 0 | 3 | 6 | 9
y- 5 |10 |20 |40
That means it contains the 4 points
(0,5), (3,10), (6,20) and (9,40)
The polynomial of least degree which has 4
coefficients is a third degree (cubic) polynomial:
y = ax³ + bx² + cx + d
Substitute each of those points in:
5 = a(0)³ + b(0)² + c(0) + d
5 = d
10 = a(3)³ + b(3)² + c(3) + d
10 = 27a + 9b + 3c + d
20 = a(6)³ + b(6)² + c(6) + d
20 = 216a + 36b + 6c + d
40 = a(9)³ + b(9)² + c(9) + d
40 = 729a + 81b + 9c + d
So you have the system of 4 equations
in 4 unknowns:
5 = d
10 = 27a + 9b + 3c + d
20 = 216a + 36b + 6c + d
40 = 729a + 81b + 9c + d
Can you solve that? If not post again.
a = 5/162, b = 0, c = 25/18, d = 5.
So the formula
y = ax³ + bx² + cx + d
becomes
y =  x³ + 0x² -  x + 5
or letting y = f(x)
f(x) = x³ - x + 5
Edwin
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