Question

use quadratic regression to find a function that fits the following points.
(1, 3) (2, -7) (-1, 11)
?x² + □x + □

Explanation:

Step1: Assume the quadratic function

Let the quadratic function be \( y = ax^2 + bx + c \).

Step2: Substitute the points into the function

For the point \((1, 3)\):
\( 3 = a(1)^2 + b(1) + c \), which simplifies to \( a + b + c = 3 \). (Equation 1)
For the point \((2, -7)\):
\( -7 = a(2)^2 + b(2) + c \), which simplifies to \( 4a + 2b + c = -7 \). (Equation 2)
For the point \((-1, 11)\):
\( 11 = a(-1)^2 + b(-1) + c \), which simplifies to \( a - b + c = 11 \). (Equation 3)

Step3: Subtract Equation 1 from Equation 3

\((a - b + c) - (a + b + c) = 11 - 3\)
\(a - b + c - a - b - c = 8\)
\(-2b = 8\), so \(b = -4\).

Step4: Substitute \(b = -4\) into Equation 1 and Equation 2

Equation 1 becomes \(a - 4 + c = 3\), so \(a + c = 7\). (Equation 4)
Equation 2 becomes \(4a + 2(-4) + c = -7\), so \(4a - 8 + c = -7\), and \(4a + c = 1\). (Equation 5)

Step5: Subtract Equation 4 from Equation 5

\((4a + c) - (a + c) = 1 - 7\)
\(4a + c - a - c = -6\)
\(3a = -6\), so \(a = -2\).

Step6: Substitute \(a = -2\) into Equation 4

\(-2 + c = 7\), so \(c = 9\).

Answer:

The quadratic function is \(-2x^2 - 4x + 9\), so the coefficient of \(x^2\) is \(-2\), the coefficient of \(x\) is \(-4\), and the constant term is \(9\). The values to fill in the boxes are \(-2\), \(-4\), and \(9\) respectively.