<s>[INST] Write the equation of the parabola that has the vertex at point $(5,0)$ and passes through the point $(7,−2)$. Write the equation of the parabola that has the vertex at point $(5,0)$ and passes through the point $(7,−2)$.
I know how to do it with the $x$ intercepts but I can't solve this.
 [/INST] The vertex form of the equation of a parabola is 
$$f(x) = a(x - h)^2 + k$$
where $(h, k)$ is the vertex of the parabola.  In this case, we are given that $(h, k) = (5, 0)$.  Hence,
\begin{align*}
f(x) & = a(x - 5)^2 + 0\\
     & = a(x - 5)^2
\end{align*}
Since we also know the parabola passes through the point $(7, -2)$, we can solve for $a$ because we know that $f(7) = -2$.
\begin{align*}
a(7 - 5)^2 & = -2\\
a(2)^2 & = -2\\
4a & = -2\\
a & = -\frac{1}{2}
\end{align*}
Thus, the given parabola has equation 
$$f(x) = -\frac{1}{2}(x - 5)^2$$ 
</s>