Given:
The graph of a quadratic function opens downward and has its vertex at (0, 1).
To find:
The quadratic function.
Solution:
The vertex form of a quadratic function is:
[tex]y=a(x-h)^2+k[/tex] ...(i)
Where, a is a constant and (h,k) is the vertex.
If a<0, then graph opens downward and if a>0, then graph opens upward.
It is given that the quadratic function opens downward and has its vertex at (0, 1). It means a must be negative.
Putting h=0 and k=1 in (i), we get
[tex]y=a(x-0)^2+1[/tex]
[tex]y=ax^2+1[/tex]
For [tex]a=-1[/tex],
[tex]y=(-1)x^2+1[/tex]
[tex]y=x^2+1[/tex]
So, option A is correct.
In other options the leading coefficient is positive it means their graphs open upward. So, options B, C and D are incorrect.
Therefore, the correct option is A.