Graph y=tan(3x)

Problem

y=tan(3*x)

Solution

1. Identify the parent function and its properties. The function is y=tan(b*x) where b=3 The standard period of tan(x) is π

2. Determine the period of the transformed function. The period P is calculated by dividing the standard period by the absolute value of b

P=π/3

3. Find the vertical asymptotes by setting the argument of the tangent function equal to the standard asymptote locations, π/2+n*π

3*x=π/2

x=π/6

3*x=−π/2

x=−π/6

4. Identify the x-intercepts by setting the function equal to zero. For tan(3*x) the intercepts occur at the center of each period.

3*x=0

x=0

5. Determine key points between the intercept and the asymptotes. Since the amplitude coefficient is 1 the function reaches y=1 and y=−1 at one-quarter and three-quarters of the period.

x=π/12⇒y=tan(3⋅π/12)=tan(π/4)=1

x=−π/12⇒y=tan(3⋅−π/12)=tan(−π/4)=−1

6. Sketch the graph by drawing dashed vertical lines at x=±π/6 plotting the point (0,0) and drawing the characteristic tangent curves passing through (π/12,1) and (−π/12,−1)

Final Answer

y=tan(3*x)* has period π/3* and asymptotes at *x=π/6+(n*π)/3

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