Question

How do you find the vertex of the parabola $y = - 2{x^2} + 12x - 13$?

Answer 1

Hint:
The above given problem is a very simple problem of coordinate geometry. The given question is related to the concept of parabola and for solving these types of questions we first need to understand the different forms of parabolas. There are four types of parabola and for each of them, the corresponding vertex and foci are as follows:

Equation	Vertex	Foci
${y^2} = 4ax$	$\left( {0,0} \right)$	$\left( {a,0} \right)$
${y^2} = - 4ax$	$\left( {0,0} \right)$	$\left( { - a,0} \right)$
${x^2} = 4by$	$\left( {0,0} \right)$	$\left( {0,b} \right)$
${x^2} = - 4by$	$\left( {0,0} \right)$	$\left( {0, - b} \right)$

Complete step by step solution:
Given is $y = - 2{x^2} + 12x - 13$ and we have to find the vertex of this parabola.
We know that the equation of a parabola in vertex form is $y = a{\left( {x - h} \right)^2} + k$, where $\left( {h,k} \right)$ are the coordinates of the vertex and $a$ is a multiplier.
In order to obtain the vertex form for the given parabola we use the method of completing the square.
Since the coefficient of ${x^2}$ must be one, so,
$y = - 2\left( {{x^2} - 6x + \dfrac{{13}}{2}} \right)$
Add or subtract\[{\left( {\dfrac{1}{2}\;coefficient\;of\;the\;x - term} \right)^2}\] to \[{x^2} - 6x\].
\[
  y = - 2\left( {{x^2} + 2\left( { - 3} \right)x + 9 - 9 + \dfrac{{13}}{2}} \right) \\
  y = - 2{\left( {x - 3} \right)^2} - 2\left( { - 9 + \dfrac{{13}}{2}} \right) \\
  y = - 2{\left( {x - 3} \right)^2} + 5 \\
 \]
So, after comparing the above obtained equation with the vertex form of equation $y = a{\left( {x - h} \right)^2} + k$, $\left( {h,k} \right)$ being the vertex, we can say that \[h = 3,k = 5\]. So, the vertex is at \[\left( {3,5} \right)\].

Hence, the vertex of the parabola is \[\left( {3,5} \right)\].

Note:
We can also find the vertex using the method of shifting of origin or origin transformation, but it is a very complex as well as a lengthy process. The solution shown above is the simplest one. In origin transformation, we shift the origin of the actual coordinate system, to some other arbitrary system, so as to simplify the equation and make it similar to that of the general form. However, to find the required point, we need to revert back to the original coordinate system.