Question 389610
y=-0.0002x^(2)+20x

To create a trinomial square on the left-hand side of the equation, add a value to both sides of the equation that is equal to the square of half the coefficient of x.  In this problem, add (-50000)^(2) to both sides of the equation.
y=-0.0002(x^(2)-100000x+2500000000)-(-0.0002)(0+2500000000)

Remove the 0 from the polynomial; adding or subtracting 0 does not change the value of the expression.
y=-0.0002(x^(2)-100000x+2500000000)-(-0.0002)(2500000000)

Factor the perfect trinomial square into (x-50000)^(2).
y=-0.0002((x-50000)^(2))-(-0.0002)(2500000000)

Factor the perfect trinomial square into (x-50000)^(2).
y=-0.0002(x-50000)^(2)-(-0.0002)(2500000000)

Multiply -0.0002 by 2500000000 to get -500000.
y=-0.0002(x-50000)^(2)-(-500000)

Multiply -1 by each term inside the parentheses.
y=-0.0002(x-50000)^(2)+500000

This is the form of a paraboloa.  Use this form to determine the values used to find vertex and x-y intercepts.
y=a(x-h)^(2)+k

Use the standard form to determine the vertex and x-y intercepts.
a=-0.0002_k=500000_h=50000

The vertex of a parabola is (h,k).
Vertex: (50000,500000)

This formula is used to find the distance from the vertex to the focus.
(1)/(4p)=a

Substitute the value of a into the formula.
(1)/(4p)=-0.0002

Solve the equation for p.
p=-1250

Add p to the vertex to find the focus.  If the parabola points up or down add p to the y-coordinate of the vertex, if it points left or right add it to the x-coordinate.
Focus=(50000,500000-1250)

Find the focus.
Focus=(50000,498750)

A parabola can also be defined as locus of points in a plane which are equidistant from a given point (the focus) and a given line (the directrix).
y=500000-(-1250)

Find the directrix.
Directrix: y=501250

The axis of symmetry is the line that passes through the vertex and focus.  The two sides of a graph on either side of the axis of symmetry look like mirror images of each other.
Axis of Symmetry: x=50000

These values represent the important values for graphing and analyzing a parabola.
Vertex: (50000,500000)_Focus: (50000,498750)_Directrix: y=501250_Axis of Symmetry: x=50000