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Which of the following equations would transform the tangent graph to the parent cotangent graph? a) y = tan(x) b) y = -tan(x) c) y = 1/tan(x) d) y = cot(x)

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The correct answer is c) y = 1/tan(x).

Here's why:

Key Relationship between Tangent and Cotangent:

- The tangent and cotangent functions are reciprocals of each other. This means that:

cot(x) = 1/tan(x)

Transforming the Tangent Graph:

- To transform the tangent graph into the cotangent graph, we need to apply a transformation that essentially "flips" the graph across the y-axis. This is achieved by taking the reciprocal of the tangent function.

Analyzing the Options:

- Option a) y = tan(x) is the original tangent function itself, so it wouldn't transform it.

- Option b) y = -tan(x) reflects the tangent graph across the x-axis, but it doesn't create the cotangent graph.

- Option c) y = 1/tan(x) is the reciprocal of the tangent function, which produces the cotangent graph.

- Option d) y = cot(x) is already the cotangent function, so it doesn't apply to transforming the tangent graph.

Visualizing the Transformation:

- Imagine the tangent graph with its vertical asymptotes at x = π/2, -π/2, 3π/2, and so on.

- When you take the reciprocal (y = 1/tan(x)), the asymptotes "switch" to horizontal positions at y = 0, creating the characteristic shape of the cotangent graph.

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