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