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=

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 $c$.

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:

1. For $(-9, 0)$: $$81a - 9b + c = 0$$
2. For $(-7, 8)$: $$49a - 7b + c = 8$$
3. For $(-5, 8)$: $$25a - 5b + c = 8$$
4. For $(-3, 0)$: $$9a - 3b + c = 0$$

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: $$\begin{bmatrix} a \\ b \\ c \end{bmatrix} = \begin{bmatrix} -1 \\ -12 \\ -27 \end{bmatrix}$$

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

$$y = -1x^2 - 12x - 27$$

Final Answer