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How to transform graph of y = tan x to the graph of y = tan (x - pi/3)?

Question:

Draw the graph of y = tan (x - pi/3)

Solution:

Find domain of the function y = tan (x - pi/3)

Let t = x - pi/3, then y = tan t

The domain of y = tan t is: t is not equal to pi/2 + k pi, in which k = 0, =-1, =-2, ...

x -pi/3 is not equal to pi/2 + k pi
x is not equal to pi/3 + pi/2 + k pi
x is not equal to 2pi/6 + 3pi/6 + k pi
x is not equal to 5pi/6 + k pi

when k = -1, x is not equal to 5pi/6 - pi -> x is not equal to - pi/6

when k = 0, x is not equal to 5pi/6

when k = 1, x is not equal to 5pi/6 + pi -> x is not equal to 11pi/6

when k = 2, x is not equal to 5pi/6 + 2pi -> x is not equal to 17pi/6

draw graph of y = tan (x - pi/3)

In the figure above, the vertical lines are asymptotes of the function y = tan (x - pi/3). The origin curve is the graph of y = tan (x - pi/3). The origin curve is never across the vertical lines. Now we find the relationship between the graph of y = tan x and the graph of y = tan (x - pi/3).

- pi/6 = - pi/2 + pi/3
5pi/6 = pi/2 + pi/3
11pi/6 = 3pi/2 + pi/3
17pi/6 = 5pi/2 + pi/3

-pi/2, pi/2, 3pi/2, 5pi/2 are asymptotes of the function y = tan x. -pi/6, 5pi/6, 11pi/6, 17pi/6 are the asymptotes of the function y = tan (x - pi/3).

So, move the asymptotes of y = tan x right pi/3 units, we can get the asymptotes of y = tan (x - pi/3). Therefore, move the graph of y = tan x right pi/3 units, we can get the graph of y = tan (x - pi/3). The graph of tangent function is never across its asymptotes.

draw graph of y = tan (x - pi/3)

In the figure above, the blue curve is the graph of y = tan x. The blue lines are the asymptotes of y = tan x. The blue curve is never across the blue lines. The origin curve is the graph of y = tan (x - pi/3). The origin lines are the asymptotes of y = tan (x - pi/3). The orange curve is never across the orange lines. By moving the graph of y = tan x right pi/3 units, we get the graph of y = tan (x - pi/3). Watch the video for more details.