# How to sketch $y=2\tan(x+\frac{\pi}{4})$ , $x \in (0,2\pi)$

How to sketch $y=2\tan(x+\frac{\pi}{4})$ , $x \in (0,2\pi)$

$2\tan$ , 2 is used to be amplitude in $\cos$ and $\sin$ graph but for the $\tan$ there is no amplitude,so where will that $2\tan$ sketch, also
$x+\frac{\pi}{4}=\pi$
$x=\frac{4\pi}{4}-\frac{\pi}{4}$
$x=\frac{3\pi}{4}$
it is right?

Can you please explain me in step by step and show me how to sketch. Thank you so much.

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Well, you weren't taught first how to draw $\tan\,x$ itself? – J. M. May 13 '12 at 9:06
yes! sorry for my lack of maths knowledge. thx – Sb Sangpi May 13 '12 at 9:08
Oh, you don't have to apologize. It does irritate me that your teachers have neglected to teach you how to draw the basic function and then ask you to draw something a bit more elaborate... their fault, not yours. – J. M. May 13 '12 at 9:11
If I were you I would ask this in general...there is nothing special about this function any you still want extremum points of y and y'... – Belgi May 13 '12 at 9:22
yes! please provide me solution step by step. Appreciate it! – Sb Sangpi May 13 '12 at 9:25

## 1 Answer

1) Open any basic calculus, or even trigonometry, book and look and understand the graph of $\tan x$

2) Now "shift" that graph by a rate of $\frac{\pi}{4}$ to the right to get $\tan\left(x+\frac{\pi}{4}\right)$ (thus for ex., for $x=0\,\,$ we'll have now the value that $\tan x$ had at $x=\frac{\pi}{4}$...

3) Finally, multiply every value of $\tan\left(x+\frac{\pi}{4}\right)$ by 2, thus "expanding the graph"

DonAntonio

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Are you sure that to get $\tan\left(x+\frac\pi4\right)$ you should shift the graph to the right? (I agree with what you wrote in parenthesis after that, but I would call that shift to the left, e.g. $\frac\pi4$ is shifted to $0$.) – Martin Sleziak May 13 '12 at 12:42
I think you may be right, it looks geometrically more logical to call that a shift to the left. – DonAntonio May 13 '12 at 22:43