Ch 4 Graphs of the Circular Functions
4.1 Graphs of the Sine and Cosine Functions
A periodic function is a function f such that
f  x   f  x  np ,
for every real number x in the domain of f,
every integer n, and some positive real
number p. The least possible positive value
of p is the period of the function.
sin x  sin  x  n  2 
 The graph is continuous over its entire
domain, (–, ).
 Its x-intercepts are of the form n, where
n is an integer.
 Its period is 2.
 The graph is symmetric with respect to
the origin, so the function is an odd
function. sin(–x) = –sin x.
cos x  cos x  n  2 
 The graph is continuous over its entire
domain, (–, ).

 Its x-intercepts are of the form 2n  1 ,
2
where n is an integer.
 Its period is 2.
 The graph is symmetric with respect to
the y-axis, so the function is an even
function. cos(–x) = cos x.
Graphing y  a sin x Vertical stretch or shrink
The amplitude of a periodic function is half
the difference between the maximum and
minimum values.
Graphing y  sin bx Horizontal stretch or shrink
For b > 0, the graph of y  sin bx will
resemble that of y  sin x , but with
2
period
.
b
Also, the graph of y  cos bx will resemble
2
that of y  cos x , but with period
.
b
Graph y  2 sin x
Graph y  sin 2 x
Graph y  2 sin 3x
Reflection across x axis
Guidelines for Sketching Graphs of Sine and
Cosine Functions
To graph y = a sin bx or y = a cos bx, with
b  0 , follow these steps.
1
2
3
4
5
2
Find the period, . Start with 0 on the
b
2
x-axis, and lay off a distance of
.
b
Divide the interval into four equal parts.
Evaluate the function for each of the five
x-values resulting from Step 2. The points
will be maximum points, minimum points,
and x-intercepts.
Plot the points found in Step 3, and join
them with a sinusoidal curve having
amplitude a .
Draw the graph over additional periods as
needed.
2
Graph y  cos x
3
Graph y  3 cos x
Determine an equation of form y  a cos bx
or y  a sin bx , b > 0, for
The average temperature in °F at Mould
Bay, Nunavut, can be approximated by


f  x   34 sin   x  4.3 , x is the month#.
6

a. 0,25by 45,45
b. Average temperature in May?
c. Average annual temperature?
4.2 Translations of the Graphs of the Sine
and Cosine Functions
y  f x  d 
Translations
The graph of the function y  f  x  d  is
translated horizontally compared to the
graph of y  f  x . The translation is d units
to the right if d  0 and is |d| units to the left
if d  0 . With circular functions, a
horizontal translation is called a phase shift.


Graph y  sin  x  
3



Graph y  3 cos x  
4

Graph y  2 cos3x    over 2 periods
Graph y  3  2 cos 3x over 2 periods
Further Guidelines for Sketching Graphs of
Sine and Cosine Functions
Method 1
1. Find an interval whose length is one
2
period by solving the three-part inequality
b
0  b x  d   2
2. Divide the interval into four equal parts.
3. Evaluate the function for each of the five
x-values resulting from Step 2. The points
will be maximum points, minimum points,
and points that intersect the line y  c .
4. Plot the points found in Step 3, and join
them with a sinusoidal curve having
amplitude a .
5 Graph over additional periods, if needed.
Method 2
1. Graph y  a sin bx or y  a cos bx . The
amplitude of the function is a , and the
2
period is
.
b
2. Use translations to graph the desired
function. The vertical translation is c units up if
c  0 and |c| units down if c  0 . The horizontal
translation (phase shift) is d units to the right if
d  0 and |d| units to the left if d  0 .
Graph y  1  2 sin 4 x    over 2 periods
The maximum average monthly temperature
in New Orleans is 83°F and the minimum is
53°F. Using only the maximum and
minimum temperatures, determine a function
of the form f  x   a sin b x  d   c where a,
b, c, and d are constants, that models the
average monthly temperature in New
Orleans.
4.3 Graphs of the Tangent and Cotangent Functions
A vertical asymptote is a vertical line that
the graph approaches but does not intersect.
As the x-values get closer and closer to the
line, the function values increase or decrease
without bound.
(1314) The line x  k is a vertical asymptote
of f  x  if f  x    or f  x    as x
approaches k from the left or the right.
tan  x    tan x
Graph of tan x


Domain:  x x  2n  1  , n is an integer
2

cot  x    cot x
Graph of cot x
Domain: x x  n , n is an integer
Guidelines for Sketching Graphs of Tangent
and Cotangent Functions

. To locate two
b
adjacent vertical asymptotes, solve the
following equations for x:


y  a tan bx : bx   and bx 
2
2
y  a cot bx : bx  0 and bx  
2. Sketch the two vertical asymptotes found
in Step 1.
3. Divide the interval formed by the vertical
asymptotes into four equal parts.
4. Evaluate the function for the first-quarter
point, midpoint, and third-quarter point,
using the x-values found in Step 3.
5. Join the points with a smooth curve,
approaching the vertical asymptotes.
Indicate additional asymptotes and periods
of the graph if needed.
1. Determine the period,
Graph y  tan 2 x
1
Graph y  3 tan x
2
1
Graph y  cot 2 x
2
Graph y  2  tan x


Graph y  2  cot  x  
4

Determine an equation for each graph
4.4 Graphs of the Secant and Cosecant Functions
secant function
f  x   sec x sec  x   sec x


Domain:  x x  2n  1  , n is an integer
2

Range:  ,1  1,  
1
sec x 
cos x
cosecant function
csc  x   csc x
f  x   csc x
Domain: x x  n , n is an integer
Range:  ,1  1,  
1
csc x 
sin x
Guidelines for Sketching Graphs of
Cosecant and Secant Functions
1. Graph the corresponding reciprocal
function as a guide, using a dashed curve.
To Graph
Use as a Guide
y = a csc bx
y = a sin bx
y = a sec bx
y = a cos bx
2. Sketch the vertical asymptotes. They will
have equations of the form x = k, where k is
an x-intercept of the graph of the guide
function.
3. Sketch the graph of the desired function
by drawing the typical U-shaped branches
between the adjacent asymptotes.
1
Graph y  2 sec x
2
3  
Graph y  csc  x  
2 
2
Determine an equation for each graph
Addition of Ordinates
New functions can be formed by adding or
subtracting other functions.
y  cos x  sin x
Find functional values (ordinates) for certain
x, then for cos x and sin x . Then add to see
cos x  sin x , and join the resulting points
with a sinusoidal curve.
4.5 Harmonic Motion
In part A of the figure, a
spring with a weight
attached to its free end
is in equilibrium (or
rest) position. If the
weight is pulled down a
units and released (part
B), the spring’s
elasticity causes the
weight to rise a units
( a  0 ) above the
equilibrium position, as
seen in part C, and then
to oscillate about the
equilibrium position.
Simple Harmonic Motion
The position of a point oscillating about an
equilibrium position at time t is modeled by
either st   a cos t or st   a sin t
where a and ω are constants, with  0 .
The amplitude of the motion is a , the period
2

is
and the frequency is
oscillations

2
per time unit.
Suppose an object is attached to a spring,
and pulled down 5cm from equilibrium.
The time for one oscillation is 4 sec.
Give an equation that models position wrt t.
Determine position at t  1.5sec
Find frequency.
Suppose an object oscillates according to the
model st   8 sin 3t . Analyze.
Damped Oscillatory Motion
Most oscillatory motions are damped by the
force of friction which causes the amplitude
of the motion to decrease gradually until the
weight comes to rest. This motion can be
modeled by st   e t sin t
 e t  e t sin t  et