Question 347560
y=-x^(2)+12x+15

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 (-6)^(2) to both sides of the equation.
y=-1(x^(2)-12x+36)-1(-15)-(-1)(0+36)

Remove the 0 from the polynomial; adding or subtracting 0 does not change the value of the expression.
y=-1(x^(2)-12x+36)-1(-15)-(-1)(36)

Factor the perfect trinomial square into (x-6)^(2).
y=-1((x-6)^(2))-1(-15)-(-1)(36)

Factor the perfect trinomial square into (x-6)^(2).
y=-1(x-6)^(2)-1(-15)-(-1)(36)

Multiply -1 by each term inside the parentheses.
y=-1(x-6)^(2)+15-(-1)(36)

Multiply -1 by 36 to get -36.
y=-1(x-6)^(2)+15-(-36)

Multiply -1 by each term inside the parentheses.
y=-1(x-6)^(2)+15+36

Add 36 to 15 to get 51.
y=-1(x-6)^(2)+51

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=-1_k=51_h=6

The vertex of a parabola is (h,k).
Vertex: (6,51)

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)=-1

Solve the equation for p.
p=-(1)/(4)

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=(6,51-(1)/(4))

Find the focus.
Focus=(6,(203)/(4))

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=51-(-(1)/(4))

Find the directrix.
Directrix: y=(205)/(4)

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=6

These values represent the important values for graphing and analyzing a parabola.
Vertex: (6,51)_Focus: (6,(203)/(4))_Directrix: y=(205)/(4)_Axis of Symmetry: x=6