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: Recall quadratic equation form

A quadratic equation is of the form \( y = ax^2 + bx + c \). We substitute the given points into this equation to form a system of equations.

For point \((-1, 23)\):
Substitute \( x=-1 \) and \( y = 23 \) into \( y=ax^2+bx + c \):
\( 23=a(-1)^2 + b(-1)+c \)
\( 23=a - b + c \) --- (1)

For point \((1, -5)\):
Substitute \( x = 1 \) and \( y=-5 \) into \( y=ax^2+bx + c \):
\( -5=a(1)^2 + b(1)+c \)
\( -5=a + b + c \) --- (2)

For point \((3, -57)\):
Substitute \( x = 3 \) and \( y=-57 \) into \( y=ax^2+bx + c \):
\( -57=a(3)^2 + b(3)+c \)
\( -57 = 9a+3b + c \) --- (3)

Step2: Subtract equation (1) from equation (2)

\((a + b + c)-(a - b + c)=-5 - 23\)
Simplify left side: \(a + b + c - a + b - c=2b\)
Right side: \(-28\)
So \(2b=-28\), then \(b=-14\)

Step3: Add equation (1) and equation (2)

\((a - b + c)+(a + b + c)=23+(-5)\)
Simplify left side: \(a - b + c+a + b + c = 2a + 2c\)
Right side: \(18\)
Divide both sides by 2: \(a + c=9\) --- (4)

Step4: Substitute \(b = - 14\) into equation (3)

\(9a+3(-14)+c=-57\)
\(9a-42 + c=-57\)
\(9a + c=-57 + 42=-15\) --- (5)

Step5: Subtract equation (4) from equation (5)

\((9a + c)-(a + c)=-15 - 9\)
Simplify left side: \(9a + c - a - c = 8a\)
Right side: \(-24\)
So \(8a=-24\), then \(a=-3\)

Step6: Find \(c\) using equation (4)

From \(a + c=9\) and \(a=-3\), substitute \(a\) into equation (4):
\(-3 + c=9\)
\(c=9 + 3=12\)

Answer:

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