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Write the equation of the parabola that passes through the points shown in the table. $(-2,0)$ $(2,-11)$ $(4,0)$

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Find the vertex of a parabola

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Q. Write the equation of the parabola that passes through the points shown in the table.

(-2,0)

(2,-11)

(4,0)

Identify Parabola Form: Identify the general form of a parabola. $\newline$ The general form of a quadratic equation is $y = ax^2 + bx + c$ . We need to find the values of $a$ , $b$ , and $c$ using the given points.
Set Up Equations: Set up equations using the points. $\newline$ Substitute the points into the general form: $\newline$ $1$ . $(-2, 0)$ : $0 = a(-2)^2 + b(-2) + c$ $\newline$ $2$ . $(2, -11)$ : $-11 = a(2)^2 + b(2) + c$ $\newline$ $3$ . $(4, 0)$ : $0 = a(4)^2 + b(4) + c$
Simplify Equations: Simplify the equations. $\newline$ $1$ . $0 = 4a - 2b + c$ $\newline$ $2$ . $-11 = 4a + 2b + c$ $\newline$ $3$ . $0 = 16a + 4b + c$
Solve System of Equations: Solve the system of equations. $\newline$ From equation $1$ and $2$ : $\newline$ Add them: $-11 = 8a + c$ $\newline$ From equation $1$ and $3$ : $\newline$ Subtract them: $11 = 12a + 2b$
Continue Solving: Continue solving. $\newline$ Divide the second equation by $2$ : $5.5 = 6a + b$ $\newline$ Substitute $b$ from this equation into the first: $-11 = 8a + c$

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