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Sagot :
Let's break down the key aspects of the function [tex]\( f(x) = -x^2 - 2x \)[/tex] step-by-step.
### Vertex
The vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is given by the formula [tex]\( \left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right) \)[/tex].
For the given function [tex]\( f(x) = -x^2 - 2x \)[/tex]:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = -2 \)[/tex]
To find the x-coordinate of the vertex:
[tex]\[ x_v = -\frac{b}{2a} = -\frac{-2}{2(-1)} = \frac{2}{-2} = -1 \][/tex]
Now, substitute [tex]\( x_v = -1 \)[/tex] back into the function to find the y-coordinate:
[tex]\[ f(-1) = -(-1)^2 - 2(-1) = -1 + 2 = 1 \][/tex]
So, the vertex of the function is [tex]\((-1, 1)\)[/tex].
### Increasing Interval
To determine where the function is increasing, consider the first derivative of the function:
[tex]\[ f'(x) = \frac{d}{dx}(-x^2 - 2x) = -2x - 2 \][/tex]
Set the first derivative equal to zero to find the critical points:
[tex]\[ -2x - 2 = 0 \][/tex]
[tex]\[ x = -1 \][/tex]
For a quadratic function with a negative leading coefficient (i.e., [tex]\( a < 0 \)[/tex]), the function is increasing to the left of the vertex.
Thus, the function is increasing for [tex]\( x \)[/tex] in the interval [tex]\( (-\infty, -1) \)[/tex].
### Decreasing Interval
For a quadratic function with a negative leading coefficient (i.e., [tex]\( a < 0 \)[/tex]), the function is decreasing to the right of the vertex.
Thus, the function is decreasing for [tex]\( x \)[/tex] in the interval [tex]\( (-1, \infty) \)[/tex].
### Domain
The domain of any polynomial function is all real numbers.
Thus, the domain of the function is [tex]\( (-\infty, \infty) \)[/tex].
### Range
Since the function opens downwards (due to the negative leading coefficient), the range is all values less than or equal to the y-coordinate of the vertex.
Thus, the range of the function is [tex]\( (-\infty, 1) \)[/tex].
### Summary
- The vertex is [tex]\(\boxed{(-1, 1)}\)[/tex].
- The function is increasing for [tex]\( \boxed{(-\infty, -1)} \)[/tex].
- The function is decreasing for [tex]\( \boxed{(-1, \infty)} \)[/tex].
- The domain of the function is [tex]\( \boxed{(-\infty, \infty)} \)[/tex].
- The range of the function is [tex]\( \boxed{(-\infty, 1)} \)[/tex].
### Vertex
The vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is given by the formula [tex]\( \left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right) \)[/tex].
For the given function [tex]\( f(x) = -x^2 - 2x \)[/tex]:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = -2 \)[/tex]
To find the x-coordinate of the vertex:
[tex]\[ x_v = -\frac{b}{2a} = -\frac{-2}{2(-1)} = \frac{2}{-2} = -1 \][/tex]
Now, substitute [tex]\( x_v = -1 \)[/tex] back into the function to find the y-coordinate:
[tex]\[ f(-1) = -(-1)^2 - 2(-1) = -1 + 2 = 1 \][/tex]
So, the vertex of the function is [tex]\((-1, 1)\)[/tex].
### Increasing Interval
To determine where the function is increasing, consider the first derivative of the function:
[tex]\[ f'(x) = \frac{d}{dx}(-x^2 - 2x) = -2x - 2 \][/tex]
Set the first derivative equal to zero to find the critical points:
[tex]\[ -2x - 2 = 0 \][/tex]
[tex]\[ x = -1 \][/tex]
For a quadratic function with a negative leading coefficient (i.e., [tex]\( a < 0 \)[/tex]), the function is increasing to the left of the vertex.
Thus, the function is increasing for [tex]\( x \)[/tex] in the interval [tex]\( (-\infty, -1) \)[/tex].
### Decreasing Interval
For a quadratic function with a negative leading coefficient (i.e., [tex]\( a < 0 \)[/tex]), the function is decreasing to the right of the vertex.
Thus, the function is decreasing for [tex]\( x \)[/tex] in the interval [tex]\( (-1, \infty) \)[/tex].
### Domain
The domain of any polynomial function is all real numbers.
Thus, the domain of the function is [tex]\( (-\infty, \infty) \)[/tex].
### Range
Since the function opens downwards (due to the negative leading coefficient), the range is all values less than or equal to the y-coordinate of the vertex.
Thus, the range of the function is [tex]\( (-\infty, 1) \)[/tex].
### Summary
- The vertex is [tex]\(\boxed{(-1, 1)}\)[/tex].
- The function is increasing for [tex]\( \boxed{(-\infty, -1)} \)[/tex].
- The function is decreasing for [tex]\( \boxed{(-1, \infty)} \)[/tex].
- The domain of the function is [tex]\( \boxed{(-\infty, \infty)} \)[/tex].
- The range of the function is [tex]\( \boxed{(-\infty, 1)} \)[/tex].
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