# 6.2 Graphs of the other trigonometric functions  (Page 2/9)

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## Graphing variations of y = tan x

As with the sine and cosine functions, the tangent    function can be described by a general equation.

$y=A\mathrm{tan}\left(Bx\right)$

We can identify horizontal and vertical stretches and compressions using values of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B.\text{\hspace{0.17em}}$ The horizontal stretch can typically be determined from the period of the graph. With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph.

Because there are no maximum or minimum values of a tangent function, the term amplitude cannot be interpreted as it is for the sine and cosine functions. Instead, we will use the phrase stretching/compressing factor when referring to the constant $\text{\hspace{0.17em}}A.$

## Features of the graph of y = A Tan( Bx )

• The stretching factor is $\text{\hspace{0.17em}}|A|.$
• The period is $\text{\hspace{0.17em}}P=\frac{\pi }{|B|}.$
• The domain is all real numbers $\text{\hspace{0.17em}}x,$ where $\text{\hspace{0.17em}}x\ne \frac{\pi }{2|B|}+\frac{\pi }{|B|}k\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is an integer.
• The range is $\text{\hspace{0.17em}}\left(\mathrm{-\infty },\infty \right).$
• The asymptotes occur at $\text{\hspace{0.17em}}x=\frac{\pi }{2|B|}+\frac{\pi }{|B|}k,\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is an integer.
• $y=A\mathrm{tan}\left(Bx\right)\text{\hspace{0.17em}}$ is an odd function.

## Graphing one period of a stretched or compressed tangent function

We can use what we know about the properties of the tangent function to quickly sketch a graph of any stretched and/or compressed tangent function of the form $\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{tan}\left(Bx\right).\text{\hspace{0.17em}}$ We focus on a single period    of the function including the origin, because the periodic property enables us to extend the graph to the rest of the function’s domain if we wish. Our limited domain is then the interval $\text{\hspace{0.17em}}\left(-\frac{P}{2},\frac{P}{2}\right)\text{\hspace{0.17em}}$ and the graph has vertical asymptotes at $\text{\hspace{0.17em}}±\frac{P}{2}\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}P=\frac{\pi }{B}.\text{\hspace{0.17em}}$ On $\text{\hspace{0.17em}}\left(-\frac{\pi }{2},\frac{\pi }{2}\right),\text{\hspace{0.17em}}$ the graph will come up from the left asymptote at $\text{\hspace{0.17em}}x=-\frac{\pi }{2},\text{\hspace{0.17em}}$ cross through the origin, and continue to increase as it approaches the right asymptote at $\text{\hspace{0.17em}}x=\frac{\pi }{2}.\text{\hspace{0.17em}}$ To make the function approach the asymptotes at the correct rate, we also need to set the vertical scale by actually evaluating the function for at least one point that the graph will pass through. For example, we can use

$f\left(\frac{P}{4}\right)=A\mathrm{tan}\left(B\frac{P}{4}\right)=A\mathrm{tan}\left(B\frac{\pi }{4B}\right)=A$

because $\text{\hspace{0.17em}}\mathrm{tan}\left(\frac{\pi }{4}\right)=1.$

Given the function $\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{tan}\left(Bx\right),\text{\hspace{0.17em}}$ graph one period.

1. Identify the stretching factor, $\text{\hspace{0.17em}}|A|.$
2. Identify $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ and determine the period, $\text{\hspace{0.17em}}P=\frac{\pi }{|B|}.$
3. Draw vertical asymptotes at $\text{\hspace{0.17em}}x=-\frac{P}{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=\frac{P}{2}.$
4. For $\text{\hspace{0.17em}}A>0,\text{\hspace{0.17em}}$ the graph approaches the left asymptote at negative output values and the right asymptote at positive output values (reverse for $\text{\hspace{0.17em}}A<0$ ).
5. Plot reference points at $\text{\hspace{0.17em}}\left(\frac{P}{4},A\right),\text{\hspace{0.17em}}$ $\left(0,0\right),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-\frac{P}{4},-A\right),\text{\hspace{0.17em}}$ and draw the graph through these points.

## Sketching a compressed tangent

Sketch a graph of one period of the function $\text{\hspace{0.17em}}y=0.5\mathrm{tan}\left(\frac{\pi }{2}x\right).$

First, we identify $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B.$

Because $\text{\hspace{0.17em}}A=0.5\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B=\frac{\pi }{2},\text{\hspace{0.17em}}$ we can find the stretching/compressing factor and period. The period is $\text{\hspace{0.17em}}\frac{\pi }{\frac{\pi }{2}}=2,\text{\hspace{0.17em}}$ so the asymptotes are at $\text{\hspace{0.17em}}x=±1.\text{\hspace{0.17em}}$ At a quarter period from the origin, we have

$\begin{array}{l}f\left(0.5\right)=0.5\mathrm{tan}\left(\frac{0.5\pi }{2}\right)\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=0.5\mathrm{tan}\left(\frac{\pi }{4}\right)\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=0.5\hfill \end{array}$

This means the curve must pass through the points $\text{\hspace{0.17em}}\left(0.5,0.5\right),$ $\left(0,0\right),$ and $\text{\hspace{0.17em}}\left(-0.5,-0.5\right).\text{\hspace{0.17em}}$ The only inflection point is at the origin. [link] shows the graph of one period of the function.

Sketch a graph of $\text{\hspace{0.17em}}f\left(x\right)=3\mathrm{tan}\left(\frac{\pi }{6}x\right).$

## Graphing one period of a shifted tangent function

Now that we can graph a tangent function that is stretched or compressed, we will add a vertical and/or horizontal (or phase) shift. In this case, we add $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ to the general form of the tangent function.

"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
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Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo