Question

use quadratic regression to find the equation for the parabola going through these 3 points. (2, 19), (6, 99), and (-1, 1) 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 \((2, 19)\):
\( 19 = a(2)^2 + b(2) + c \)
\( 19 = 4a + 2b + c \) --- (1)

For point \((6, 99)\):
\( 99 = a(6)^2 + b(6) + c \)
\( 99 = 36a + 6b + c \) --- (2)

For point \((-1, 1)\):
\( 1 = a(-1)^2 + b(-1) + c \)
\( 1 = a - b + c \) --- (3)

Step2: Subtract equations to eliminate \( c \)

Subtract equation (1) from equation (2):
\( (36a + 6b + c) - (4a + 2b + c) = 99 - 19 \)
\( 32a + 4b = 80 \)
Divide by 4: \( 8a + b = 20 \) --- (4)

Subtract equation (3) from equation (1):
\( (4a + 2b + c) - (a - b + c) = 19 - 1 \)
\( 3a + 3b = 18 \)
Divide by 3: \( a + b = 6 \) --- (5)

Step3: Solve for \( a \) and \( b \)

Subtract equation (5) from equation (4):
\( (8a + b) - (a + b) = 20 - 6 \)
\( 7a = 14 \)
\( a = 2 \)

Substitute \( a = 2 \) into equation (5):
\( 2 + b = 6 \)
\( b = 4 \)

Step4: Solve for \( c \)

Substitute \( a = 2 \) and \( b = 4 \) into equation (3):
\( 1 = 2 - 4 + c \)
\( 1 = -2 + c \)
\( c = 3 \)

Answer:

\( y = 2x^2 + 4x + 3 \)