On a coordinate plane, a parabola opens down. It goes through (negative 3, negative 4), has a vertex at (negative 1, 0), and goes through (1, negative 4).

Question

The graph of the function f(x) = –(x + 1)2 is shown. Use the drop-down menus to describe the key aspects of the function.

The vertex is the
.

The function is positive
.

The function is decreasing
.

The domain of the function is
.

The range of the function is

Answer 1

Let's analyze the given parabola and its properties.

1. The vertex is the: The vertex is given as \((-1, 0)\).

   • Answer: (–1, 0)

2. The function is positive: Since the vertex is at \((–1, 0)\) and the parabola opens downwards, the function will be positive for values of \(x\) that are less than \(-1\) but will be zero at the vertex and negative for values greater than \(-1\).

   • Answer: for x < –1

3. The function is decreasing: The function decreases as it moves from the vertex to the left, which is from \(-1\) going leftwards. The function also decreases from the vertex to the right until it reaches the end of the interval where it returns to its negative values.

   • Answer: on the interval (–∞, –1) and for x > –1 (but it only decreases until it reaches (1, -4) in this context).

4. The domain of the function is: Since it's a parabola that continues indefinitely in both directions on the x-axis, the domain is all real numbers.

   • Answer: (–∞, ∞)

5. The range of the function is: The maximum point of the parabola is at the vertex \((–1, 0)\), and since the parabola opens downwards, the range includes all values from \(-\infty\) to \(0\).

   • Answer: (–∞, 0]

Summarizing the answers in the drop-down menu structure:

1. The vertex is the (–1, 0).
2. The function is positive for x < –1.
3. The function is decreasing on the interval (–∞, –1).
4. The domain of the function is (–∞, ∞).
5. The range of the function is (–∞, 0].