Question

How do you find the vertex of \[y={{x}^{2}}+8x-9\] ?

Answer 1

Hint: These types of problems are very simple and easy to solve. The problem is of coordinate geometry of sub-topic parabola, and for solving these types of problems we need to first understand what the different general forms of parabolas are possible in geometry. There are four general 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 answer:
Now, starting off with the solution of our given problem, we can first of all rearrange the left and right hand side of the given parabolic equation to make it look like one of the standardized form of general equation. So rearranging all the terms we get,
\[y+9={{x}^{2}}+8x\]
Now, since the \[x\] term of the equation has coefficient \[8\] , which is basically \[2\times 4\] , we add \[{{4}^{2}}\], which is basically \[16\], on both sides of the equation, so that the right hand side of the equation becomes a perfect square. We do the operation as,
\[\Rightarrow y+9+16={{x}^{2}}+8x+16\]
Evaluating the values, we write,
\[\Rightarrow y+25={{x}^{2}}+8x+16\] ,
Now, representing the right hand side of the equation as a perfect square, we get,
\[\Rightarrow y+25={{\left( x+4 \right)}^{2}}\]
Rearranging the terms we get,
\[\Rightarrow {{\left( x+4 \right)}^{2}}=y+25\]

We can clearly observe that the above equation is of the form \[{{x}^{2}}=4by\] , where we can say, \[b=\dfrac{1}{4}\] .
We can however rewrite the general parabolic equation as,
\[{{\left( x-0 \right)}^{2}}=4b\left( y-0 \right)\] and from this it is very clear that for this equation \[\left( 0,0 \right)\] is the vertex.
From the formed equation we can get the values of the \[x\] and \[y\] coordinates of vertex as,
\[\begin{align}
  & \Rightarrow x+4=0 \\
 & \Rightarrow x=-4 \\
\end{align}\] and \[\begin{align}
  & \Rightarrow y+25=0 \\
 & \Rightarrow y=-25 \\
\end{align}\]

Thus, we can very clearly say that the vertex of the given parabola is, \[\left( -4,-25 \right)\].

Note: We can solve the above given problem using another method, known as the method of shifting of origin or origin transformation, but it is a very complex as well as a long process. In such a process, we transfer the parabolic equation to some other coordinate system. We then find the value of the coordinates in the new system. After that, we revert back to our original coordinate system and evaluate the coordinates in our original system. The solution shown above is the simplest one.