Respuesta :
Given:
There are given the vertex and point :
[tex]\begin{gathered} vertex:(2,1) \\ point:(3,-2) \end{gathered}[/tex]Explanation:
According to the question:
We need to find the quadratic function in the form of vertex:
So,
From the vertex form of the equation:
[tex]y=a(x-h)^2+k[/tex]Where,
[tex]\begin{gathered} h=2 \\ k=1 \end{gathered}[/tex]Then,
Put both the value into the given vertex form:
So,
[tex]\begin{gathered} y=a(x-h)^{2}+k \\ y=a(x-2)^2+1...(1) \end{gathered}[/tex]Now,
We need to find the value of a:
So,
Put 3 for x and -2 for y into the equation (1):
Then,
[tex]\begin{gathered} \begin{equation*} y=a(x-2)^2+1 \end{equation*} \\ -2=a(3-2)^2+1 \\ -2=a(1)^2+1 \\ -2=a+1 \\ a=-3 \end{gathered}[/tex]Then,
Put the value of an into the equation (1):
So,
[tex]\begin{gathered} \begin{equation*} y=a(x-2)^2+1 \end{equation*} \\ y=-3(x-2)^2+1 \end{gathered}[/tex]Final answer:
Hence, the vertex form of the quadratic function is shown below:
[tex]y=-3(x-2)^{2}+1[/tex]