The graph of the function $f(x)=-(x+1)^2$ is shown. Use the dropdown 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 $\square$
Answer (1)
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 . This is a quadratic function, and its graph is a parabola. The negative sign in front of the parenthesis indicates that the parabola opens downwards. We need to find the vertex, determine where the function is positive, where it is decreasing, and find its domain and range.
Finding the Vertex The vertex form of a parabola is f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex. In our case, f ( x ) = − ( x + 1 ) 2 = − 1 ( x − ( − 1 ) ) 2 + 0 . Therefore, the vertex is at ( − 1 , 0 ) .
Determining Where the Function is Positive Since the parabola opens downwards and the vertex is on the x-axis, the function is never positive. It is either zero (at the vertex) or negative.
Determining Where the Function is Decreasing The parabola opens downwards, so the function is increasing to the left of the vertex and decreasing to the right of the vertex. Therefore, the function is decreasing for -1"> x > − 1 .
Finding the Domain The domain of a quadratic function is all real numbers, since we can plug in any real number for x . So the domain is ( − ∞ , ∞ ) .
Finding 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 of the function is all real numbers less than or equal to 0, which is ( − ∞ , 0 ] .
Examples
Understanding the properties of quadratic functions like f ( x ) = − ( x + 1 ) 2 is crucial in various real-world applications. For instance, consider the trajectory of a projectile, such as a ball thrown into the air. The height of the ball over time can be modeled by a quadratic function. Knowing the vertex (maximum height), the range (possible heights), and where the function is increasing or decreasing (when the ball is going up or down) helps in predicting the ball's motion and optimizing its trajectory for activities like sports or launching mechanisms.
