Question

use quadratic regression to find the equation for the parabola going through these 3 points. (1, -12), (2, -7), and (5, 80) y = ?x² + x +

Explanation:

Step1: Recall quadratic equation form

A quadratic equation is of the form \( y = ax^2 + bx + c \). We have three points \((1, -12)\), \((2, -7)\), and \((5, 80)\). Substitute each point into the equation.

For \((1, -12)\): \( -12 = a(1)^2 + b(1) + c \) ⇒ \( a + b + c = -12 \) (Equation 1)

For \((2, -7)\): \( -7 = a(2)^2 + b(2) + c \) ⇒ \( 4a + 2b + c = -7 \) (Equation 2)

For \((5, 80)\): \( 80 = a(5)^2 + b(5) + c \) ⇒ \( 25a + 5b + c = 80 \) (Equation 3)

Step2: Subtract Equation 1 from Equation 2

\((4a + 2b + c) - (a + b + c) = -7 - (-12)\)

Simplify: \( 3a + b = 5 \) (Equation 4)

Step3: Subtract Equation 2 from Equation 3

\((25a + 5b + c) - (4a + 2b + c) = 80 - (-7)\)

Simplify: \( 21a + 3b = 87 \) (Equation 5)

Step4: Solve Equation 4 and Equation 5

From Equation 4: \( b = 5 - 3a \)

Substitute \( b = 5 - 3a \) into Equation 5:

\( 21a + 3(5 - 3a) = 87 \)

\( 21a + 15 - 9a = 87 \)

\( 12a + 15 = 87 \)

\( 12a = 72 \) ⇒ \( a = 6 \)

Step5: Find b using a = 6

Substitute \( a = 6 \) into Equation 4: \( 3(6) + b = 5 \) ⇒ \( 18 + b = 5 \) ⇒ \( b = -13 \)

Step6: Find c using a = 6 and b = -13

Substitute \( a = 6 \) and \( b = -13 \) into Equation 1: \( 6 - 13 + c = -12 \) ⇒ \( -7 + c = -12 \) ⇒ \( c = -5 \)

Answer:

\( y = 6x^2 - 13x - 5 \)