What transformations were applied to the graph of the parent function y = tangent(x) to produce the function graphed below? On a coordinate plane, the x axis ranges from -2π to 2π with an interval of π/4 units, and the y axis ranges from -6 to 6 with an interval of 2 units. It shows a curve that oscillates between positive and negative infinity as x approaches multiples of π. The curve has vertical asymptotes at x = ±nπ. The curve crosses the x axis at (7π/6, 0), (π/6, 0), and (-5π/5, 0).

To understand the transformations applied to the parent function y = tan ( x ) to produce the graphed function, we should carefully observe the modifications occurring in its key characteristics such as the asymptotes and x-intercepts.
Step-by-Step Analysis:

Parent Function Characteristics :

The parent function of tangent, y = tan ( x ) , has vertical asymptotes at x = 2 π ​ + nπ , where n is an integer. It crosses the x-axis at x = nπ .


Transformed Function Observations :

The given function shows vertical asymptotes at x = ± nπ .
The function crosses the x-axis at x = 6 7 π ​ , 6 π ​ , − 6 5 π ​ .


Horizontal Shift Analysis :

In the parent function, x-intercepts are at x = nπ . Here, x-intercepts at x = 6 π ​ and x = 6 7 π ​ indicate that the graph has been horizontally shifted .
The intercepts at x = 6 π ​ suggest a horizontal shift to the right by 6 π ​ .


Transformation Description :

The graph of y = tan ( x ) is horizontally translated to y = tan ( x − 6 π ​ ) .
This shift causes the new x-intercepts to be at x = 6 π ​ + nπ , aligning with the observed points.



In conclusion, to obtain the function graphed, a horizontal shift by 6 π ​ to the right was applied to the parent function y = tan ( x ) .
Remember, analyzing key features such as asymptotes and intercepts helps in understanding function transformations.

Answered by ElijahBenjaminCarter | 2025-07-07

The function graphed is a transformation of the parent function y = tan ( x ) involving a horizontal shift. Specifically, it has been shifted to the right by 6 π ​ . This transformation aligns the new x-intercepts and asymptotes with the observed values.
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Answered by ElijahBenjaminCarter | 2025-07-07