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Section 7.3 The Graphs of the Tangent, Cosecant, Secant, and Cotangent Functions.
OBJECTIVE 1: Understanding the Graph of the Tangent Function and its Properties
Recall that
€
y = tan x can equivalently be written
€
y =sin xcosx
. Recall also that division by zero is never
allowed. Therefore, the domain of
€
f (x) = tan x is not the set of all real numbers.!
Complete the tables below.
x 0
€
π2
€
π
€
3π2
€
2π
€
y = tan x
!
x
€
π4
€
3π4
€
5π4
€
7π4
€
y = tan x
Using information above and the graph of
€
f (x) = tan x , write the characteristics of the Tangent Function:
• The domain is ___________________ • The range is _____________________(interval notation) • The function is periodic with a period of P = ________________. • The principal cycle (includes 0) of the graph occurs on the interval ___________________ • The function has infinitely many vertical asymptotes with equations ___________________ • The y-intercept is ___________ • For each cycle there is one center point. The x-coordinates of the center points are also the
x-intercepts or zeros are of the form ______ where n is ___________ • For each cycle there are two halfway points. The halfway point to the left of the x-intercept has a
y-coordinate of ________, and the halfway point to the right of the x-intercept has a y-coordinate of ___________
• The function is __________ (odd or even or neither), which means
€
tan(−x) =__________. The graph is symmetric about ______________________.
• The graph of each cycle of
€
y = tan x is one-to-one (this will take on more meaning in 7.4)
Draw a set of axes. Determine appropriate tic marks for x and y, plot points, dash and label vertical asymptotes (where
€
cos x = 0) and sketch the graph of
€
f (x) = tan x .
OBJECTIVE 2: Sketching Functions of the Form
€
y = A tan(Bx −C)+ D
Steps&for&Sketching&Functions&of&the&Form& ( )= − +tany A Bx C D !
Step&1:! If 0<B ,!rewrite!the!function!in!an!equivalent!form!such!that! 0>B .!!Use!the!odd!property!of!the!tangent!function.!!
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&We&now&use&this&new&form&to&determine&A,#B,&C,#and&D.!Step&2:!!Determine!the!interval!and!the!equations!of!the!vertical!asymptotes!of!the!principal!cycle.!!The!interval!
for!the!principal!cycle!can!be!found!by!solving!the!inequality!2 2π π
− < − <Bx C .!!The!vertical!asymptotes!of!the!
principal!cycle!occur!at!the!“endpoints”!of!the!interval!of!the!principal!cycle.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Step&3:!!!The!period!is! =PBπ
.!
Step&4:!!Determine!the!center!point!of!the!principal!cycle!of! ( )tan= − +y A Bx C D .!!The!x#coordinate!of!the!
center!point!is!located!midway!between!the!vertical!asymptotes!of!the!principal!cycle.!!The!y#coordinate!of!the!center!point!is!D.!!Note!that!when! 0=D ,!the!x#!coordinate!of!the!center!point!is!the!x#intercept.!
Step&5:! !!Determine!the!coordinates!of!the!two!halfway!points!of!the!principal!cycle!of ( )tan= − +y A Bx C D .!!
Each!xCcoordinate!of!a!halfway!point!is!located!halfway!between!the!x#coordinate!of!the!center!point!and!a!vertical!asymptote.!The!y#coordinates!of!these!points!are A times!the!y#coordinate!of!the!corresponding!halfway!point!of tan=y x !plus!D.! ! ! ! ! ! ! ! ! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Step&6:!!Sketch!the!vertical!asymptotes,!plot!the!center!point,!and!plot!the!two!halfway!points.!!Connect!these!points!with!a&smooth!curve.!Complete!the!sketch!showing!appropriate!behavior!of!the!graph!as!it!approaches!each!asymptote.!
For each example, determine the interval for the principal cycle. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
7.3.12 y = 2 tan(−3x)
7.3.16 y = 4 tan(−2x +π )+1
OBJECTIVE 3: Understanding the Graph of the Cotangent Function and its Properties
Recall that
€
y = cot x can equivalently be written
€
y =cosxsin x
. Recall also that division by zero is never
allowed. Therefore, the domain of
€
f (x) = cot x is not the set of all real numbers.!
Complete the tables below.
x 0
€
π2
€
π
€
3π2
€
2π
€
y = cot x
!
x
€
π4
€
3π4
€
5π4
€
7π4
€
y = cot x
Using information above and the graph of
€
f (x) = cot x , write the characteristics of the Cotangent Function:
• The domain is ___________________ • The range is _____________________(interval notation) • The function is periodic with a period of P = ________________. • The principal cycle (includes 0) of the graph occurs on the interval ___________________ • The function has infinitely many vertical asymptotes with equations ___________________
• The y-intercept is ___________ • For each cycle there is one center point. The x-coordinates of the center points are also the
x-intercepts or zeros are of the form ______ where n is ___________ • For each cycle there are two halfway points. The halfway point to the left of the x-intercept has a
y-coordinate of ________, and the halfway point to the right of the x-intercept has a y-coordinate of ___________
• The function is __________ (odd or even or neither), which means
€
cot(−x) =__________. The graph is symmetric about ______________________.
• The graph of each cycle of
€
y = tan x is one-to-one (this will take on more meaning in 7.4)
Draw a set of axes. Determine appropriate tic marks for x and y, plot points, dash and label vertical asymptotes (where
€
sin x = 0) and sketch the graph of
€
f (x) = cot x .
