How is the graph of the parent function of [tex]$y=\sqrt[3]{x}$[/tex] transformed to produce the graph [tex]$y=\sqrt[3]{\frac{1}{2}} x$[/tex] ?
A. It is horizontally stretched by a factor of [tex]$\frac{1}{2}$[/tex].
B. It is vertically stretched by a factor of [tex]$\frac{1}{2}$[/tex].
C. It is translated left by [tex]$\frac{1}{2}$[/tex] unit.
D. It is translated right by [tex]$\frac{1}{2}$[/tex] unit.
Answer (2)
Rewrite the transformed function: y = 3 2 1 x .
Identify the horizontal stretch: The function represents a horizontal stretch by a factor of 2.
Analyze the options: None of the options directly state a horizontal stretch by a factor of 2.
Conclude: The closest interpretation, though not perfectly accurate, relates to the horizontal transformation of the x-coordinate. The correct answer should be a horizontal stretch by a factor of 2, but since that is not an option, we acknowledge the discrepancy.
There is no correct answer among the options provided.
There is no \boxed{} notation to provide, as none of the options are correct.
Explanation
Understanding the Problem We are given the parent function y = 3 x and the transformed function y = 3 2 1 x . We want to determine how the graph of the parent function is transformed to produce the graph of the transformed function.
Rewriting the Transformed Function We can rewrite the transformed function as y = 3 2 1 x = 3 2 1 3 x . This form suggests a vertical compression by a factor of 3 2 1 ≈ 0.7937 .
Horizontal Stretch Alternatively, we can view the transformation as a horizontal stretch. The general form for a horizontal stretch by a factor of k is y = f ( k 1 x ) . In our case, we have y = 3 2 1 x , which means k 1 = 2 1 , so k = 2 . This indicates a horizontal stretch by a factor of 2.
Determining the Transformation Comparing the options, we see that the transformation is a horizontal stretch by a factor of 2. Since the question asks how the graph is transformed, we should choose the option that describes a horizontal stretch by a factor of 2. The option 'It is horizontally stretched by a factor of 2 1 ' is incorrect. However, since 3 2 1 x can be seen as a horizontal stretch by a factor of 2, we can rewrite the transformed function as y = 3 2 1 x . This corresponds to a horizontal stretch by a factor of 2.
Analyzing the Options The correct transformation is a horizontal stretch by a factor of 2. However, none of the options directly state this. Let's analyze the given options:
It is horizontally stretched by a factor of 2 1 . (Incorrect)
It is vertically stretched by a factor of 2 1 . (Incorrect)
It is translated left by 2 1 unit. (Incorrect)
It is translated right by 2 1 unit. (Incorrect)
Since y = 3 2 1 x can be interpreted as a horizontal stretch by a factor of 2, none of the provided options are correct. However, if the question meant y = 2 1 3 x , then the answer would be 'It is vertically stretched by a factor of 2 1 '. But that is not the given equation. The given equation is y = 3 2 1 x .
Final Answer The graph of y = 3 x is transformed to produce the graph y = 3 2 1 x by a horizontal stretch by a factor of 2. However, this option is not available. Thus, the closest answer is that it is horizontally stretched by a factor of 2 1 if we consider the transformation x → 2 1 x . But this is a compression, not a stretch. So, the correct answer should be a horizontal stretch by a factor of 2. Since that is not an option, we must assume there is a typo in the question or the options.
Examples
Understanding transformations of functions is crucial in many fields. For example, in image processing, stretching or compressing an image can be represented mathematically as a transformation of coordinates. Similarly, in physics, scaling time or space coordinates can be described using function transformations. These transformations help us analyze and manipulate data in a predictable way.
The graph of y = 3 x is transformed to y = 3 2 1 x by a horizontal stretch by a factor of 2. However, this specific option is not listed among the choices. Therefore, none of the given options correctly describe the transformation.
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