Question

Find the vertical asymptotes (if any) of the graph of the function: tan(2x)

Answer 1

can i get steps?

Answer 2

Juan, do you know what the period of the tangent function is?
Are you able to sketch the graph of the tangent function?

Comparing the graphs of tan x and tan 2x, the period of the tan x function is TWICE the period of the tan 2x function.

Please edit my sketch in the Draw utility by typing or drawing in the left and the right endpoints of one cycle of tan x. Note that when x=0, tan x=0 also, and that the graph of tan x goes thru the origin.

Again, you will need to know the period of the tangent function to do this properly.

Answer 3

What are the equations of the vertical asymptotes of the tan x function?

What are the equations of the vert. asy. of the tan 2x function?

Answer 4

I'm having trouble graphing it, and I don't know what is a period

Answer 5

Time to do some reading, Juan. You'll learn the most if you yourself actually look up this information. Try googling "period of sine function," "period of tangent function," and so on. Any reputable source will tell you what the period of each of these functions is.

Make a list: sin x, cos x, tan x.
Under each heading, write in the period, the domain and range.

You MUST know and understand this information about the trig functions to do well in trig.

Answer 6

For now, copy the graphs you find thru your research.
If you have a graphing calculator and know how to use it, try graphing sin x.

Answer 7

Specific hints pertaining to the tangent function:

The period of sin 2x is HALF the period of sin x.

Similarly, the period of tan 2x is HALF the period of tan x.

tan x has a longer period than tan 2x.

You must know the period of tan x when the graph is centered on (0,0).

Answer 8

ok I have graphed it but I am confused with the points and the setup of my window

Answer 9

The calculator won't tell you that. It merely follows YOUR input.

Have you done a quick Internet search for "graph of tan x?"

If not, please do so.

If you find a good source, it will tell you the period of the tan x function.

Answer 10

Please see:

https://www.google.com/search?q=graph+of+tan+x&oq=graph+of+tan+x&aqs=chrome..69i57j0l5.2416j0j7&sourceid=chrome&ie=UTF-8

Answer 11

^He is very smart. I say you trust him. xD

Answer 12

sorry for the delay I have searched the period of tanx and have seen the graph.
can you resume your explanation? @mathmale

Answer 13

OK. If you look very carefully at this graph:

https://www.google.com/search?q=graph+of+tan+x&oq=graph+of+tan+x&aqs=chrome..69i57j0l5.2416j0j7&sourceid=chrome&ie=UTF-8

you'll see that the period of the function y=tan x is pi. No, it's not 2pi, which is for the sine and cosine, secant and cosecant.

So you now have the function y=tan x, or, rewritten, y=tan 1x. The period of this graph is pi.

If, on the other hand, you had y=tan 2x, the period is HALF of what it was before: pi/2.
If you had y=tan 3x, the period of this function is pi/3.

Note this pattern; you will need it later.

In summary:

The period of the tangent function y =tan x is pi.
The period of the tangent function with an argument of 2x is pi/2.

Your graph of y=tan 2x must start to the right of the vertical line (asymptote) x=-pi/4 and end just before the vertical line (asymptote) x=+pi/4.

Hope this helps. If you have further questions, please ask them.

Answer 14

wait how did you get pi?

Answer 15

sorry for my ignorance

Answer 16

Never ignorance in math, just need understanding...while Mathmales way will DEFINITELY allow you to dive deeper into the understanding of the methods...there are other ways!

For example...Do you know that \(\large tan(x) = \frac{sin(x)}{cos(x)}\) ?

Answer 17

yes

Answer 18

great, so then you agree that we can write

\[\large tan(2x) = \frac{sin(2x)}{cos(2x)}\]

Now let me ask you...when do vertical asymptotes occur?

Answer 19

when the denominator is 0?

Answer 20

Exactly! So looking at what we have here...when does that happen?

Meaning: When does \(\large cos(2x) = 0\) ?

Answer 21

Yeah, I think here is where I get stuck

Answer 22

is there a simple way to find out?

Answer 23

Okay, not a problem

Well, there are ways to figure it out...we just need to know when a cosine function equals 0

This can be done with the unit circle (GREATLY suggest getting familiar with this)...or just a graph...for example:

Answer 24

Looking at that graph of \(\large cos(x)\) when does it appear to cross the x-axis?

Answer 25

pi/2 and 3pi/2 ------ negative of these as well

Answer 26

Right, it seems the cosine function equals 0 at odd multiples of \(\large \pi/2\) right? *There is a reason for this, again...unit circle is your friend!*

But for now...just knowing that is good enough

So now we just need to apply that knowledge to what we have here!

\(\large cos(x) = 0\) at \(\large \frac{(2n-1)\pi}{2}\) *notice if you put in any integer there for 'n' you will arrive at the same result

So if THAT is true...when does \(\large cos(2x) = 0\) ? Just give me 1 answer for that

Answer 27

wait where did you get (2n-1)Pi/(2) from

Answer 28

Observe that cos 2x = 0 for pi/2, 3pi/2, 5pi/2, and so on.

The trick here is to summarize these examples into a single formula, which here is (2n-1)(pi/2), for all integer values of n: ... , -3, -2, -1, 0, 1, 2, 3, 4, ...

Answer 29

ok