OBJECTIVE 4: Sketching Functions of the Form
€
y = Acot(Bx −C)+ D
Steps&for&Sketching&Functions&of&the&Form&
€
y = Acot(Bx −C)+ D !
Step&1:! If 0<B ,!rewrite!the!function!in!an!equivalent!form!such!that! 0>B .!!Use!the!odd!property!of!the!tangent!function.!!
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&We&now&use&this&new&form&to&determine&A,#B,&C,#and&D.!Step&2:!!Determine!the!interval!and!the!equations!of!the!vertical!asymptotes!of!the!principal!cycle.!!The!interval!for!the!principal!cycle!can!be!found!by!solving!the!inequality!
€
0 < Bx −C < π .!!The!vertical!asymptotes!of!the!principal!cycle!occur!at!the!“endpoints”!of!the!interval!of!the!principal!cycle.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Step&3:!!!The!period!is! =PBπ
.!
Step&4:!!Determine!the!center!point!of!the!principal!cycle!of!
€
y = Acot(Bx −C)+ D .!!The!x#coordinate!of!the!center!point!is!located!midway!between!the!vertical!asymptotes!of!the!principal!cycle.!!The!y#coordinate!of!the!center!point!is!D.!!Note!that!when! 0=D ,!the!x#!coordinate!of!the!center!point!is!the!x#intercept.!Step&5:! !!Determine!the!coordinates!of!the!two!halfway!points!of!the!principal!cycle!of
€
y = Acot(Bx −C)+ D .!!Each!xCcoordinate!of!a!halfway!point!is!located!halfway!between!the!x#coordinate!of!the!center!point!and!a!vertical!asymptote.!The!y#coordinates!of!these!points!are A times!the!y#coordinate!of!the!corresponding!halfway!point!of
€
y = cot x !plus!D.! ! ! ! ! ! ! ! ! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Step&6:!!Sketch!the!vertical!asymptotes,!plot!the!center!point,!and!plot!the!two!halfway!points.!!Connect!these!points!with!a&smooth!curve.!Complete!the!sketch!showing!appropriate!behavior!of!the!graph!as!it!approaches!each!asymptote.!
7.3.35 Given the function y = 3cot(2x +π )−1 !determine the interval for the principal cycle. Then for the principal cycle, determine the equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of the halfway points. Sketch the graph.
OBJECTIVE 5: Understanding the Graphs of the Cosecant and Secant Functions and Their Properties
Rewrite
€
y = csc x as
€
y =1sin x
and use knowledge of the graph of the sine function to graph the cosecant
function.
For
€
f (x) = csc x ,
• The domain is ___________________ • The range is _____________________(interval notation) • The function is periodic with a period of P = ________________. • The function has infinitely many vertical asymptotes with equations ___________________ • The function obtains a relative maximum at ________________ where n is a _____________
o The relative maximum value is ________________ • The function obtains a relative minimum at ________________ where n is a _____________
o The relative minimum value is ________________ • The function is __________ (odd or even or neither), which means
€
csc(−x) =__________. The graph is symmetric about ______________________.
Draw a set of axes. Determine appropriate tic marks for x and y, plot points, dash and label vertical asymptotes (where
€
sin x = 0) and sketch the graph of
€
f (x) = csc x .
Rewrite
€
y = sec x as
€
y =1
cosx and use knowledge of the graph of the cosine function to graph the secant
function.
For
€
f (x) = sec x ,
• The domain is ___________________ • The range is _____________________(interval notation) • The function is periodic with a period of P = ________________. • The function has infinitely many vertical asymptotes with equations ___________________ • The function obtains a relative maximum at ________________ where n is a _____________
o The relative maximum value is ________________ • The function obtains a relative minimum at ________________ where n is a _____________
o The relative minimum value is ________________ • The function is __________ (odd or even or neither), which means
€
sec(−x) =__________. The graph is symmetric about ______________________.
Draw a set of axes. Determine appropriate tic marks for x and y, plot points, dash and label vertical asymptotes (where
€
cos x = 0) and sketch the graph of
€
f (x) = sec x .
To sketch functions of the form ( )cscy A Bx C D= − + and ( )secy A Bx C D= − + , first sketch the
graph of the corresponding reciprocal function.
Steps for Sketching Functions of the Form ( )cscy A Bx C D= − + and ( )secy A Bx C D= − +
Step 1: Lightly sketch at least two cycles of the corresponding reciprocal function using the process outlined in Section 7.2. If 0≠D , lightly sketch two reciprocal functions, one with 0=D and one with 0≠D .
Step 2: Sketch the vertical asymptotes. The vertical asymptotes will correspond to the x-intercepts of the
reciprocal function ( )sin= −y A Bx C or ( )cos= −y A Bx C .
Step 3: Plot all maximum and minimum points on the graph of ( )sin= − +y A Bx C D or
( )cos= − +y A Bx C D .
Step 4: Draw smooth curves through each point from Step 3, making sure to approach the vertical asymptotes.
In the examples below, determine the equations of the vertical asymptotes and all relative maximum and relative minimum points of two cycles of each function and then sketch its graph.
7.3.45 y = 2sec(3x +π )
7.3.46 y = −3csc(2x +π )
7.3.49 y = 5− 2sec(x −π )