Transcript Graphing the Sine Function

Start Up Day 26
1. Graph each function from -2π to 2π
g(x) = cos ( x )
The range of y = sin x is _____________.
The range of the cosine curve is
________________.
2. Find a polynomial function with zeros
of: 0, -4, 3i
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f (x) = sin ( x )
OBJECTIVE: SWBAT use trigonometric graphs to
define and interpret features such as domain, range,
intercepts, periods, amplitude, phase shifts, vertical
shifts and asymptotes. SWBAT to graph Sine and
Cosine functions with and without graphing technology.
EQ: What are the key points for the basic Sine
and Cosine parent graphs & where do they
come from? How do the values of “a” ,”b”, “h”
and “k” affect the graphs of Sine and Cosine?
HOME LEARNING: Worksheet#1
Graphing Sine Functions
Graphing Trigonometric Functions
Sine Waves: The Movers
and the Shapers
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The UNIT CIRCLE—unwrapped!
• Y=sin x is the basic curve: one hill, one valley,
starting at 0 and ending at 2 π
• 5 KEY Points: 0, 1, 0, -1 , 0
• Y=cos x is the basic curve:
an upside down

bell shape, starting at 0 and ending at 2 π
• 5 KEY Points: 1, 0, -1, 0 ,1
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“h” AND “k” OR “c” & “d”,
respectively: The “Movers”
• The “h”or the “c” is the most difficult to see!
(although it is always in the parentheses!)
– The “h” causes a “phase shift” OR
“HORIZONTAL TRANSLATION”
– You have to factor out your “b” in order to see
your “h” for the HT!
• The “k”or the “d” is much more obvious!
– When looking to the “k”, you get exactly what you
see!
– The “k” causes a VERTICAL TRANSLATION
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

y  sin  x    1
2

a=
Amplitude:
b=
Period/Wavelength:
h=
Start:
k=
End:
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A and B: The Shapers!
• The “b” value: FREQUENCY = b
– Horizontal Compression—If the “b” is greater than
1
– Horizontal Stretch –If the “b” is less than one.
– Period OR Wavelength = 2pi/b
• The “a” value: AMPLITUDE = Absolute Value of a
– Vertical stretch--If the absolute value of “a” is
greater than one.
– Vertical Compression--If the absolute value of “a”
is less than one.
– Reflection over the x-axis—If the “a” is a negative
value.
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Where to begin? y=a sin b(x-h)+k
1. First-- think of the basic wave and the
5 KEY POINTS
Sine: (0,1, 0,-1, 0) OR
Cosine:(1, 0, -1, 0, 1)
2. Next--Identify the values of
a,b,”h”(c) and “k” (d)
3. Finally--determine amp, frequency,
period or wave length, horizontal
and/or vertical translations
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Sketching y=a sin b(x-h)+k
1) Dash in a horizontal axis at “k” or“d”
2) If c=o then start at o and end at 2π/b, otherwise
continue with begin/end
– BEGIN: Set your (bx-bh) =o and solve for x—
this will be your starting point.
– END: Set your (bx-bh) =2π and solve for x
again—this will be your ending spot.
3) Divide your “wavelength” into FOUR equal
spaces— making room for your FIVE KEY
PLACES!
4) Let your “a” wipe the “ones” away and sketch
your wave!
– Remember that a “-a” causes a reflection over
the x-axis
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y  2 sin 2 x     1
A=-2
START:
2
3p
2x - p = 2p ;x =
2
END:
B=2
2x - p = 0;x =
p
Period/Wavelength:
C=-π/2
Amplitude:
2p 2p
=
=p
b
2
a = -2 = 2
D=1
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y  2 sin 2 x     1
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Let’s apply it—Now you try it! Sketch
each over 2 complete periods.
()
#3. y = -4sin x
æ 2x ö
#16. y = -4sin ç ÷
è 3ø
æxö
# 27. y = 4sin ç ÷
è4ø
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Start Up
Day 27
Problems #58 & 61 from p.358
Construct a sine function (sinusoid) with the given
constraints:
#58 Amplitude 2, period 3π, point (0, 0)
“a” and “b”: The Shapers!
Y = a cos b(x – c) +d
“a”
Amplitude = IaI
“b”
Frequency = b
Period (wavelength)=
2π/b
½ the distance from the
Max to the min.
a
Reflection over
x-axis
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IaI> 1,
Vertical Stretch
IaI < 1,
Vertical Compression
IbI>1
Horizontal
Compression
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IbI<1
Horizontal
Stretch
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“h” and “k” OR “c” & “d” respectively—
The Movers
y=a cos b(x-h)+k
The Movers
”h”
Horizontal Translation= “h” or “c”
Always opposite of what you see
in the ( ) and don’t
Forget to factor out your “b” in
order to see your
REAL “h”OR set your () = 0 and
Solve for “x”.
Changes the starting point of the
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curve!
“k” or “d”
K = Vertical Translation
k lifts or lowers the base lin
of the curve
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