To determine the domain and range of the given parabola [tex]\( f(x) = -2(x+3)^2 - 6 \)[/tex], we need to proceed step by step.
### Step 1: Determining the Domain
The domain of a quadratic function (parabola) is the set of all possible input values (x-values). Since this is a parabola, and parabolas extend infinitely in both the positive and negative directions, the domain is all real numbers. In interval notation, this is written as:
[tex]\[
\text{Domain: } (-\infty, \infty)
\][/tex]
### Step 2: Determining the Range
To find the range, we need to analyze the vertex of the parabola and the direction in which it opens.
- The given function is [tex]\( f(x) = -2(x + 3)^2 - 6 \)[/tex].
- This is in the standard vertex form [tex]\( f(x) = a(x - h)^2 + k \)[/tex], where [tex]\( a \)[/tex], [tex]\( h \)[/tex], and [tex]\( k \)[/tex] are constants.
- Here, [tex]\( a = -2 \)[/tex], [tex]\( h = -3 \)[/tex], and [tex]\( k = -6 \)[/tex].
From the vertex form, the vertex [tex]\((h, k)\)[/tex] is [tex]\((-3, -6)\)[/tex].
- When [tex]\( a < 0 \)[/tex] (which is the case here since [tex]\( a = -2 \)[/tex]), the parabola opens downwards.
- The vertex represents the maximum point of the parabola because it opens downwards.
Therefore, the maximum value of [tex]\( f(x) \)[/tex] is the y-coordinate of the vertex, which is [tex]\(-6\)[/tex]. Since the parabola opens downwards, the function values decrease from [tex]\(-6\)[/tex] to negative infinity.
So, the range of the parabola is all values less than or equal to [tex]\(-6\)[/tex]. In interval notation, this is written as:
[tex]\[
\text{Range: } (-\infty, -6]
\][/tex]
### Final Answer
Combining these results, we have:
[tex]\[
\text{Domain: } (-\infty, \infty)
\][/tex]
[tex]\[
\text{Range: } (-\infty, -6]
\][/tex]
