Section 7.3 The Graphs Of The Tangent, Cosecant, Secant .
Section 7.3 The Graphs of the Tangent, Cosecant, Secant, and CotangentFunctions.OBJECTIVE 1:Understanding the Graph of the Tangent Function and its Propertiessin x. Recall also that division by zero is nevercos xallowed. Therefore, the domain of f (x) tan x is not the set of all real numbers.!Recall that y tan x can equivalently be written y Completethe tables below. x 0 π3π2π2y tan x ! x 3π4π42π5π47π4y tan x Using information above and the graph of f (x) tan x , write the characteristics of the Tangent Function: The domain isThe range is (interval notation) The function is periodic with a period of P .The principal cycle (includes 0) of the graph occurs on the intervalThe function has infinitely many vertical asymptotes with equationsThe y-intercept isFor each cycle there is one center point. The x-coordinates of the center points are also thex-intercepts or zeros are of the form where n isFor each cycle there are two halfway points. The halfway point to the left of the x-intercept has ay-coordinate of , and the halfway point to the right of the x-intercept has a y-coordinateofThe function is (odd or even or neither), which means tan( x) . Thegraph 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 verticalasymptotes (where cos x 0 ) and sketch the graph of f (x) tan x .
OBJECTIVE 2:Sketching Functions of the Form y Atan(Bx C) DSteps&for&Sketching&Functions&of&the&Form& y A tan ( Bx C ) D ! Step&1:! If B 0 that! B 0 g!the!inequality! π2 Bx C !!!!!!!!Step&3:!!!The!period!is! P incipal!cycle!of! y A tan ( Bx C ) D t!is!D.!!Note!that!when! D 0 ntercept.!Step&5:! nts!of!the!principal!cycle!of y A tan ( Bx C ) D !these!points!are A y!point!of y tan x raph!as!it!approaches!each!asymptote.!For each example, determine the interval for the principal cycle. Then for the principal cycle, determinethe equations of the vertical asymptotes, the coordinates of the center point, and the coordinates of thehalfway points. Sketch the graph.7.3.12 y 2 tan( 3x)
7.3.16 y 4 tan( 2x π ) 1OBJECTIVE 3:Understanding the Graph of the Cotangent Function and its Propertiescos x. Recall also that division by zero is neversin xallowed. Therefore, the domain of f (x) cot x is not the set of all real numbers.!Recall that y cot x can equivalently be written y Completethe tables below. x 0 π3π2π2y cot x ! x 3π4π42π5π47π4y cot x Using information above and the graph of f (x) cot x , write the characteristics of the CotangentFunction: 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 intervalThe function has infinitely many vertical asymptotes with equations
The y-intercept isFor each cycle there is one center point. The x-coordinates of the center points are also thex-intercepts or zeros are of the form where n isFor each cycle there are two halfway points. The halfway point to the left of the x-intercept has ay-coordinate of , and the halfway point to the right of the x-intercept has a y-coordinateofThe function is (odd or even or neither), which means cot( x) . Thegraph 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 verticalasymptotes (where sin x 0 ) and sketch the graph of f (x) cot x . OBJECTIVE 4: Sketching Functions of the Form y Acot(Bx C) DSteps&for&Sketching&Functions&of&the&Form& y Acot(Bx C) D ! Step&1:! If B 0 that!B 0 .!!Use!the!odd!property!of!the!tangent!function.!! g!the!inequality! 0 Bx C π !!!!!Step&3:!!!The!period!is! P πB.! al!cycle!of! y Acot(Bx C) D t!is!D.!!Note!that!when! D 0 ntercept.!Step&5:! nts!of!the!principal!cycle!ofy Acot(Bx C) D .!! ese!points!are A y!point!of y cot x !plus!D.!!!!!!! oaches! each!asymptote.!
7.3.35 Given the function y 3cot(2x π ) 1 !determine the interval for the principal cycle. Then for theprincipal 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:PropertiesUnderstanding the Graphs of the Cosecant and Secant Functions and TheirRewrite y csc x as y 1and use knowledge of the graph of the sine function to graph the cosecantsin xfunction. 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 ao The relative maximum value is The function obtains a relative minimum at where n is ao The relative minimum value is The function is (odd or even or neither), which means csc( x) . Thegraph 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 1and use knowledge of the graph of the cosine function to graph the secantcos xfunction. 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 ao The relative maximum value is The function obtains a relative minimum at where n is ao The relative minimum value is The function is (odd or even or neither), which means sec( x) . Thegraph 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 y A csc ( Bx C ) D and y A sec ( Bx C ) D , first sketch thegraph of the corresponding reciprocal function.Steps for Sketching Functions of the Form y A csc ( Bx C ) D and y A sec ( Bx C ) DStep 1: Lightly sketch at least two cycles of the corresponding reciprocal function using the process outlined inSection 7.2. If D 0 , lightly sketch two reciprocal functions, one with D 0 and one with D 0 .Step 2: Sketch the vertical asymptotes. The vertical asymptotes will correspond to the x-intercepts of thereciprocal function y A sin ( Bx C ) or y A cos ( Bx C ) .Step 3: Plot all maximum and minimum points on the graph of y A sin ( Bx C ) D ory A cos ( 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 andrelative 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 )
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 tanx can equivalently be written y sin x cosx. Recall also that division by zero is never allowed. Therefore, the domain of
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Math 6 NOTES Name _ Types of Graphs: Different Ways to Represent Data Line Graphs Line graphs are used to display continuous data. Line graphs can be useful in predicting future events when they show trends over time. Bar Graphs Bar graphs are used to display categories of data.
difierent characterizations of pushdown graphs also show the limited expres-siveness of this formalism for the deflnition of inflnite graphs. Preflx Recognizable Graphs. The equivalence of pushdown graphs to the graphs generated by preflx rewriting systems on words leads to a natural extension of pushdown graphs.
