Which of the following could be the graph of [tex]f(x)=-a(x+b)^{1 / 3}[/tex] if both a and b are positive numbers?
A.
B.
Answer (2)
The function f ( x ) = − a ( x + b ) 1/3 with 0"> a > 0 and 0"> b > 0 is a cube root function that has been shifted horizontally and reflected vertically. The graph should:
Pass through ( − b , 0 ) .
Be decreasing.
Resemble a reflected cube root function shifted to the left. Without the graphs, we can't choose one, but these characteristics should help identify the correct graph.
Explanation
Analyzing the Function We are given the function f ( x ) = − a ( x + b ) 1/3 , where 0"> a > 0 and 0"> b > 0 . We need to determine which of the provided graphs could represent this function. Let's analyze the function's properties to narrow down the possibilities.
Horizontal Shift The basic cube root function y = x 1/3 passes through the origin (0,0) and is an increasing function. The term ( x + b ) inside the cube root shifts the graph horizontally. Since 0"> b > 0 , the graph shifts to the left by b units. This means the graph will pass through the point ( − b , 0 ) .
Vertical Reflection and Stretch The term − a in front of the cube root reflects the graph across the x-axis and stretches it vertically. Since 0"> a > 0 , − a is negative, so the graph is reflected across the x-axis. This means that as x increases, f ( x ) will decrease.
Expected Graph Shape Combining these transformations, we expect the graph to pass through ( − b , 0 ) and to be decreasing. The graph should resemble a reflected cube root function shifted to the left.
Conclusion Based on this analysis, we need to look for a graph that is decreasing and passes through a point on the negative x-axis (since 0"> b > 0 ). Without the actual graphs to choose from, we can describe the key features to look for: a decreasing function that intersects the x-axis at x = − b , where b is a positive number.
Examples
Understanding the transformations of functions is crucial in many fields. For example, in physics, the motion of an object can be modeled by functions, and shifts and reflections can represent changes in initial position or direction. In economics, demand and supply curves can be modeled by functions, and understanding how these curves shift can help predict market changes. In computer graphics, transformations of functions are used to manipulate objects in 2D and 3D space, allowing for realistic animations and simulations.
The graph of f ( x ) = − a ( x + b ) 1/3 is a decreasing curve that passes through the point ( − b , 0 ) and reflects the cube root function downward due to the negative coefficient. It shows a horizontal shift to the left by b units and decreases overall as x increases. Therefore, look for a graph that fits these features.
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