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Understanding horizontal reflection is crucial when transforming the graph of a function. For a parent function like \( y = \frac{1}{x} \), a horizontal reflection means flipping the graph across the y-axis. This transformation can be identified by the presence of a negative sign in front of the variable within the function's expression.In the step-by-step solution, the function is rewritten as \( y = \frac{1}{-(x-1)} \). The negative sign here causes the graph's branches to move from the first and third quadrants to the second and fourth quadrants, effectively reflecting it across the y-axis. To summarize:

• Reflection over the y-axis is initiated by a negative sign in front of \( x \).
• The graph of \( y = \frac{1}{x} \) flips horizontally.

This reflection is part of understanding how different transformations affect the graph's shape and position.

Vertical asymptotes are the invisible boundaries where the function approaches but never touches or crosses. They occur where the denominator of a rational function equals zero, leading to undefined values for the function.In this exercise, the transformation indicated a shift in the vertical asymptote from its original position at \( x=0 \) to \( x=1 \). This shift occurs because the expression \( 1-x \) equals zero at \( x=1 \). Understanding this:

• The vertical asymptote for \( y = \frac{1}{1-x} \) is at \( x=1 \).
• The hyperbola never crosses this vertical line.

Being aware of where vertical asymptotes are positioned helps us accurately understand and predict the behavior of graphs.

Inverse functions play a significant role in understanding graph transformations, even though not directly applied here. It is often useful to think of inverse functions as reflections across the line \( y = x \). While \( y = \frac{1}{x} \) is not reversed in this exercise, recognizing that transformations can relate to inverse operations can greatly aid comprehension:

• If \( f(x) \) is the original function, then \( f^{-1}(x) \) would be its inverse, flipping the input and output.
• The graph of \( y = f^{-1}(x) \) is a reflection of \( y = f(x) \) across the line \( y = x \).

Understanding inverse functions can deepen understanding of symmetry and reflections, which are key in graph transformations.

Graphing utilities are powerful tools for visualizing mathematical functions and their transformations. They allow us to plot functions quickly and verify our sketches or theoretical predictions.When using a graphing utility for the equation \( y = \frac{1}{1-x} \), it confirms both the position of the vertical asymptote and the effects of the horizontal reflection and translation:

• Graphs can be plotted to check the shapes and asymptotes.
• Adjusting function parameters directly shows transformation outcomes.

By employing a graphing utility, we can ensure our understanding and graphical representations are accurate, providing instant feedback when exploring functions.