Sure, let's break down the given function:
[tex]\[ y = -2|x + 4| - 1 \][/tex]
### Vertex
To identify the vertex of the function, we can recognize that this is an absolute value function in the form:
[tex]\[ y = a|x - h| + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex.
In our equation, the form is:
[tex]\[ y = -2|x + 4| - 1 \][/tex]
Comparing this to the standard form, we see:
- [tex]\( a = -2 \)[/tex]
- Inside the absolute value, we have [tex]\((x + 4)\)[/tex]. This can be rewritten as [tex]\((x - (-4))\)[/tex], so [tex]\( h = -4 \)[/tex].
- Constant term outside the absolute value is [tex]\( -1 \)[/tex], so [tex]\( k = -1 \)[/tex].
Thus, the vertex is:
[tex]\[ (-4, -1) \][/tex]
### Opens
The function opens downwards because the coefficient of the absolute value, [tex]\( a = -2 \)[/tex], is negative.
### Relation to Parent Function
The parent function for an absolute value is [tex]\( y = |x| \)[/tex]. Here, the coefficient [tex]\( |a| = 2 \)[/tex], which is greater than 1, means the function is narrower than the parent function [tex]\( y = |x| \)[/tex].
### Domain
The domain of any absolute value function is all real numbers. This is because there are no restrictions on the values that [tex]\( x \)[/tex] can take in [tex]\( |x + 4| \)[/tex].
The domain is:
[tex]\[ \text{All Real Numbers} \][/tex]
### Range
The range of the function [tex]\( y = -2|x + 4| - 1 \)[/tex] can be determined by considering the vertex and the direction the function opens. Since the function opens downwards and the vertex is the maximum point at [tex]\( y = -1 \)[/tex], the maximum value of [tex]\( y \)[/tex] is [tex]\( -1 \)[/tex]. The function can only output values less than or equal to [tex]\(-1\)[/tex].
The range is:
[tex]\[ y \leq -1 \][/tex]
### Final Answer in the Given Format
- Vertex:
- [tex]\(-4\)[/tex]
- [tex]\(\text{-1}\)[/tex]
- Opens:
- [tex]\(\text{Down}\)[/tex]
- Relation to Parent Function:
- [tex]\(\text{Narrower}\)[/tex]
- Domain:
- [tex]\(\text{All Real Numbers}\)[/tex]
- Range:
- [tex]\( y \leq -1 \)[/tex]
