8.2 Graphs of the other trigonometric functions (Page 3/9)

Page 3 / 9

f (x) = A \tan (B x - C) + D

The graph of a transformed tangent function is different from the basic tangent function $\tan x$ in several ways:

a general note label

Features of the graph of y = A Tan( Bx − C )+ D

The stretching factor is $| A | .$
The period is $\frac{π}{| B |} .$
The domain is $x \neq \frac{C}{B} + \frac{π}{| B |} k,$ where $k$ is an integer.
The range is $(−∞, \infty) .$
The vertical asymptotes occur at $x = \frac{C}{B} + \frac{π}{2 | B |} k,$ where $k$ is an odd integer.
There is no amplitude.
$y = A \tan (B x)$ is and odd function because it is the qoutient of odd and even functions(sin and cosine perspectively).

How To Feature

Given the function $y = A \tan (B x - C) + D,$ sketch the graph of one period.

Express the function given in the form $y = A \tan (B x - C) + D .$
Identify the stretching/compressing factor , $| A | .$
Identify $B$ and determine the period, $P = \frac{π}{| B |} .$
Identify $C$ and determine the phase shift, $\frac{C}{B} .$
Draw the graph of $y = A \tan (B x)$ shifted to the right by $\frac{C}{B}$ and up by $D .$
Sketch the vertical asymptotes, which occur at $x = \frac{C}{B} + \frac{π}{2 | B |} k,$ where $k$ is an odd integer.
Plot any three reference points and draw the graph through these points.

Graphing one period of a shifted tangent function

Graph one period of the function $y = −2 \tan (π x + π) −1.$

Step 1. The function is already written in the form $y = A \tan (B x - C) + D .$
Step 2. $A = −2,$ so the stretching factor is $| A | = 2.$
Step 3. $B = π,$ so the period is $P = \frac{π}{| B |} = \frac{π}{π} = 1.$
Step 4. $C = - π,$ so the phase shift is $\frac{C}{B} = \frac{- π}{π} = −1.$
Step 5-7. The asymptotes are at $x = - \frac{3}{2}$ and $x = - \frac{1}{2}$ and the three recommended reference points are $(−1.25, 1),$ $(−1, −1),$ and $(−0.75, −3) .$ The graph is shown in [link] .

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How would the graph in [link] look different if we made $A = 2$ instead of $−2 ?$

It would be reflected across the line $y = - 1,$ becoming an increasing function.

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How To Feature

Given the graph of a tangent function, identify horizontal and vertical stretches.

Find the period $P$ from the spacing between successive vertical asymptotes or x -intercepts.
Write $f (x) = A \tan (\frac{π}{P} x) .$
Determine a convenient point $(x, f (x))$ on the given graph and use it to determine $A .$

Identifying the graph of a stretched tangent

Find a formula for the function graphed in [link] .

A graph of two periods of a modified tangent function, with asymptotes at x=-4 and x=4. — A stretched tangent function

The graph has the shape of a tangent function.

Step 1. One cycle extends from –4 to 4, so the period is $P = 8.$ Since $P = \frac{π}{| B |},$ we have $B = \frac{π}{P} = \frac{π}{8} .$
Step 2. The equation must have the form $f (x) = A \tan (\frac{π}{8} x) .$
Step 3. To find the vertical stretch $A,$ we can use the point $(2, 2) .$
$2 = A \tan (\frac{π}{8} \cdot 2) = A \tan (\frac{π}{4})$

Because $\tan (\frac{π}{4}) = 1,$ $A = 2.$

This function would have a formula $f (x) = 2 \tan (\frac{π}{8} x) .$

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Find a formula for the function in [link] .

A graph of four periods of a modified tangent function, Vertical asymptotes at -3pi/4, -pi/4, pi/4, and 3pi/4.

$g (x) = 4 \tan (2 x)$

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Analyzing the graphs of y = sec x And y = csc x

The secant was defined by the reciprocal identity $\sec x = \frac{1}{\cos x} .$ Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at $\frac{π}{2},$ $\frac{3 π}{2},$ etc. Because the cosine is never more than 1 in absolute value, the secant, being the reciprocal, will never be less than 1 in absolute value.

We can graph $y = \sec x$ by observing the graph of the cosine function because these two functions are reciprocals of one another. See [link] . The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the secant function increases. Where the graph of the cosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is undefined.

Questions & Answers

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.

Widad Reply

can you send the book attached ?

Ariel

What is economics

Widad Reply

the study of how humans make choices under conditions of scarcity

AI-Robot

U(x,y) = (x×y)1/2 find mu of x for y

Desalegn Reply

U(x,y) = (xÃy)1/2 find mu of x for y

Desalegn

what is ecnomics

Jan Reply

this is the study of how the society manages it's scarce resources

Belonwu

what is macroeconomic

John Reply

macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.

husaini

etc

husaini

difference between firm and industry

husaini Reply

what's the difference between a firm and an industry

Abdul

firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅

Abdulraufu

Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?

Toofiq Reply

explain standard reason why economic is a science

innocent Reply

factors influencing supply

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what is economic.

Milan Reply

scares means__________________ends resources. unlimited

Jan

economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses

Jan

calculate the profit maximizing for demand and supply

Zarshad Reply

Why qualify 28 supplies

Milan

what are explicit costs

Nomsa Reply

out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials

AI-Robot

concepts of supply in microeconomics

David Reply

economic overview notes

Amahle Reply

identify a demand and a supply curve

Salome Reply

i don't know

Parul

there's a difference

Aryan

Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods

Israr

Hi Sir please how do u calculate Cross elastic demand and income elastic demand?

Abari

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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