Question 347560: which direction does the parabola for the quadratic equation y = -x2 + 12x + 15 open
Answer by haileytucki(390)   (Show Source):
You can put this solution on YOUR website! 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
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