General Formula: y=& a(x- h )^2 + k
Equation: y=& 2(x-( -2))^2+( -1)
We can see that a= 2, h= - 2, and k= - 1.
Vertex
The vertex of a quadratic function written in vertex form is the point ( h, k). For this exercise, we have h= -2 and k= -1 . Therefore, the vertex of the given equation is ( -2, - 1).
Axis of Symmetry
The axis of symmetry of a quadratic function written in vertex form is the vertical line with equation x= h. As we have already noticed, for our function, this is h= -2. Therefore, the axis of symmetry is the line x= -2.
Maximum or Minimum Value
Before we determine the maximum or minimum recall that, if a>0, the parabola opens upwards. Conversely, if a<0, the parabola opens downwards.
In the given function, we have a= 2, which is greater than 0. Therefore, the parabola opens upwards and we will have a minimum value. The minimum or maximum value of a parabola is always the y-coordinate of the vertex, k. For this function, it is k= - 1. Therefore, the minimum value is (-2,- 1).
Domain and Range
Unless there is a specific restriction given in the context of the problem, the domain of a quadratic function is all real numbers. In this case, there is no restriction on the value of x. Since the minimum value of the function is - 1, the range is all real numbers greater than or equal to - 1.
Range: y≥- 1
