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 $( 2,3 )$ .
A) $f ( x ) = \frac { 13 } { 32 } \left( x - \frac { 3 } { 2 } \right) ^ { 2 } + \frac { 1 } { 9 }$
B) $f ( x ) = \frac { 13 } { 32 } \left( x + \frac { 2 } { 3 } \right) ^ { 2 } + \frac { 1 } { 9 }$
C) $f ( x ) = \frac { 13 } { 4 } \left( x - \frac { 2 } { 3 } \right) ^ { 2 } - \frac { 1 } { 9 }$
D) $f ( x ) = \frac { 13 } { 8 } \left( x - \frac { 3 } { 2 } \right) ^ { 2 } - \frac { 1 } { 9 }$
E) $f ( x ) = \frac { 13 } { 4 } \left( x + \frac { 2 } { 3 } \right) ^ { 2 } + \frac { 1 } { 9 }$

Question

Quizplus · Accepted Answer

The standard form of a parabola's equation is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex. Given the vertex $\left( - \frac { 2 } { 3 } , \frac { 1 } { 9 } \right)$, the equation starts as $f(x) = a\left(x + \frac{2}{3}\right)^2 + \frac{1}{9}$. To find $a$, use the point $(2,3)$ that the parabola passes through. Substituting $x = 2$ and $f(x) = 3$ into the equation and solving for $a$ gives $a = \frac{13}{32}$. Thus, the correct equation is $f(x) = \frac{13}{32}\left(x + \frac{2}{3}\right)^2 + \frac{1}{9}$.

Write the Standard Form of the Equation of the Parabola (−23,19)\left( - \frac { 2 } { 3 } , \frac { 1 } { 9 } \right)(−32​,91​)

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