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

 Page 2 / 9

## 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.

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
how do you get the (1.4427)^t in the carp problem?
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
hello
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
rotation by 80 of (x^2/9)-(y^2/16)=1
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
what is the standard form if the focus is at (0,2) ?
a²=4