<s>[INST] How can I find the quadratic equation from highest point on parabola? If I have a parabola, where the vertex is in $P=(2,0)$, and a point on the parabola is $Q=(-1,-6)$, how can I find the quadratic equation of the form:
$$f(x) = ax^2 + bx + c$$
 [/INST] Vertex form of a quadratic is: $f(x)=a(x-h)^2+k$, where $(h,k)$ is the vertex.  So, in our case the vertex is $(2,0)$ so $h=2$ and $k=0$.  So we have, $f(x)=a(x-2)^2$.  Now, find $a$ by using the point $(-1,-6$).  So we have, $-6=a(-1-2)^2$ which gives us $-6=9a$ so $a=-\frac{2}{3}$.  So our equation is $f(x)=-\frac{2}{3}(x-2)^2$.  Now, multiply this out to get $$f(x)=-\frac{2}{3}x^2+\frac{8}{3}x-\frac{8}{3}$$
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