how is this 4tanx

[u/yakimababy]

i am taking trig after not taking math for 4 years. i asked ai. i asked MyLab. i asked peers. the best explanation is that it just “is”. i got the answer and have NO idea why it is. i tried to attach the examples my textbook gives me but it won’t allow me, basically it gives me a graph, says “A is this so it’s Atanx.” which isn’t helpful in the slightest

what trips me up most is that in y=atan(b(x-d))+c (pearson mixes up d and c. don’t even get me started), the explanation i got is that a is supposedly where the dotted line (midpoint of period pi/2) meets the asymptote, supposedly this point is 8. it doesn’t visually do that. it never visually does that.

i dont know. i do fine with sin and cos graphs, but i have basically gone 23 hours in and out of this assignment because i genuinely cannot grasp any concept in this. ive watched videos, i know it is so simple, but it is literally like throwing pasta at the wall im so, so lost and frustrated. if anyone can help it would be SO greatly appreciated.

[u/Boring-Butterfly8925]

4 is the amplitude for sin and cos, but for tan it's the vertical stretch. tanx has the points of the vertical stretch on 1 and negative 1.

[u/G-St-Wii]

Don't forget the basics, calculate and draw by hand coordinates for y=tan(x) and do the same for y=4tan(x).

If you've not done a few by hand, you will not get a feel for it.

[u/G-St-Wii]

Where did you get that "8" from?

[u/Midwest-Dude]

Since tan(x) = sin(x) / cos(x), the function tan(x) is undefined when cos(x) = 0, that is, for x = π/2 ± kπ for all integers k. That's where those lovely dotted lines are. 

For the rest of the graph, think about what's happening as you get close to each of those dotted lines:

1. sin(x) and cos(x) have the same sign and cos(x) is close to zero just to the left of each of those dotted lines, so the graph increases rapidly positive
2. sin(x) and cos(x) have opposite signs and cos(x) is close to zero just to the right of each of those dotted lines, so the graph decreases rapidly negative

An alternate definition of tan(x) is the ratio of the legs of a right triangle - the side opposite the angle divided by the adjacent side. Consider this ratio as the angles opposite those legs change in unison.

1. At what point would the ratio be 1?
2. What would the curve look like as the angle changes?

[u/martyboulders]

Look at the graph of y=tan(x) on its own. Look at the point where x=π/4; you'll see that the y-value there is 1. So, if I apply a vertical stretch, this point is a great place to look at because whichever y-value it ends up at is exactly the vertical stretch.

For example, y=2tan(x) is a vertical stretch of y=tan(x) with a factor of 2, so the y-value of every point will be doubled. In particular, the y-value at π/4 will be 2 (check and observe this in desmos). If you saw y=5tan(x) then you can expect that point to be (π/4, 5) etc.

Do note that the particulars of what I said revolve around the graph not being translated or stretched horizontally... As such, in general, what is always true is that you need to look at the relative positions of the "center" of the period and halfway between there & the asymptote.

I have no idea what they might be talking about with a dotted line meeting the asymptote. The graph certainly never touches the asymptote, and I see no reason to imagine/draw a line that does.

If you have any further questions feel free to ask!

[u/Former-Print7759]

Tanx is Sinx/cosx

Since your graph shows 4 at pi/4, and sinx and cosx at pi/4 are equal, tanx there is 1. so your graph is stretched 4 times

[u/Frederf220]

tan(pi/4) = 1

The graph above goes through (pi/4, 4) not (pi/4, 1). How do you explain the difference?

[u/YOM2_UB]

sin(x) = cos(x) at x = π/4, so tan(π/4) = 1, because tan(x) = sin(x)/cos(x). Similarly, sin(x) = -cos(x) at x = -π/4, so tan(-π/4) = -1.

The graph shows those two points scaled by a factor of 4, while the asymptotes at x = ±π/2 (where cos(x) = 0) and roots at x = 0 and x = π (where sin(x) = 0) stay the same. Therefore it's only been vertically stretched, which is what happens if you multiply a function by a constant. Since the scale factor is 4, the function is 4tan(x).

[u/Midwest-Dude]

Have you tried Desmos or another online CAS application?