we know that
watching the graph
the x-coordinate of the vertex is 0.5
the y-coordinate of the vertex is < -1
the axis of symmetry is the coordinate x of the vertex
so
x=0.5
the only options with those three answers is the option B and option D
check option B
vertex (0.5,-1.25)
f(x)=x²-x-1
find the value of f(x) for x=0.5 and compare with the y-coordinate of the vertex
f(0.5)=0.5²-0.5-1-------> f(0.5)=0.25-0.5-1----> f(0.5)=-1.25
-1.25 is equal to -1.25-----------> this is the function
check option D
vertex (0.5,-1.25)
find the value of f(x) for x=0.5 and compare with the y-coordinate of the vertex
f(x)=x²+x-1
f(0.5)=0.5²+0.5-1-------> f(x)=-0.25
-0.25 is not equal to -1.25-----------> Is not the function
therefore
the answer is the option B