How is the graph of the parent function [tex]y=\frac{1}{x}[/tex] transformed to create the graph of [tex]y=-\frac{1}{3 x}[/tex] ?
A. It is translated 3 units down and reflected over the [tex]y[/tex]-axis.
B. It is horizontally compressed by a factor of 3 and reflected over the [tex]x[/tex]-axis.
C. It is horizontally stretched by a factor of 3 and reflected over the [tex]y[/tex]-axis.
D. It is translated 3 units down and reflected over the [tex]x[/tex]-axis.
Answer (2)
The transformed function can be written as y = − 3 1 ⋅ x 1 .
Replacing x with 3 x in the parent function represents a horizontal compression by a factor of 3 1 .
The negative sign represents a reflection over the x-axis.
Alternatively, the graph is horizontally stretched by a factor of 3 and reflected over the x-axis. The final answer is: It is horizontally stretched by a factor of 3 and reflected over the x -axis.
Explanation
Understanding the Problem We are given the parent function y = x 1 and the transformed function y = − 3 x 1 . We need to determine how the graph of the parent function is transformed to obtain the graph of the transformed function.
Rewriting the Transformed Function We can rewrite the transformed function as y = − 3 1 ⋅ x 1 . This indicates two transformations: a vertical compression by a factor of 3 1 and a reflection over the x-axis. Alternatively, we can rewrite the transformed function as y = − 3 x 1 = − 3 1 ⋅ x 1 = − x / ( 1/3 ) 1 . This can be interpreted as a horizontal stretch by a factor of 3 (since x is replaced by x / ( 1/3 ) = 3 x ) and a reflection over the x-axis.
Analyzing the Transformations The transformation y = 3 x 1 can be seen as replacing x with 3 x in the parent function y = x 1 . This represents a horizontal compression by a factor of 3 1 . The negative sign in y = − 3 x 1 represents a reflection over the x-axis. Therefore, the graph of the parent function is horizontally compressed by a factor of 3 1 and reflected over the x-axis. However, the options given do not have 3 1 , so we need to consider the alternative interpretation.
Alternative Interpretation Consider the transformation y = − 3 x 1 . We can rewrite this as y = − 3 1 ⋅ x 1 = − x / ( 1/3 ) 1 . Replacing x with 1/3 x is equivalent to a horizontal stretch by a factor of 3. The negative sign represents a reflection over the x-axis. Thus, the graph of the parent function is horizontally stretched by a factor of 3 and reflected over the x-axis.
Conclusion The correct answer is that the graph of the parent function is horizontally stretched by a factor of 3 and reflected over the x-axis.
Examples
Understanding transformations of functions is crucial in many fields. For example, in physics, understanding how graphs of motion equations transform can help predict the behavior of objects under different conditions. Similarly, in economics, transformations of supply and demand curves can illustrate the impact of taxes or subsidies on market equilibrium. This problem demonstrates how changing the input or output of a function affects its graphical representation, a fundamental concept in mathematical modeling.
The graph of the parent function y = x 1 is transformed into y = − 3 x 1 through a horizontal stretch by a factor of 3 and a reflection over the x-axis. This is due to the horizontal manipulation of the input and the negative multiplier affecting the output. Therefore, the correct option is that the transformation is indicated by option B.
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