Which of the following is the graph of [tex]$f(x)=-0.5|x+3|-2$[/tex]?
Answer (2)
The graph of f ( x ) = − 0.5∣ x + 3∣ − 2 is a transformation of the absolute value function, shifted 3 units left, 2 units down, reflected across the x-axis, and vertically compressed. Its vertex is at ( − 3 , − 2 ) and it opens downwards. Look for the graph that matches these features.
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The function is a transformation of the absolute value function y = ∣ x ∣ .
The graph is horizontally translated 3 units to the left, reflected across the x-axis, vertically compressed by a factor of 0.5, and vertically translated 2 units down.
The vertex of the graph is at ( − 3 , − 2 ) .
The graph opens downwards and is wider than the basic absolute value function.
The final answer is the graph with vertex ( − 3 , − 2 ) that opens downwards. The graph is f ( x ) = − 0.5∣ x + 3∣ − 2 .
Explanation
Analyzing the Function We are given the function f ( x ) = − 0.5∣ x + 3∣ − 2 and we need to identify its graph. This is an absolute value function that has been transformed. Let's analyze the transformations.
Identifying Transformations The basic absolute value function is y = ∣ x ∣ . The given function f ( x ) = − 0.5∣ x + 3∣ − 2 has undergone the following transformations:
Horizontal Translation: The term ( x + 3 ) inside the absolute value shifts the graph 3 units to the left.
Vertical Compression and Reflection: The term − 0.5 multiplies the absolute value. The 0.5 causes a vertical compression by a factor of 0.5, and the negative sign reflects the graph across the x-axis, so it opens downwards.
Vertical Translation: The term − 2 at the end shifts the graph 2 units down.
Finding the Vertex The vertex of the absolute value function y = ∣ x ∣ is at ( 0 , 0 ) . Due to the transformations, the vertex of f ( x ) = − 0.5∣ x + 3∣ − 2 will be at ( − 3 , − 2 ) . This is because the horizontal translation shifts the vertex 3 units to the left, and the vertical translation shifts the vertex 2 units down.
Determining the Direction and Compression Since the coefficient of the absolute value term is negative, the graph opens downwards. The vertical compression by a factor of 0.5 means that the graph is wider than the basic absolute value function.
Identifying the Graph Based on the vertex ( − 3 , − 2 ) , the fact that the graph opens downwards, and the vertical compression, we can identify the correct graph.
Examples
Absolute value functions are used in various real-world applications, such as modeling distances, tolerances in engineering, and error margins in statistics. For example, in manufacturing, the acceptable deviation from a target measurement can be modeled using an absolute value function. If a machine part is supposed to be 5 cm long, a tolerance of 0.1 cm can be expressed as ∣ x − 5∣ ≤ 0.1 , where x is the actual length of the part. This ensures that the part is within acceptable limits.
