Use cubic regression to find a function that fits the following points. $$ (-2,-16)(1,5)(3,59)(6,440) $$ $$ 2 x^{3}+[?] x+\square $$

Question

UpStudy AI · Accepted Answer

Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}a\left(-2\right)^{3}+b\left(-2\right)+c=-16\a\times 1^{3}+b\times 1+c=5\a\times 3^{3}+b\times 3+c=59\a\times 6^{3}+b\times 6+c=440\end{array}\right.$$
- step1: Calculate:
  $$\left\{ \begin{array}{l}-8a-2b+c=-16\a+b+c=5\27a+3b+c=59\216a+6b+c=440\end{array}\right.$$
- step2: Solve the equation:
  $$\left\{ \begin{array}{l}c=-16+8a+2b\a+b+c=5\27a+3b+c=59\216a+6b+c=440\end{array}\right.$$
- step3: Substitute the value of $$c:$$
  $$\left\{ \begin{array}{l}a+b-16+8a+2b=5\27a+3b-16+8a+2b=59\216a+6b-16+8a+2b=440\end{array}\right.$$
- step4: Simplify:
  $$\left\{ \begin{array}{l}9a+3b-16=5\35a+5b-16=59\224a+8b-16=440\end{array}\right.$$
- step5: Solve the equation:
  $$\left\{ \begin{array}{l}a=\frac{7-b}{3}\35a+5b-16=59\224a+8b-16=440\end{array}\right.$$
- step6: Substitute the value of $$a:$$
  $$\left\{ \begin{array}{l}35\times \frac{7-b}{3}+5b-16=59\224\times \frac{7-b}{3}+8b-16=440\end{array}\right.$$
- step7: Simplify:
  $$\left\{ \begin{array}{l}\frac{35\left(7-b\right)}{3}+5b-16=59\\frac{224\left(7-b\right)}{3}+8b-16=440\end{array}\right.$$
- step8: Solve the equation:
  $$\left\{ \begin{array}{l}b=1\\frac{224\left(7-b\right)}{3}+8b-16=440\end{array}\right.$$
- step9: Substitute the value of $$b:$$
  $$\frac{224\left(7-1\right)}{3}+8\times 1-16=440$$
- step10: Simplify:
  $$440=440$$
- step11: Substitute the value of $$b:$$
  $$a=\frac{7-1}{3}$$
- step12: Simplify:
  $$a=2$$
- step13: Substitute the values of $$a,b:$$
  $$c=-16+8\times 2+2\times 1$$
- step14: Simplify:
  $$c=2$$
- step15: Calculate:
  $$\left\{ \begin{array}{l}a=2\b=1\c=2\end{array}\right.$$
- step16: Check the solution:
  $$\left\{ \begin{array}{l}a=2\b=1\c=2\end{array}\right.$$
- step17: Rewrite:
  $$\left(a,b,c\right) = \left(2,1,2\right)$$
To find a cubic regression function that fits the given points $$(-2,-16)$$, $$(1,5)$$, $$(3,59)$$, and $$(6,440)$$, we can express the function in the form:

$$
f(x) = ax^3 + bx + c
$$

Given the points, we can set up a system of equations based on the function:

1. For the point $$(-2, -16)$$:
   $$
   a(-2)^3 + b(-2) + c = -16
   $$
   This simplifies to:
   $$
   -8a - 2b + c = -16 \quad \text{(1)}
   $$

2. For the point $$(1, 5)$$:
   $$
   a(1)^3 + b(1) + c = 5
   $$
   This simplifies to:
   $$
   a + b + c = 5 \quad \text{(2)}
   $$

3. For the point $$(3, 59)$$:
   $$
   a(3)^3 + b(3) + c = 59
   $$
   This simplifies to:
   $$
   27a + 3b + c = 59 \quad \text{(3)}
   $$

4. For the point $$(6, 440)$$:
   $$
   a(6)^3 + b(6) + c = 440
   $$
   This simplifies to:
   $$
   216a + 6b + c = 440 \quad \text{(4)}
   $$

Now, we can solve this system of equations. The solution gives us:

$$
(a, b, c) = (2, 1, 2)
$$

Thus, the cubic regression function that fits the points is:

$$
f(x) = 2x^3 + 1x + 2
$$

In the format you provided, it can be expressed as:

$$
2x^3 + [1]x + \square
$$

where $$\square = 2$$.
The cubic regression function that fits the points is $$ f(x) = 2x^3 + x + 2 $$.

Use cubic regression to find a function that fits the following points. \( (-2,-16)(1,5)(3,59)(6,440) \) \( 2 x^{3}+[?] x+\square \)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Algebra Questions