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Sagot :
Sure, let's describe the key aspects of the function [tex]\( f(x) = -x^2 - 2x - 1 \)[/tex] step by step.
1. Vertex (Maximum Value):
- The function [tex]\( f(x) = -x^2 - 2x - 1 \)[/tex] is a quadratic equation in the form [tex]\( ax^2 + bx + c \)[/tex] with [tex]\( a = -1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -1 \)[/tex].
- For a quadratic function [tex]\( ax^2 + bx + c \)[/tex], the vertex is found using the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( x = -\frac{-2}{2 \cdot (-1)} = 1 \)[/tex].
- To find the vertex's y-coordinate, substitute [tex]\( x = -1 \)[/tex] back into the function:
[tex]\[ f(-1) = -(-1)^2 - 2(-1) - 1 = -1 + 2 - 1 = 0 \][/tex]
- Thus, the vertex is at [tex]\((-1, 0)\)[/tex], and since the coefficient of [tex]\( x^2 \)[/tex] is negative, this vertex is the maximum value [tex]\( v \)[/tex].
2. Increasing Interval:
- A quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] with [tex]\( a < 0 \)[/tex] is upward-facing (concave down), meaning it decreases to the vertex and then increases after the vertex.
- Since our vertex is at [tex]\( x = -1 \)[/tex], the function is increasing on the interval [tex]\( (-1, \infty) \)[/tex].
3. Decreasing Interval:
- Similarly, the function is decreasing before the vertex. Therefore, it is decreasing on the interval [tex]\( (-\infty, -1] \)[/tex].
4. Domain:
- The domain of any quadratic function is all real numbers, which is [tex]\( (-\infty, \infty) \)[/tex].
5. Range:
- The range is determined by the vertex and the direction the parabola opens. Since the vertex is [tex]\((-1, 0)\)[/tex] and the parabola opens downwards, the range is from [tex]\( 0 \)[/tex] (the y-coordinate of the vertex) to negative infinity.
- Hence, the range of the function is [tex]\( (-\infty, 0] \)[/tex].
In summary:
- The vertex is the maximum value [tex]\( v \)[/tex].
- The function is increasing (choose the interval) [tex]\( (-1, \infty) \)[/tex].
- The function is decreasing (choose the interval) [tex]\( (-\infty, -1] \)[/tex].
- The domain of the function is [tex]\( (-\infty, \infty) \)[/tex].
- The range of the function is [tex]\( (-\infty, 0] \)[/tex].
1. Vertex (Maximum Value):
- The function [tex]\( f(x) = -x^2 - 2x - 1 \)[/tex] is a quadratic equation in the form [tex]\( ax^2 + bx + c \)[/tex] with [tex]\( a = -1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -1 \)[/tex].
- For a quadratic function [tex]\( ax^2 + bx + c \)[/tex], the vertex is found using the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( x = -\frac{-2}{2 \cdot (-1)} = 1 \)[/tex].
- To find the vertex's y-coordinate, substitute [tex]\( x = -1 \)[/tex] back into the function:
[tex]\[ f(-1) = -(-1)^2 - 2(-1) - 1 = -1 + 2 - 1 = 0 \][/tex]
- Thus, the vertex is at [tex]\((-1, 0)\)[/tex], and since the coefficient of [tex]\( x^2 \)[/tex] is negative, this vertex is the maximum value [tex]\( v \)[/tex].
2. Increasing Interval:
- A quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] with [tex]\( a < 0 \)[/tex] is upward-facing (concave down), meaning it decreases to the vertex and then increases after the vertex.
- Since our vertex is at [tex]\( x = -1 \)[/tex], the function is increasing on the interval [tex]\( (-1, \infty) \)[/tex].
3. Decreasing Interval:
- Similarly, the function is decreasing before the vertex. Therefore, it is decreasing on the interval [tex]\( (-\infty, -1] \)[/tex].
4. Domain:
- The domain of any quadratic function is all real numbers, which is [tex]\( (-\infty, \infty) \)[/tex].
5. Range:
- The range is determined by the vertex and the direction the parabola opens. Since the vertex is [tex]\((-1, 0)\)[/tex] and the parabola opens downwards, the range is from [tex]\( 0 \)[/tex] (the y-coordinate of the vertex) to negative infinity.
- Hence, the range of the function is [tex]\( (-\infty, 0] \)[/tex].
In summary:
- The vertex is the maximum value [tex]\( v \)[/tex].
- The function is increasing (choose the interval) [tex]\( (-1, \infty) \)[/tex].
- The function is decreasing (choose the interval) [tex]\( (-\infty, -1] \)[/tex].
- The domain of the function is [tex]\( (-\infty, \infty) \)[/tex].
- The range of the function is [tex]\( (-\infty, 0] \)[/tex].
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