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The General
Sine Function:
Data table for the standard sine function
over 360˚
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0˚ |
90˚ |
180˚ |
270˚ |
360˚ |
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sin
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0 |
1 |
0 |
-1 |
0 |
Graph of the standard sine function over 360˚
Properties:
Amplitude is the distance
from any maximum or minimum point to the x-axis.
It is represented by the letter “a” in the
function y = a sin. Consider
the following when graphing transformations
involving the amplitude.
- When the amplitude is
less than 0, there is a reflection on the
x-axis (ex. -1)
- When the amplitude is
less than 1 but greater than 0, there is a
vertical compression (ex. ½)
- When the amplitude is
greater than 1, there is a vertical
expansion (ex. 2)
Period represents the
factor by which the graph has been compressed
horizontally or expanded horizontally. It is
represented by  in
the function y = sin k.
Consider the following when graphing functions
involving period changes.
- When the period is
greater than 1, the graph is horizontally
compressed. For example if the k value is 2,
then the graph is horizontally compressed by
a factor of 2
- When the period is
less than 1, the graph is vertically
compressed. For example if the k value is ¼,
then the graph is horizontally expanded by a
factor of 4
In this chapter you may
also encounter questions that ask for Min / Max
values of a function. The maximum is the highest
y-value on the graph that the function touches.
The minimum is the lowest y-value on the graph
that the function touches.
Transformations:
Vertical displacement is
the distance the graph has translated
vertically. It is represented by the letter “q”
in the function y = sin + q.
Consider the following when graphing
transformations involving a vertical
displacement.
- When the vertical
displacement is positive, the graph
undergoes an upward translation. For example
in function y = sin + 1, the
graph will translate 1 unit upwards
- When the vertical
displacement is negative, the graph
undergoes a downward translation. For
example in the function y = sin – 1, the
graph will translate 1 unit downwards
Phase shift is the distance
the graph has been translated horizontally. It
is represented by the letter “p” in the function
y = sin ( – p).
Consider the following when graphing functions
involving a phase shift.
- When the phase shift
is negative, for example… y = sin ( – ¼ π), d
indicates a horizontal translation of ¼ π
units to the right.
- When the phase shift
is positive, for example… y = sin ( + ¼ π), d
indicates a horizontal translation of ¼ π
units to the left.
Domain & Range
In this unit, you may
encounter questions that ask for the domain and
range of the function. Domain is the set of
x-values on the graph. For example, if a graph
is between -360˚ to 360˚ then the domain is
-360˚ ≤ ≤ 360˚ …
this indicates that ranges from
-360˚ to 360˚. Range is the set of y-values on
the graph; it is the point from the maximum
position of the graph to the minimum position.
For example in the function y = sin (see above
for the graph), the max position is 1 and the
min position is -1, therefore the range of the
graph is -1 ≤ y ≤ 1
Sample Questions:
Graph y = 3 sin 2 (x – 45)
+ 3 over 360o. State the amplitude,
period, phase shift, vertical displacement,
domain and range.
Amplitude = 3
Period =  =
180˚
Phase Shift = 45o
[right]
Vertical Displacement = 3
[up]
Domain = 0o ≤
≤ 360o,
ε r
Range = 0 ≤ y ≤ 6, y ε r
Max = 6
Min = 0
Walkthrough of the
process:
When you are graphing these
types of problems, it is most efficient
adjusting the changes in order. I usually start
by graphing the standard sine function to get a
general shape of the graph, then I graph the
amplitude, in this function you have noticed
that the amplitude is 3, so start this function
by graphing y = 3 sin . After that
you may want to adjust the period by following
the formula:  this
will give you the new period. At this point you
may graph the period changes by adjusting the
original sine function to include the period
change, and the amplitude change y = 3 sin 2
θ. After the period and amplitude adjustments,
you will have to adjust the phase shift; the
original equation indicated that the function
has a phase shift of 45o to the
right, which means you will have to move the
entire function over by 45o. I
usually do this by taking key points and moving
45˚ to the right, then redrawing the line. The
last adjustment to make is the vertical
displacement, this equation has a vertical
displacement of +3, and this means that you have
to move the entire function up by 3 units. So
again, take the individual points and move them
up, then redraw the line.
In this chapter, students
may also encounter questions that give a sine
function and ask for an equation to describe it.
For example:
Write a sinusoidal equation
that describes the following graph:
In these types of
questions, you must work backwards. For example,
start with the amplitude… by looking at the
graph we can tell that there is amplitude of 3,
[by using this equation: ],
we can also conclude that the period is 360o,
since the function repeats itself over 360o.
The phase shift is -90o since the
original sine function has moved over 90o
to the right from its initial position (you have
to picture the original sine function, then see
how much it has moved). Since there is no
vertical displacement we can conclude that the
sine function for this graph is y = 3 sin ( – 90)
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The following
summarizes the transformations that can be
applied to the standard sine function.
y = a sin k (
– p) + q
a = amplitude
=
period
d = phase shift
c = vertical
displacement
Steps to
graphing a function:
i)
Draw the general sine function.
ii) Adjust
amplitude if necessary; if negative then graph
its reflection on the x-axis.
iii) Adjust
period if necessary.
iv) Apply the
phase shift.
v) Apply the
vertical displacement.
Strategies for
Application Questions:
i)
Find your period, and divide it into four equal
sections
ii) Your phase
shift is where your function first peaks on your
graph.
iii) Your rest
position is the vertical displacement
iv) Amplitude
is equal to (peak – rest)
v) Vertical
translation is the mean value
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Questions |
Complete Solutions |
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1. Graph y = 3 sin
over
two cycles. Determine the domain and
range, phase shift, amplitude, vertical
displacement, period and its max / min
values.
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Domain: 0o
≤ ≤ 360o,
ε r
Range: -3 ≤ y ≤ 3,
y ε r
Phase Shift: 0
Amplitude: 3
Vertical
Displacement: 0
Period=  =
360o
Max: 3
Min: -3
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2. Describe what
happens to the graph of each function.
a) y =a sin 2 (– 90) – 1,
as a varies
b) y =3 sin k (–
45) + 1, as k varies
c) y =2 sin 3 (– p) – 2, as
p varies
d) y =4 sin 2 (– 30) + q,
as q varies
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a) If the ‘a’ value
increases then the amplitude of the
graph will increase, and undergo a
vertical expansion. If the ‘a’ value
decreases then the amplitude of the
graph will decrease, and undergo a
vertical compression.
b) If the ‘k’ value
increases then the period of the graph
will decrease and undergo a horizontal
compression, if the ‘k’ value decreases
then the period of the graph will
increase, and the graph will undergo a
horizontal expansion
c) As the ‘p’ value
increases then the graph will move
farther to the right, as the ‘d’ value
decreases then the graph will move
farther to the left.
d) As the ‘q’ value
increases the graph will move farther
up, as the ‘c’ value decreases then the
graph will move farther down. |
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3. Graph y = 4 sin
(- 30o)
-1 over one cycle.
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4. For the function
y = 2 sin 5 ( – 45) + 4…
Determine the phase shift, amplitude,
vertical displacement, period and its
max / min values.
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Phase Shift: 45o
[right]
Amplitude: 2
Period:  =
72o
Vertical
Displacement: 4
Max: 6
Min: 4 |
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