Questions: Find the quadratic function (y=ax^2+bx+c) whose graph passes through the given points. ((1,1),(-1,-9),(-3,5)) (y=)

Transcript text: Find the quadratic function $y=a x^{2}+b x+c$ whose graph passes through the given points. \[ (1,1),(-1,-9),(-3,5) \] \[ \mathrm{y}=\square \]

Solution

Solution Steps

Step 1: Set up the system of equations

Given the points ($1, 1$), ($-1, -9$), ($-3, 5$), we substitute them into the general form $y = ax^2 + bx + c$ to get:

$a(1)^2 + b(1) + c = 1$
$a(-1)^2 + b(-1) + c = -9$
$a(-3)^2 + b(-3) + c = 5$

Step 2: Solve the system of equations

We use the matrix method to solve the system, setting up the matrix equation $AX = Y$, where: $A = \begin{pmatrix} 1^2 & 1 & 1 \\ -1^2 & -1 & 1 \\ -3^2 & -3 & 1 \end{pmatrix}$, $X = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$, $Y = \begin{pmatrix} 1 \\ -9 \\ 5 \end{pmatrix}$. Solving for $X$, we find the coefficients of the quadratic function.