Question

Use cubic regression to find a function that fits the following points.

Answer 1

y = ax³ + bx² +cx +d (Main eq.)

Solve for a, b, c and d.

Solution:

y = 2x³ + x² -4x +6

Explanation:

i) Solving for d:

Given point (0,6), at this point your x value is 0 and your y value is 6. Substitute the values on the Main Eq. above and you will find the d value to be 6.

ii) Solving for a, b & c.

Using points (-1,9),(1,5) and (2,18)

Substitute the given points on the Main Eq. above and you will have 3 equations with 3 unknown variables and solve the equations simultaneously to find your a, b & c.

Answer 2

y = 2x³ + x² - 4x + 6

Explanation:

General formula:

y = ax³ + bx² + cx + d

If you replace point (0, 6) in the general formula, you get:

6 = a(0)³ + b(0)² + c(0) + d

6 = d

If you replace points (-1, 9), (1, 5) and (2, 18) in the general formula, you get the following system of equations:

9 = a(-1)³ + b(-1)² + c(-1) + 6

9 = -a + b - c + 6 (eq. 1)

5 = a(1)³ + b(1)² + c(1) + 6

5 = a + b + c + 6 (eq. 2)

18 = a(2)³ + b(2)² + c(2) + d

18 = 8a + 4b + 2c + 6 (eq. 3)

Adding equation 1 to equation 2:

14 = 2b + 12

(14 - 12)/2 = b

b = 1

Multiplying equation 1 by 2:

18 = -2a + 2b - 2c + 12 (eq. 4)

Adding equation 4 to equation 3, and replacing with b value:

36 = 6a + 6b + 18

36 = 6a + 6 + 18

(36 - 6 - 18)/6 = a

a = 2

Replacing a and b values in equation 1:

9 = -2 + 1 - c + 6

c = -2 + 1 + 6 - 9

c = -4