Question

use quadratic regression to find the equation for the parabola going through these 3 points. (-1, 23), (1, -5), and (3, -57) y = ?x² + x +

Explanation:

Step1: Substitute points into quadratic equation

The general form of a quadratic equation is \(y = ax^{2}+bx + c\).
Substitute \((-1,23)\): \(23=a(-1)^{2}+b(-1)+c=a - b + c\)
Substitute \((1,-5)\): \(-5=a(1)^{2}+b(1)+c=a + b + c\)
Substitute \((3,-57)\): \(-57=a(3)^{2}+b(3)+c = 9a+3b + c\)

Step2: Solve the system of equations

Subtract the second equation from the first equation:
\((a - b + c)-(a + b + c)=23-(-5)\)
\(a - b + c - a - b - c=28\)
\(-2b = 28\), so \(b=-14\)

Substitute \(b = - 14\) into the first two - equation:
First equation: \(a-(-14)+c=23\), i.e., \(a + c=9\)
Second equation: \(a+(-14)+c=-5\), i.e., \(a + c=9\) (same as above)
Substitute \(b=-14\) into the third equation: \(9a+3(-14)+c=-57\)
\(9a-42 + c=-57\)
\(9a + c=-15\)

Subtract \(a + c=9\) from \(9a + c=-15\):
\((9a + c)-(a + c)=-15 - 9\)
\(9a + c - a - c=-24\)
\(8a=-24\), so \(a=-3\)

Substitute \(a=-3\) into \(a + c=9\), we get \(-3 + c=9\), so \(c = 12\)

Answer:

\(y=-3x^{2}-14x + 12\)