Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
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].
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
