Write the standard form of the equation of the parabola that has a vertex at $\left( \frac { - 2 } { 3 } , \frac { 1 } { 9 } \right)$ and passes through the point $( 3 , - 4 )$.
A) $f ( x ) = - \frac { 37 } { 11 } \left( x + \frac { 2 } { 3 } \right) ^ { 2 } + \frac { 1 } { 9 }$
B) $f ( x ) = - \frac { 37 } { 121 } \left( x - \frac { 3 } { 2 } \right) ^ { 2 } + \frac { 1 } { 9 }$
C) $f ( x ) = - \frac { 37 } { 121 } \left( x + \frac { 2 } { 3 } \right) ^ { 2 } + \frac { 1 } { 9 }$
D) $f ( x ) = - \frac { 37 } { 11 } \left( x - \frac { 2 } { 3 } \right) ^ { 2 } - \frac { 1 } { 9 }$
E) $f ( x ) = - \frac { 37 } { 25 } \left( x - \frac { 3 } { 2 } \right) ^ { 2 } - \frac { 1 } { 9 }$

Question

Quizplus · Accepted Answer

The vertex form of a parabola is $ f(x) = a(x-h)^2 + k $. Given the vertex $\left( \frac{-2}{3}, \frac{1}{9} \right)$, the equation becomes $ f(x) = a\left(x + \frac{2}{3}\right)^2 + \frac{1}{9} $. Using the point $(3, -4)$ to find $a$, we solve $-4 = a\left(3 + \frac{2}{3}\right)^2 + \frac{1}{9}$. Solving for $a$ gives $a = -\frac{37}{121}$. Thus, the equation is $ f(x) = -\frac{37}{121}\left(x + \frac{2}{3}\right)^2 + \frac{1}{9} $.

Write the Standard Form of the Equation of the Parabola (−23,19)\left( \frac { - 2 } { 3 } , \frac { 1 } { 9 } \right)(3−2​,91​)

Write the Standard Form of the Equation of the Parabola $\left( \frac { - 2 } { 3 } , \frac { 1 } { 9 } \right)$