Write the standard form of the equation of the parabola that has a vertex at $( 3,8 )$ and passes through the point $( 5 , - 2 )$.
A) $f ( x ) = - \frac { 1 } { 2 } ( x - 3 ) ^ { 2 } + 5$
B) $f ( x ) = - \frac { 5 } { 2 } ( x - 3 ) ^ { 2 } + 8$
C) $f ( x ) = - \frac { 10 } { 9 } ( x + 3 ) ^ { 2 } + 8$
D) $f ( x ) = \frac { 8 } { 5 } ( x - 8 ) ^ { 2 } - 2$
E) $f ( x ) = \frac { 8 } { 9 } ( x - 8 ) ^ { 2 } - 5$

Question

Quizplus · Accepted Answer

The vertex form of a parabola is given by $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex. Substituting the values given, we get $f(x) = a(x-3)^2 + 8$. Now, we use the fact that the parabola passes through the point $(5,-2)$ to solve for $a:$
\begin{align*}
-2 &= a(5-3)^2 + 8 \
-2 &= 4a + 8\
a &= -\frac{5}{2}
\end{align*}
Substituting this value for $a$ into our equation gives $f(x) = -\frac{5}{2}(x-3)^2 + 8$, which is in standard form. Hence, the answer is $\boxed{	ext{(B)}}.$

Write the Standard Form of the Equation of the Parabola (3,8)( 3,8 )(3,8)

Write the Standard Form of the Equation of the Parabola $( 3,8 )$