Question

use quadratic regression to find a function that fits the following points. (1, 3) (2, -7) (-1, 11) x² + x +

Explanation:

Step1: Set up the system of equations

A quadratic function is \(y = ax^{2}+bx + c\).
Substitute \((x = 1,y = 3)\): \(a(1)^{2}+b(1)+c=3\), so \(a + b + c=3\).
Substitute \((x = 2,y=-7)\): \(a(2)^{2}+b(2)+c=-7\), so \(4a + 2b + c=-7\).
Substitute \((x=-1,y = 11)\): \(a(-1)^{2}+b(-1)+c=11\), so \(a - b + c=11\).

Step2: Solve the system of equations

Subtract the first equation \(a + b + c=3\) from the third equation \(a - b + c=11\):
\((a - b + c)-(a + b + c)=11 - 3\)
\(a - b + c-a - b - c=8\)
\(-2b=8\), so \(b=-4\).

Substitute \(b = -4\) into the first equation: \(a-4 + c=3\), so \(a + c=7\) (Equation A).
Substitute \(b=-4\) into the second equation: \(4a+2(-4)+c=-7\), \(4a-8 + c=-7\), so \(4a + c=1\) (Equation B).

Subtract Equation A (\(a + c=7\)) from Equation B (\(4a + c=1\)):
\((4a + c)-(a + c)=1 - 7\)
\(4a + c-a - c=-6\)
\(3a=-6\), so \(a=-2\).

Substitute \(a=-2\) into Equation A (\(a + c=7\)): \(-2 + c=7\), so \(c = 9\).

Answer:

\(-2x^{2}-4x + 9\)