Questions: The table represents a quadratic function. Write an equation of the function in standard form. x -9 -7 -5 -3 y 0 8 8 0 y=

Transcript text: The table represents a quadratic function. Write an equation of the function in standard form. \begin{tabular}{|c|c|c|c|c|} \hline$x$ & -9 & -7 & -5 & -3 \\ \hline$y$ & 0 & 8 & 8 & 0 \\ \hline \end{tabular} \[ y= \]

Solution

Solution Steps

To find the equation of a quadratic function in standard form $ y = ax^2 + bx + c $ given a set of points, we can use the method of solving a system of equations. We will substitute each pair of $ (x, y) $ values into the quadratic equation to form a system of equations. Then, we will solve this system to find the coefficients $ a $, $ b $, and $ c $.

Step 1: Set Up the System of Equations

Given the points $(-9, 0)$, $(-7, 8)$, $(-5, 8)$, and $(-3, 0)$, we substitute these into the quadratic equation $y = ax^2 + bx + c$ to form a system of equations:

For $(-9, 0)$: \[ 81a - 9b + c = 0 \]
For $(-7, 8)$: \[ 49a - 7b + c = 8 \]
For $(-5, 8)$: \[ 25a - 5b + c = 8 \]
For $(-3, 0)$: \[ 9a - 3b + c = 0 \]

Step 2: Solve the System of Equations

Using the first three equations, we solve for the coefficients $a$, $b$, and $c$:

The matrix $A$ and vector $B$ are: \[ A = \begin{bmatrix} 81 & -9 & 1 \\ 49 & -7 & 1 \\ 25 & -5 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 8 \\ 8 \end{bmatrix} \]
Solving the system $A \cdot \begin{bmatrix} a \\ b \\ c \end{bmatrix} = B$, we find: \[ \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \begin{bmatrix} -1 \\ -12 \\ -27 \end{bmatrix} \]

Step 3: Write the Quadratic Equation

Substitute the coefficients $a = -1$, $b = -12$, and $c = -27$ into the standard form of the quadratic equation:

\[ y = -1x^2 - 12x - 27 \]