Select the graph of [tex]f(x)=-0.5|x+2|-1[/tex]

The function f ( x ) = − 0.5∣ x + 2∣ − 1 has a vertex at ( − 2 , − 1 ) and opens downwards. The slopes to the right and left of the vertex are − 0.5 and 0.5 , respectively. Thus, when selecting the graph, look for these characteristics.
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Answered by Anonymous | 2025-07-04

The function is an absolute value function with vertex at ( − 2 , − 1 ) .
The graph opens downwards because the coefficient of the absolute value term is negative.
The slope of the right side of the vertex is − 0.5 , and the slope of the left side is 0.5 .
The graph of f ( x ) = − 0.5∣ x + 2∣ − 1 is an absolute value function with vertex at ( − 2 , − 1 ) that opens downwards, with the specified slopes. Therefore, we can select the correct graph based on these features.

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Explanation

Understanding the Function We are given the function f ( x ) = − 0.5∣ x + 2∣ − 1 and we need to select its graph.

Finding the Vertex The function is an absolute value function of the form f ( x ) = a ∣ x − h ∣ + k , where a = − 0.5 , h = − 2 , and k = − 1 . The vertex of the absolute value function is at the point ( h , k ) . In this case, the vertex is at ( − 2 , − 1 ) .

Determining the Direction and Slope Since a = − 0.5 is negative, the graph opens downwards. The slope of the right side of the vertex is − 0.5 , and the slope of the left side of the vertex is 0.5 .

Conclusion The graph of f ( x ) = − 0.5∣ x + 2∣ − 1 is an absolute value function with vertex at ( − 2 , − 1 ) that opens downwards. The right side of the graph has a slope of − 0.5 , and the left side has a slope of 0.5 .


Examples
Absolute value functions are used in many real-world applications, such as finding the distance between two points, modeling the behavior of electrical circuits, and optimizing the design of lenses. For example, in engineering, absolute value functions can be used to model the tolerance of a component, ensuring that it falls within acceptable limits. In finance, they can be used to model the risk associated with an investment, providing a measure of the potential loss or gain.

Answered by GinnyAnswer | 2025-07-04