Question

determine the domain and range of the quadratic function f (x) = -6(x + 1) ^2 - 9

Answer 1

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.

1. **Identify the vertex:**
The vertex can be identified from the vertex form. Here, \( a = -6 \), \( h = -1 \), and \( k = -9 \). Thus, the vertex is \( (-1, -9) \).

2. **Determine the direction of opening:**
Since \( a = -6 < 0 \), the parabola opens downward.

3. **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:** \((- \infty, -9]\)