# Write a cubic function that passes through the points (-2:0) (2:0) (6:0) (3:15)... please explain your work.

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The general cubic function is y = ax^3 + bx^2 + cx + d.

This function passes through the points (-2,0), (2,0), (6,0) and (3,15).

Substituting the values of the x and y coordinates from the points in the cubic function gives:

0 = a*(-2)^3 + b*(-2)^2 + c*(-2) + d

=> a*-8 + b*4 + c*-2 + d = 0 ...(1)

0 = a*(2)^3 + b*(2)^2 + c*(2) + d

=> 8a + 4b + 2c + d = 0 ...(2)

0 = a*(6)^3 + b*(6)^2 + c*(6) + d

=> 216a + 36b + 6c + d = 0 ...(3)

15 = a*(3)^3 + b*3^2 + c*3 + d

=> 27a + 9b + 3c + d = 15 ...(4)

(1) + (2)

8b + 2d = 0

=> d = -4b

3 - 3*(2)

192a + 24b - 2d = 0

Substituting d = -4b

192a + 32b = 0

6a + b = 0

Substitute a = -b/6 and d = -4b in 8a + 4b + 2c + d = 0

(-b/6)*8 + 4b + 2c - 4b = 0

-4b + 12b + 6c - 12b = 0

c = (2/3)*b

Substitute d = -4b, a = -b/6 and c = (2/3)*b in 27a + 9b + 3c + d = 15

27*(-b/6) + 9b + 3*(2/3)*b -4b = 15

=> 9*(-b/2) + 9b + 2b -4b = 15

=> -9b + 18b + 4b - 8b = 30

=> 5b = 30

=> b = 6

As a = -b/6, a = -1

c = (2/3)*b = 4

d = -4b = -24

**The required cubic function is y = -x^3 + 6x^2 + 4x - 24**

The points (-2:0) (2:0) (6:0) are x-intercepts and, thus, "zeros" of the function. That means that:

from (-2,0), "x - (-2)" is a factor for the function, or "x+2"

from (2,0), "x - (2)" is a factor for the function, or "x-2"

from (6,0), "x - 6" is a factor for the function, or "x-6"

Multiplying those factors together, we get:

(x-2)(x+2)(x-6)

(x^2 - 4)(x-6)

x^3 - 6x^2 - 4x + 24

So, "so far", we have:

y = x^3 - 6x^2 - 4x + 24

And, if you plug in -2, 2, or 6 in for x, you will "0" for y each time.

Now, for the fourth point, (3,15). If you plug in the 3, we get "-15" for y. But, we need "+15". We can't just add 30 to the 24, getting "54" at the end of the function. Because, then, the first 3 points don't work. So, we have to multiply the right side, in effect "each side", of the function by something, not unheard of since the general form for a cubic function is "y = ax^3 + bx^2 + cx + d", and we have 1 for "a" right now, and that isn't working. But, if we multiply each side by -1, then we get our fourth point, (3,15), and the first 3 points still work.

So, the function is:

y = (-1)(x^3 - 6x^2 - 4x + 24)

= -1x^3 + 6x^2 + 4x - 24