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.