Question

determine the domain and range of the quadratic function f (x) = -6(x+1)^2 - 9
a domain R; range y > 9
b domain R; range y < 9
c domain R; range y > -9
d domain R; range y < -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.

1. **Domain**:
The domain of any polynomial function, including quadratics, is all real numbers. Therefore, the domain is:
\[
\text{Domain: } \mathbb{R}
\]

2. **Range**:
The function \( f(x) = -6(x+1)^2 - 9 \) is a downward-opening parabola (since the coefficient of the squared term, -6, is negative).

To find the vertex of the parabola, we can rewrite the function. The vertex form of a quadratic function is \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex.

In this case, we can identify \( h = -1 \) and \( k = -9 \), which means the vertex of the parabola is at the point \( (-1, -9) \).

The maximum value of the function occurs at the vertex, which is \( -9 \) (the highest point) because the parabola opens downward. Thus, the range of the function is all values less than or equal to \( -9 \).

Therefore, the range is:
\[
\text{Range: } y \leq -9
\]

From the options given, the correct choice regarding the domain and range is:
- **Domain: \( \mathbb{R} \)**
- **Range: \( y < -9 \)**

So the answer is:
**d domain R; range y < -9.**