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4.5 THE GRAPH OF SINE AND
COSINE REVIEWED
Graphing with
translations and phase
shifts.
SKETCH THE GRAPH OF THE FOLLOWING
Amp:
Period:
= 2 sin 2 +
X
గ
ଶ
Y
Phase Shift:
Increment:
Starting point:
note: the amplitude
doesn’t change the x
values, it changes the y values.
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A QUICK SLIDE ON VERTICAL SHIFT
Vertical translations are caused by the constant in the equations
= + asin( + )
Label each section of the general equation to show what it affects on the
graph.
‫ ݀ = ݕ‬+ asin(ܾ‫ ݔ‬+ ܿ)
SKETCH THE GRAPH OF THE FOLLOWING
Amp:
Period:
Phase Shift:
= 1 − 0.5sin(0.5 − )
X
Increment:
Starting point:
Y
Note that this graph also has
a vertical shift upward 1 unit
and a reflection over the x axis.
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ON YOUR OWN: FIND THE KEY POINTS AND
GRAPH
Graph the following function by hand. = 2 + 3 cos 2
Amp:
Period:
Phase Shift:
X
Y
4.6 OTHER TRIG FUNCTIONS
Students will know how
to sketch the graphs of
trigonometric functions.
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GRAPH OF THE TANGENT FUNCTION
Information about the graph of a tangent function.
1) The tangent function is odd and periodic with period , calculated
గ
௕
2) The function consists of multiple vertical asymptotes, for the base
గ
ୱ୧୬ ௫
function that would be at + , since tan =
.
∈
ଶ
ୡ୭ୱ ௫
3) The key points for the graph of the tangent function are the
asymptotes at the ends of the period and the intercepts in the middle.
Basic information
ߨ
2
Period:
Domain:
Range:
X-intercepts:
−
Identify the key points:
ߨ
2
GRAPH THE TANGENT FUNCTION
Graph the following function. = tan X
Y
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SKETCHING OTHER TANGENT FUNCTIONS
AMP:
PERIOD:
Sketch = −3 tan 2
X
Y
OYO SKETCH = 22
AMP:
X
PERIOD:
Y
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THE GRAPH OF COTANGENT
This is very similar to the graph of tangent. The cotangent function is an
odd function.
గ
ୡ୭ୱ ௫
, therefore, the period is: , (This is because it
cot =
௕
ୱ୧୬ ௫
starts repeating after The domain of the function is:
The range of the function is:
The vertical asymptotes are at:
Intercepts?
Identify the key points:
ߨ
0
SKETCHING THE GRAPH OF COTANGENT
AMP:
PERIOD:
Graph the following function: = cot X
Y
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OYO TRY TO GRAPH Y = −22
AMP:
X
PERIOD:
Y
THE GRAPH OF RECIPROCAL FUNCTIONS
We will be looking at the graph of csc which is the reciprocal of sin and at sec which is the reciprocal of cos .
There are certain patterns that exist within these functions that stem
from the original functions. Note: ∈ = csc The period:
The domain:
‫ = ݕ‬sec ‫ݔ‬
The period:
The domain:
The range:
Vertical Asymptotes:
Symmetry:
The range:
Vertical Asymptotes:
Symmetry:
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HOW TO GRAPH THE RECIPROCAL FUNCTION:
STEP BY STEP
Step 1: Identify the reciprocal and graph it as a
ଵ
, so graph sin dashed line:
EX: csc =
ୱ୧୬ ௫
Step 2: At each − of the reciprocal, we put a vertical
ଵ
ଵ
= is undefined)
asymptote (because
ୱ୧୬ ௫
଴
Step 3: Each max and min of the reciprocal are the vertex of a parabola
going in the opposite direction. Draw it in!
Step 4: You are done.
THE GRAPHS: CSC X
Sinx
cscx
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THE GRAPHS: SECX
X
cosx
secx
LETS GRAPH SOME ON THE SAME GRAPH
Graph
x
గ
గ
= 2 sin + ସ and = 2 csc + ସ
2Sinx
2cscx
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DAMPED TRIGONOMETRIC FUNCTIONS
This is when two functions collide, one being a trig function and one not
being a trig function.
A product of two function can be graph using properties of the individual
function. For example = sin .
To graph this use the non trig function as boundaries and graph the trig
function within those boundaries. − ≤ sin ≤ ||. Therefore we graph
within the boundaries of x and –x graphs.
GRAPHING DAMPED GRAPHS
Graph the following = sin x
xSinx
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A HARDER ONE
= ଶ + 1
ିଵ sin x
OYO: YOU TRY
ଵ
ଶ
= ( + 2) sin 2
x
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HOMEWORK
P311 #9-27odd
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