Download Exponential Functions

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6-7
Graphs of other Trig Functions
MA
The Tangent Curve:
The Graph of y= tan x and Its Characteristics
Period: 
y
Domain: All real
numbers
except  /2 + k ,
k an integer
1
– –2 
5
2
–
3
–
2
–1

2
Range: All real numbers
2

0

–
2
3
2
5
2
x
Symmetric with respect
to the origin
Vertical asymptotes at
odd multiples of  /2
y = A tan (Bx – C)
Graphing y = A tan (Bx – C)
Bx – C =  /2
1. Find two consecutive asymptotes by
setting the variable expression in the
x
tangent equal to -/2 and /2 and solving
Bx – C = -/2 and Bx – C = /2
2. Identify an x-intercept, midway
between consecutive asymptotes.
x-intercept between
3. Find the points on the graph 1/4 and
asymptotes
3/4 of the way between and x-intercept
and the asymptotes. These points have ycoordinates of –A and A.
4. Use steps 1-3 to graph one full period
of the function. Add additional cycles to
the left or right as needed.
Bx – C = - /2
1
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Example 1
Graph y = 2 tan x/2 for – < x < 3 
1. Two consecutive asymptotes occur at x = -  and x = 
Bx – C = /2
½ x = /2
x=
Bx – C = -/2
½x = -/2
x=-
2. Identify any x-intercepts, midway between consecutive
asymptotes. Midway between x = - and x =  is x = 0. An
x-intercept is 0 and the graph passes through (0, 0).
3. Find points on the graph 1/4 and 1/4 of the way between an xintercept and the asymptotes. These points have y-coordinates
of –A and A.
Because A, the coefficient of the tangent, is 2, these points have ycoordinates of -2 and 2.
4. Use steps 1-3 to graph one full period of the function.
We use the two consecutive asymptotes, x = - and x = ,
an x-intercept of 0, and points midway between the xintercept and asymptotes with y-coordinates of –2 and 2.
We graph one full period of
y = 2 tan x/2 from – to .
In order to graph for – < x < 3 , we continue the pattern
and extend the graph another full period on the right.
y = 2 tan x/2
y
4
2
˝
-˝
3˝
x
-2
-4
The Cotangent Curve:
The Graph of y = cot x and Its Characteristics
The Graph of y = cot x and Its Characteristics
Period: 
y
Domain: All real numbers except
integral multiples of 
4
Range: All real numbers
Vertical asymptotes:
2
at integral multiples of 
xAn x-intercept occurs midway
˝
-  /2
3  /2 2 
-
 /2
between each pair of consecutive
-2
asymptotes.
-4
Odd function with origin
symmetry Points on the graph
1/4 and 3/4 of the way between
consecutive asymptotes have
y-coordinates of –1 and 1.
2
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The Cosecant Curve:
The Graph of y = csc x and Its Characteristics
y
-  /2
-2
-3  /2
-˝
1
3  /2
 /2
-1
2
˝
x
Characteristics
Period: 2
Domain: All real numbers except integral multiples of 
Range: All real numbers y such that y < -1 or y > 1
Vertical asymptotes: at integral multiples of 
Odd function with origin symmetry
Example 2
Use the graph of y = 2 sin 2x to obtain the graph of y = 2 csc 2x.
y
2
x
˝
-˝
-2
The x-intercepts of y = 2 sin 2x correspond to the vertical asymptotes of
y = 2 csc 2x. Draw vertical asymptotes through the x-intercepts.
Use the asymptotes as guides, we sketch the graph of y = 2 csc 2x.
y
2
˝
x
-2
3