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 regression formula

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 \( y = ax^2 + bx + c \), we get:
\( 23 = a(-1)^2 + b(-1) + c \)
Simplify: \( 23 = a - b + c \) --- (1)

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

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

Step2: Solve the system of equations

First, 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, \( 2b = -28 \)
Divide both sides by 2: \( b = -14 \)

Now, add equation (1) and equation (2):
\( (a - b + c) + (a + b + c) = 23 + (-5) \)
Simplify left side: \( 2a + 2c \)
Right side: \( 18 \)
Divide both sides by 2: \( a + c = 9 \) --- (4)

Substitute \( b = -14 \) into equation (3):
\( -57 = 9a + 3(-14) + c \)
Simplify: \( -57 = 9a - 42 + c \)
Rearrange: \( 9a + c = -57 + 42 = -15 \) --- (5)

Now, subtract equation (4) from equation (5):
\( (9a + c) - (a + c) = -15 - 9 \)
Simplify left side: \( 8a \)
Right side: \( -24 \)
Divide both sides by 8: \( a = -3 \)

Substitute \( a = -3 \) into equation (4):
\( -3 + c = 9 \)
Add 3 to both sides: \( c = 12 \)

Step3: Write the quadratic equation

Now that we have \( a = -3 \), \( b = -14 \), and \( c = 12 \), the quadratic equation is \( y = -3x^2 -14x + 12 \)

Answer:

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