Question

The following set of points belong to a specific function:
{(-3,0)(-2,4), (-1,0), (0,-6),(1,-8), (2,0),(3,24)}
Based on the set of points answer the following questions:
a) What type of function does it produce? Justify your answer.

b) Write an equation

Answer 1

Through n +1 points passing polynomial of degree n.

In this case you have 7 points.

Polynomial will be degree 6.


Write the polynomial in this form :

a * x ^ 6 + b * x ^ 5 + c * x ^ 4 + d * x ^ 3 + e * x ^ 2 + f *x + g = y


Now put values of x an y in this equation.


a * ( - 3 ) ^ 6 + b * ( - 3 ) ^ 5 + c * ( - 3 ) ^ 4 + d * ( - 3 ) ^ 3 + e * ( - 3 ) ^ 2 + f * ( - 3 ) + g = 0

a * ( - 2 ) ^ 6 + b * ( - 2 ) ^ 5 + c * ( - 2 ) ^ 4 + d * ( - 2 ) ^ 3 + e * ( - 2 ) ^ 2 + f * ( - 2 ) + g = 4

a * ( - 1 ) ^ 6 + b * ( - 1 ) ^ 5 + c * ( - 1 ) ^ 4 + d * ( - 1 ) ^ 3 + e * ( - 1 ) ^ 2 + f * ( - 1 ) + g = 0

a * 0 ^ 6 + b * 0 ^ 5 + c * 0 ^ 4 + d * 0 ^ 3 + e * 0 ^ 2 + f * 0 + g = - 6

a * 1 ^ 6 + b * 1 ^ 5 + c * 1 ^ 4 + d * 1 ^ 3 + e * 1 ^ 2 + f * 1 + g = - 8

a * 1 ^ 6 + b * 1 ^ 5 + c * 1 ^ 4 + d * 1 ^ 3 + e * 1 ^ 2 + f * 1 + g = - 8

a * 2 ^ 6 + b * 2 ^ 5 + c * 2 ^ 4 + d * 2 ^ 3 + e * 2 ^ 2 + f * 2 + g = 0

a * 3 ^ 6 + b * 3 ^ 5 + c * 3 ^ 4 + d * 3 ^ 3 + e * 3 ^ 2 + f * 3 + g = 24


OR


729 a - 243 b + 81 c - 27 d + 9 e - 3 f + g = 0

64 a - 32 b + 16 c - 8 d + 4 e - 2 f + g = 4

a - b + c - d + e - f + g = 0

0 + 0 + 0 + 0 + 0 + 0 + g = - 6

a + b + c + d + e + f + g = - 8

64 a + 32 b + 16 c + 8 d + 4 e + 2 f + g = 0

729 a + 243 b + 81 c + 27 d + 9 e + 3 f + g = 24


Solutions of this system of equations are :

a = 0

b = 0

c = 0

d = 1

e = 2

f = - 5

g = - 6


Your Polynomial :

y = 0 * x ^ 6 + 0 * x ^ 5 + 0 * x ^ 4 + 1 * x ^ 3 + 2 * x ^ 2 + ( - 5 ) * x + ( - 6 )

y = x ^ 3 + 2 x ^ 2 - 5 x - 6

That is cubic parabola.

Answer 2

If you want to see graph go on :

wolframalpha dot com

When page be open in rectangle type :

intepolating polynomial {(-3,0),(-2,4), (-1,0), (0,-6),(1,-8), (2,0),(3,24)}

and click option =

You will see graph