Graph y=2tan(3x)

Problem

y=2*tan(3*x)

Solution

1. Identify the parent function and its properties. The function is of the form y=A*tan(B*x) where the parent function is y=tan(x)

2. Determine the period of the function. The period of a tangent function is calculated using the formula π/|B|

Period=π/3

3. Find the vertical asymptotes by setting the argument of the tangent function equal to the standard asymptotes of y=tan(x) which are x=π/2+n*π

3*x=π/2+n*π

x=π/6+(n*π)/3

For the first cycle around the origin, the asymptotes are x=−π/6 and x=π/6

4. Determine the vertical stretch factor. The coefficient A=2 indicates a vertical stretch by a factor of 2 This means the points halfway between the center and the asymptotes, which are usually at y=±1 will now be at y=±2

5. Identify key points for one cycle. The center point is at (0,0) The points at one-quarter and three-quarters of the period are:

(−π/12,−2)

(π/12,2)

6. Sketch the graph by drawing the vertical asymptotes, plotting the key points, and drawing the characteristic tangent curves that approach the asymptotes.

Final Answer

y=2*tan(3*x)

The graph has a period of π/3 vertical asymptotes at x=π/6+(n*π)/3 and passes through (0,0) (π/12,2) and (−π/12,−2)

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