Answer:
[tex]y=(x-2)^{2}+1[/tex]
Step-by-step explanation:
The general form for the parabolic function on its vertex form is:
[tex]y=a(x-h)^2+k[/tex]
where:
[tex](h,k)=vertex\\[/tex]
the vertex will be where the parabolic function stops going down and starts to go up towars infintiy. In the problem this vertex is given as described and it is
[tex](2,1)[/tex]
we have left to find the [tex]a[/tex]. We can find it by using the point that was given to us in the problem thet the parabole goes through [tex](0,5)[/tex] and replacing it in the equation for [tex]x[/tex] and [tex]y[/tex] like this:
[tex]y=a(x-h)^2+k\\5=a(0-2)^2+1\\5=a(-2)^2+1\\5=a(4)+1\\a(4)=5-1\\a(4)=4\\a=\frac{4}{4}\\ a=1[/tex]
now we replace [tex]a[/tex] and [tex](h,k)[/tex] in the general form:
[tex]y=1(x-2)^2+1\\y=(x-2)^2+1[/tex]
