Answer:
Part A) The vertex is the point [tex](0.5,-0.5)[/tex]
Part B) The axis of symmetry is [tex]x=0.5[/tex]
Part C) The graph in the attached figure
Step-by-step explanation:
we know that
The equation of a vertical parabola into vertex form is equal to
[tex]y=a(x-h)^{2}+k[/tex]
where
(h,k) is the vertex of the parabola
and the axis of symmetry is equal to the x-coordinate of the vertex
so
[tex]x=h[/tex] -----> equation of the axis of symmetry
In this problem we have
[tex]f(x)=-2x^{2}+2x-1[/tex]
Convert to vertex form
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex]f(x)+1=-2x^{2}+2x[/tex]
Factor the leading coefficient
[tex]f(x)+1=-2(x^{2}-x)[/tex]
Complete the square. Remember to balance the equation by adding the same constants to each side
[tex]f(x)+1-(1/2)=-2(x^{2}-x+(1/4))[/tex]
[tex]f(x)+(1/2)=-2(x^{2}-x+(1/4))[/tex]
Rewrite as perfect squares
[tex]f(x)+(1/2)=-2(x-(1/2))^{2}[/tex]
[tex]f(x)=-2(x-(1/2))^{2}-(1/2)[/tex] -----> equation in vertex form
The vertex of the parabola is the point [tex](0.5,-0.5)[/tex]
Is a vertical parabola open downward
The axis of symmetry is equal to
[tex]x=0.5[/tex]
see the attached figure to better understand the problem
