To determine the domain and range of the quadratic function \( f(x) = -6(x + 1)^2 - 9 \), we can analyze the function step-by-step.
Domain:
The domain of a quadratic function is typically all real numbers unless specified otherwise by constraints (like square roots or logarithms). Since this function is a polynomial, its domain is:
\[ \text{Domain} = (-\infty, \infty) \]
Range:
Next, we need to find the range. The function is in vertex form \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola, and \( a \) indicates the direction of the opening.
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Identify the vertex: The vertex can be identified from the vertex form. Here, \( a = -6 \), \( h = -1 \), and \( k = -9 \). Thus, the vertex is \( (-1, -9) \).
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Determine the direction of opening: Since \( a = -6 < 0 \), the parabola opens downward.
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Find the range: Because the parabola opens downward, the maximum value of \( f(x) \) is at the vertex \( k = -9 \). The function can take any value that is less than or equal to this maximum.
Thus, the range is:
\[ \text{Range} = (-\infty, -9] \]
Summary:
- Domain: \((- \infty, \infty)\)
- Range: \((- \infty, -9]\)
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