Select the graph of [tex]f(x)=-0.5|x+2|-1[/tex].
Answer (2)
The function f ( x ) = − 0.5∣ x + 2∣ − 1 has a vertex at ( − 2 , − 1 ) and opens downwards. The negative coefficient indicates that the graph is an upside-down V shape, and the vertical compression by − 0.5 makes the sides less steep. Thus, the graph will reflect these characteristics visually.
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The function is an absolute value function with transformations.
The vertex of the absolute value function is found by setting the expression inside the absolute value to zero, which gives x = − 2 .
The y-coordinate of the vertex is f ( − 2 ) = − 1 , so the vertex is at ( − 2 , − 1 ) .
The graph opens downwards because the coefficient of the absolute value term is negative, so the final answer is a V-shaped graph with vertex at ( − 2 , − 1 ) opening downwards.
Explanation
Understanding the Function We are given the function f ( x ) = − 0.5∣ x + 2∣ − 1 . Our goal is to understand how the different parts of this function affect its graph, particularly the vertex and the direction it opens.
Transformations of the Absolute Value Function The absolute value function ∣ x ∣ has a V shape with its vertex at the origin (0,0). The function ∣ x + 2∣ shifts the graph 2 units to the left, so its vertex is at x = − 2 . The term − 0.5∣ x + 2∣ reflects the graph across the x-axis (because of the negative sign) and compresses it vertically by a factor of 0.5. Finally, the term − 1 shifts the entire graph 1 unit down.
Finding the Vertex The vertex of the transformed function f ( x ) = − 0.5∣ x + 2∣ − 1 is at x = − 2 . To find the y-coordinate of the vertex, we evaluate f ( − 2 ) = − 0.5∣ − 2 + 2∣ − 1 = − 0.5 ( 0 ) − 1 = − 1 . So, the vertex is at the point ( − 2 , − 1 ) . Since the coefficient of the absolute value term is negative (-0.5), the graph opens downwards, forming an upside-down V shape.
Conclusion Therefore, the graph of f ( x ) = − 0.5∣ x + 2∣ − 1 is a V-shaped graph that opens downwards, with its vertex at ( − 2 , − 1 ) .
Examples
Absolute value functions are used in various real-world applications, such as modeling distances, tolerances in engineering, and error bounds in measurements. For example, in manufacturing, the dimensions of a product might be specified with a tolerance, like 5 ± 0.1 cm. This means the actual dimension can be between 4.9 cm and 5.1 cm. The absolute value function helps define these boundaries and ensure the product meets the required specifications. Similarly, in physics, absolute values are used to calculate the magnitude of vectors, regardless of their direction.
