Which of the following is the graph of [tex]f(x)=-0.5|x+3|-2[/tex]?
Answer (1)
The function is an absolute value function with vertex at ( − 3 , − 2 ) .
The graph opens downwards because of the negative coefficient − 0.5 .
The slopes are 0.5 to the left of the vertex and − 0.5 to the right.
The graph of f ( x ) = − 0.5∣ x + 3∣ − 2 has a vertex at ( − 3 , − 2 ) and opens downwards with slopes of 0.5 and − 0.5 on either side of the vertex.
Explanation
Understanding the Function The given function is f ( x ) = − 0.5∣ x + 3∣ − 2 . We need to identify the graph of this absolute value function.
Identifying Transformations The function is a transformation of the basic absolute value function ∣ x ∣ . The transformations include a horizontal shift, a vertical stretch/compression, a reflection, and a vertical shift.
Finding the Vertex The vertex of the absolute value function is the point where the expression inside the absolute value is zero. In this case, x + 3 = 0 , so x = − 3 . The y -coordinate of the vertex is f ( − 3 ) = − 0.5∣ ( − 3 ) + 3∣ − 2 = − 0.5 ( 0 ) − 2 = − 2 . Thus, the vertex is at ( − 3 , − 2 ) .
Determining the Direction and Compression The coefficient of the absolute value term is − 0.5 , which is negative. This means the graph opens downwards. The factor of 0.5 indicates a vertical compression by a factor of 2.
Determining the Slopes To the right of the vertex ( -3"> x > − 3 ), the function is f ( x ) = − 0.5 ( x + 3 ) − 2 = − 0.5 x − 1.5 − 2 = − 0.5 x − 3.5 . The slope is − 0.5 . To the left of the vertex ( x < − 3 ), the function is f ( x ) = − 0.5 ( − ( x + 3 )) − 2 = 0.5 ( x + 3 ) − 2 = 0.5 x + 1.5 − 2 = 0.5 x − 0.5 . The slope is 0.5 .
Identifying the Graph Based on the vertex ( − 3 , − 2 ) and the fact that the graph opens downwards, we can identify the correct graph. The graph should have a vertex at ( − 3 , − 2 ) and open downwards with slopes of 0.5 and − 0.5 on either side of the vertex.
Final Answer The graph of f ( x ) = − 0.5∣ x + 3∣ − 2 is an absolute value function with vertex at ( − 3 , − 2 ) that opens downwards.
Examples
Absolute value functions are used in many real-world applications, such as modeling distances, tolerances, and error bounds. For example, in engineering, absolute value functions can be used to define the acceptable range of values for a component's dimensions. In finance, they can be used to model the risk associated with an investment. Understanding the transformations of absolute value functions allows us to analyze and interpret these models effectively. The function f ( x ) = − 0.5∣ x + 3∣ − 2 can model a scenario where the profit decreases linearly with the distance from a certain point, with a fixed initial loss.
