We have a written in . This kind of equation can give us a lot of information about the by observing the values of a, b, and c.
s(x)=-2x2−8x−5 ⇔ s(x)=-2x2+(-8)x+(-5)
We see that for the given equation, a=-2, b=-8, and c=-5.
x-value of the Vertex
Consider the point at which the curve of the parabola changes direction.
This point is the of the parabola, and defines the . If we want to calculate the
x-value of this point, we can substitute the given values of
a and
b into the expression
-2ab and simplify.
The axis of symmetry is the through the vertex and divides the parabola into two mirror images. Since every point on this line will have the same
x-coordinate as the vertex, we can form its equation.
x=-2 y-value of the Vertex
The point at which the graph of a parabola changes direction also defines the maximum or minimum point of the graph. Whether the parabola has a minimum or maximum is determined by the value of a.
Since the given value of
a is
negative, the parabola has a maximum value at the . To find this value, think of
y as a
function of
x, y=s(x). By substituting the
x-value of the vertex into the given equation and simplifying, we will get the
y-value of the vertex.
y=-2x2−8x−3
y=-2(-2)2−8(-2)−3
y=-2(4)−8(-2)−3
y=-8−8(-2)−3
y=-8+16−3
y=5
The vertex is the point
(-2,5).