Question Sketch the graph of the tangent curve in the interval from 0 to 2. y = tan (–2Θ)

Answer 1

To sketch the graph of y = tan(-2Θ) in the interval from 0 to 2, we can first start by finding key points and the behavior of the tangent function.

Key points:
- When Θ = 0, tan(-2Θ) = tan(0) = 0
- When Θ = π/4, tan(-2Θ) = tan(-π/2) = undefined
- When Θ = π/2, tan(-2Θ) = tan(-π) = 0
- When Θ = 3π/4, tan(-2Θ) = tan(-3π/2) = undefined

The graph of y = tan(Θ) has vertical asymptotes at odd multiples of π/2 (π/2, 3π/2, 5π/2, etc.) and it repeats its pattern every π. Since we have -2Θ as the argument, the graph will be compressed horizontally by a factor of 2.

In the interval from 0 to 2, we can sketch the graph as follows:
- At Θ = 0, y = 0
- Between Θ = 0 and Θ = π/4, the graph approaches a vertical asymptote at Θ = π/4
- At Θ = π/2, y = 0
- Between Θ = π/2 and Θ = 3π/4, the graph approaches a vertical asymptote at Θ = 3π/4

Therefore, the graph of y = tan(-2Θ) in the interval from 0 to 2 will have two vertical asymptotes at Θ = π/4 and Θ = 3π/4, and it will oscillate between positive and negative values.