Use the drop-down menus to describe the key aspects of the function [tex]f(x)=-x^2-2 x-1[/tex].

The vertex is the $\square$
The function is increasing $\square$
The function is decreasing $\square$
The domain of the function is $\square$
The range of the function is $\square$

Find the vertex of the quadratic function f ( x ) = − x 2 − 2 x − 1 by completing the square or using the formula h = − 2 a b ​ and k = f ( h ) , which gives the vertex ( − 1 , 0 ) .
Determine that the parabola opens downward since the coefficient of x 2 is negative, indicating a maximum at the vertex.
Identify the increasing interval as ( − ∞ , − 1 ) and the decreasing interval as ( − 1 , ∞ ) .
State the domain as ( − ∞ , ∞ ) and the range as ( − ∞ , 0 ] .

Vertex: ( − 1 , 0 ) , Increasing: ( − ∞ , − 1 ) , Decreasing: ( − 1 , ∞ ) , Domain: ( − ∞ , ∞ ) , Range: ( − ∞ , 0 ] ​
Explanation

Problem Analysis We are given the quadratic function f ( x ) = − x 2 − 2 x − 1 . Our goal is to determine its vertex, the intervals where it is increasing and decreasing, its domain, and its range.

Finding the Vertex To find the vertex, we can complete the square or use the formula h = − 2 a b ​ and k = f ( h ) , where the vertex is ( h , k ) . In our case, a = − 1 , b = − 2 , and c = − 1 . Thus, h = − 2 ( − 1 ) − 2 ​ = − − 2 − 2 ​ = − 1. Now we find k by plugging h into the function: k = f ( − 1 ) = − ( − 1 ) 2 − 2 ( − 1 ) − 1 = − 1 + 2 − 1 = 0. So, the vertex is ( − 1 , 0 ) .

Determining the Parabola's Orientation Since the coefficient of the x 2 term is negative ( a = − 1 < 0 ), the parabola opens downward. This means the vertex is a maximum point. The axis of symmetry is the vertical line x = − 1 .

Increasing and Decreasing Intervals Because the parabola opens downward, the function is increasing to the left of the vertex and decreasing to the right of the vertex. Therefore, the function is increasing on the interval ( − ∞ , − 1 ) and decreasing on the interval ( − 1 , ∞ ) .

Determining the Domain The domain of any quadratic function is all real numbers. So, the domain of f ( x ) is ( − ∞ , ∞ ) .

Determining the Range Since the parabola opens downward and the vertex is the maximum point, the range of the function is all real numbers less than or equal to the y-coordinate of the vertex. Thus, the range is ( − ∞ , 0 ] .

Final Answer In summary:



The vertex is ( − 1 , 0 ) .
The function is increasing on ( − ∞ , − 1 ) .
The function is decreasing on ( − 1 , ∞ ) .
The domain is ( − ∞ , ∞ ) .
The range is ( − ∞ , 0 ] .

Examples
Understanding the properties of quadratic functions like f ( x ) = − x 2 − 2 x − 1 is crucial in various real-world applications. For instance, consider a scenario where you're designing a parabolic arch for a bridge. Knowing the vertex helps determine the maximum height of the arch, while understanding the increasing and decreasing intervals aids in ensuring structural stability. Similarly, in physics, projectile motion follows a parabolic path, and analyzing the quadratic function that describes the trajectory allows us to find the maximum height and range of the projectile. These concepts are also applicable in economics, where quadratic functions can model cost and revenue curves, helping businesses optimize their operations.

Answered by GinnyAnswer | 2025-07-05