Transform the graph of y = tan 2x to the graph of y = tan(2x - pi/3)

Solution:

y = tan (2x - pi/3)

= tan [2x - (pi × 2)/(3 × 2)]

= tan (2x - 2pi/6)

= tan 2(x - pi/6)

Therefore, to get the graph of y = tan (2x - pi/3), we need to move the graph of y = tan 2x right pi/6.

In the figure above, the blue curve is the graph of y = tan 2x. The vertical lines are asymptotes. Now we move the asymptotes of y = tan 2x right pi/6 to get the asymptotes of y = tan (2x - pi/3)

- pi/4 + pi/6 = (3 × -pi)/4)/(3 × 4) + (2 × pi)/(2 × 6) = -3pi/12 + 2pi/12 = -pi/12. So, move asymptotes x = -pi/4 right pi/6 to get x = -pi/12.

pi/4 + pi/6 = (3 × pi)/(3 × 4) + (2 × pi)/(2 × 6) = 3pi/12 + 2pi/12 = 5pi/12. So, move the asymptotes x = 3pi/4 right pi/6 to get x = 5pi/12.

3pi/4 + pi/6 = (3 × 3pi)/(3 × 4) + (2 × pi)/(2 × 6) = 9pi/12 + 2pi/12 = 11pi/12. So, move the asymptotes x = 3pi/4 right pi/6 to get x = 11pi/12.

5pi/4 + pi/6 = (3 × 5pi)/(3 × 4) + (2 × pi)/(2 × 6) = 15pi/12 + 2pi/12 = 17pi/12. So, move the asymptotes x = 5pi/12 right pi/6 to get x = 17pi/12.

Now, we get the asymptotes of y = tan (2x - pi/3).

Then move each point of the graph of y = tan 2x right pi/6 to get the graph of y = tan (2x - pi/3)

The graph of y = tan (2x - pi/3) is never across its asymptotes.

In the figure above, the blue curve is the graph of y = tan 2x. the orange curve is the graph of y = tan (2x - pi/3). The graph of y = tan (2x - pi/3) getting from move each point of
y = tan 2x right pi/6. Watch the video for more details.