To analyze the function \( f(x) = -x^2 - 2x - 1 \), let's determine each aspect step by step.
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The vertex: The function is a quadratic in the form \( f(x) = ax^2 + bx + c \). The vertex can be found using the formula \( x = -\frac{b}{2a} \). Here, \( a = -1 \) and \( b = -2 \). Thus:
\[ x = -\frac{-2}{2 \cdot -1} = 1 \]
Now we compute \( f(1) \):
\[ f(1) = -1^2 - 2 \cdot 1 - 1 = -1 - 2 - 1 = -4 \]
Therefore, the vertex is at \( (1, -4) \).
The vertex is the (1, -4).
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The function is increasing: The function decreases to the left of the vertex (to the left of \( x = -1 \)), and increases to the right of the vertex.
The function is increasing \( x > -1 \).
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The function is decreasing: The function increases up to the vertex and decreases after.
The function is decreasing \( x < -1 \).
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The domain of the function: Since \( f(x) \) is a polynomial function, its domain is all real numbers.
The domain of the function is \( (-\infty, \infty) \).
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The range of the function: As the parabola opens downwards (since the coefficient of \( x^2 \) is negative), its maximum value is the y-coordinate of the vertex, \( -4 \). The function can take any value \( y \) less than or equal to this maximum.
The range of the function is \( (-\infty, -4] \).
To summarize:
- The vertex is the (1, -4).
- The function is increasing \( x > -1 \).
- The function is decreasing \( x < -1 \).
- The domain of the function is \((- \infty, \infty)\).
- The range of the function is \((- \infty, -4]\).