Answer:
a = 3 b = -18 c = 11 d = 0
Step-by-step explanation:
The table tells you what value the function will product (y) if you input the corresponding value of x.
So, for example, if you input the value of x = 1 into the function, you will get y = -4.
If we substitute x = 0 and y = 0 into the function we can find the value of d immediately:
0 = a(0)³ + b(0)² +c(0) + d
0 = 0 + 0 + 0 + d
0 = d
As d = 0, we can now eliminate d and write the function as:
y = ax³ + bx² + cx
Now substitute the remaining pairs of values into y = ax³ + bx² + cx to create 3 equations:
1) When x = 1, y = -4
⇒ a(1)³ + b(1)² + c(1) = -4
⇒ a + b + c = -4
2) When x = 2, y = -26
⇒ a(2)³ + b(2)² + c(2) = -26
⇒ 8a + 4b + 2c = -26
⇒ 4a + 2b + c = -13
3) When x = 3, y = -48
⇒ a(3)³ + b(3)² + c(3) = -48
⇒ 27a + 9b + 3c = -48
⇒ 9a + 3b + c = -16
Now solve the system of 3 equations simultaneously to find the value of a, b and c. To do this, reduce this to 2 equations with 2 unknowns by eliminating one of the unknowns from two pairs of equations:
Eliminate c by subtracting equation 1) from equation 2) to create equation 4):
2) 4a + 2b + c = -13
1) a + b + c = -4
4) 3a + b = -9
Eliminate c by subtracting equation 2) from equation 3) to create equation 5):
3) 9a + 3b + c = -16
2) 4a + 2b + c = -13
5) 5a + b = -3
Now use equations 4) and 5) to find a and b, by subtracting equation 4) from equation 5) to find a:
5) 5a + b = -3
4) 3a + b = -9
6) 2a = 6
⇒ a = 6 ÷ 2 = 3
and substituting the found value of a into equation 4) or 5) to find b:
5) 5(3) + b = -3
15 + b = -3
⇒ b = -18
Now substitute the found values for a and b into one of the original 3 equations to find c:
1) a + b + c = -4
3 - 18 + c = -4
⇒ c = 11
Therefore, the function is y = 3x³ - 18x² + 11x
