On a coordinate plane, a parabola opens down. It has an x-intercept at (negative 5, 0), a vertex at (negative 1, 16), a y-intercept at (0, 15), and an x-intercept at (3, 0).

Question

The function f(x) = –x2 − 2x + 15 is shown on the graph. What are the domain and range of the function?

The domain is all real numbers. The range is {y|y < 16}.
The domain is all real numbers. The range is {y|y ≤ 16}.
The domain is {x|–5 < x < 3}. The range is {y|y < 16}.
The domain is {x|–5 ≤ x ≤ 3}. The range is {y|y ≤ 16}.

Answer 1

To determine the domain and range of the function \( f(x) = -x^2 - 2x + 15 \), let's analyze the characteristics of the parabola.

1. Domain: Since the parabola is a quadratic function, it is defined for all real numbers. Therefore, the domain is: \[ \text{Domain: all real numbers.} \]

2. Range: The given parabola opens downwards, and its vertex is at the point (-1, 16). This means the maximum value of the function occurs at the vertex with a value of \( f(-1) = 16 \). Since the parabola opens downward, the function values will decrease from this maximum.

   Hence, the range of the function is: \[ \text{Range: } {y \mid y \leq 16} \]

Therefore, the correct choice is: The domain is all real numbers. The range is {y | y ≤ 16}.