Let's analyze the function [tex]\( f(x) = -(x+1)^2 \)[/tex] step-by-step to describe its key aspects.
1. Vertex of the function:
The function [tex]\( f(x) = -(x+1)^2 \)[/tex] is a quadratic function in the form [tex]\( f(x) = a(x-h)^2 + k \)[/tex], where the vertex is given by the coordinates [tex]\( (h, k) \)[/tex].
In this case, the quadratic term is [tex]\((x+1)^2\)[/tex], which means [tex]\( x \)[/tex] is replaced by [tex]\( (x+1) \)[/tex]. Thus, the vertex form here is centered at [tex]\( x = -1 \)[/tex]. Because there is no additional constant term added to [tex]\( -(x+1)^2 \)[/tex], the [tex]\( y \)[/tex]-coordinate of the vertex is 0.
So, the vertex of the function is [tex]\((-1, 0)\)[/tex].
2. Function positivity:
Examining [tex]\( f(x) = -(x+1)^2 \)[/tex], we see that it is a downward-opening parabola due to the negative sign in front of the squared term. A downward-opening parabola implies that the function takes negative or zero values only.
Since the function is always zero or negative, it is never positive.
3. Intervals of Increase/Decrease:
For [tex]\( f(x) = -(x+1)^2 \)[/tex], the function decreases before it reaches the vertex and increases after. However, for it to be decreasing just needs to be checked in the context where it is strictly on one side without crossing the vertex.
Given the vertex is at [tex]\( (-1, 0) \)[/tex], the function is decreasing for [tex]\( x < -1 \)[/tex].
4. Domain of the function:
The domain of a quadratic function [tex]\[ f(x) = -(x+1)^2 \][/tex] is typically all real numbers because there are no restrictions on the values that [tex]\( x \)[/tex] can take.
Thus, the domain of the function [tex]\( f(x) = -(x+1)^2 \)[/tex] is all real numbers.
5. Range of the function:
The range of a quadratic function [tex]\[ f(x) = -(x+1)^2 \][/tex] is given by the y-values it can take. Given that the function opens downward and its maximum value (at the vertex) is 0, the function can only take values from [tex]\(-∞\)[/tex] to 0.
Therefore, the range of the function is all real numbers less than or equal to 0.
So, summarizing the key aspects of the function:
1. The vertex is [tex]\((-1, 0)\)[/tex].
2. The function is positive: never.
3. The function is decreasing: for [tex]\( x < -1 \)[/tex].
4. The domain of the function is all real numbers.
5. The range of the function is all real numbers [tex]\(\leq 0\)[/tex].
