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 $\square$ .
The function is positive $\square$
The function is decreasing $\square$
The domain of the function is $\square$
The range of the function is $\square$

The vertex of the function f ( x ) = − ( x + 1 ) 2 is ( − 1 , 0 ) .
The function is never positive.
The function is decreasing for -1"> x > − 1 .
The domain of the function is ( − ∞ , ∞ ) .
The range of the function is ( − ∞ , 0 ] .

Explanation

Analyzing the Function The given function is f ( x ) = − ( x + 1 ) 2 . We need to identify the vertex, where the function is positive, where it is decreasing, its domain, and its range.

Finding the Vertex The vertex of a parabola in the form f ( x ) = a ( x − h ) 2 + k is ( h , k ) . Here, f ( x ) = − ( x + 1 ) 2 = − 1 ( x − ( − 1 ) ) 2 + 0 . Therefore, the vertex is ( − 1 , 0 ) .

Determining Where the Function is Positive Since the coefficient of the ( x + 1 ) 2 term is negative, the parabola opens downwards. This means the function is never positive. It is zero only at the vertex x = − 1 .

Finding Where the Function is Decreasing For a parabola opening downwards, the function is decreasing to the right of the vertex. Thus, f ( x ) is decreasing for -1"> x > − 1 .

Determining the Domain The domain of a quadratic function (a polynomial) is all real numbers, which can be written as ( − ∞ , ∞ ) .

Determining the Range Since the parabola opens downwards and the vertex is at ( − 1 , 0 ) , the maximum value of the function is 0. Therefore, the range is all real numbers less than or equal to 0, which can be written as ( − ∞ , 0 ] .


Examples
Understanding the properties of quadratic functions like f ( x ) = − ( x + 1 ) 2 is crucial in various real-world applications. For instance, designing a parabolic reflector for a flashlight or satellite dish involves knowing the vertex and the direction in which the parabola opens. Similarly, in physics, the trajectory of a projectile under gravity (ignoring air resistance) follows a parabolic path, and analyzing its equation helps determine the maximum height and range of the projectile. These applications demonstrate how a solid grasp of quadratic functions can aid in solving practical problems in engineering and science.

Answered by GinnyAnswer | 2025-07-04

The vertex of the function is ( − 1 , 0 ) . The function is never positive, decreasing for -1"> x > − 1 , has a domain of ( − ∞ , ∞ ) , and a range of ( − ∞ , 0 ] .
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Answered by Anonymous | 2025-07-14