Certainly! Let's analyze the properties of the quadratic function to find its domain and range.
### Domain of a Quadratic Function
A quadratic function is generally represented in the form:
[tex]\[ y = ax^2 + bx + c \][/tex]
The domain of any quadratic function is the set of all real numbers because a quadratic function will produce a valid output [tex]\(y\)[/tex] for any input [tex]\(x\)[/tex]. Hence, the domain is:
[tex]\[ (-\infty, \infty) \][/tex]
### Range of a Quadratic Function
The range of a quadratic function depends on its vertex and the direction in which the parabola opens.
Given the vertex of the quadratic function: [tex]\((-2, -3)\)[/tex]
1. Vertex Form:
A quadratic function can be expressed in vertex form:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex. In this case, the vertex form would look like:
[tex]\[ y = a(x + 2)^2 - 3 \][/tex]
where [tex]\((h, k) = (-2, -3)\)[/tex]
2. Direction of the Parabola:
To determine the range, we need to know if the parabola opens upwards or downwards:
- If [tex]\(a > 0\)[/tex], the parabola opens upwards, meaning it has a minimum value at the vertex.
- If [tex]\(a < 0\)[/tex], the parabola opens downwards, meaning it has a maximum value at the vertex.
Assuming the parabola opens upwards ([tex]\(a > 0\)[/tex]), the minimum value of the quadratic function is at the vertex [tex]\(y = -3\)[/tex]. Therefore, the function can take any value greater than or equal to [tex]\(-3\)[/tex].
As a result, the range is:
[tex]\[ [-3, \infty) \][/tex]
### Summary:
To summarize, the domain and range of the given quadratic function with vertex [tex]\((-2, -3)\)[/tex] are as follows:
- Domain:
[tex]\[ (-\infty, \infty) \][/tex]
- Range:
[tex]\[ [-3, \infty) \][/tex]
These results comprehensively outline the behavior of the quadratic function in terms of its domain and range.
