Sinusoidal Graphing

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Sinusoidal Graphing

• By Ms. Hales
• Trigonometry

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Cosine Graphs

• f(x) a cos (b(xc)) d

a -- amplitude
b -- period 2p/b
c -- horizontal phase shift
d -- vertical phase shift
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Sine Graphs

• f(x) a sin (b(xc)) d

a -- amplitude
b -- period 2p/b
c -- horizontal phase shift
d -- vertical phase shift
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Amplitude

• Height of curve
• Distance from midline to maximum(crest)
• Distance from midline to minimum(trough)

• Vertical stretch or shrink

Amplitude
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Period

• Horizontal stretch or shrink

• Period 2p/b

Period
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Horizontal Phase Shift

• cosine curve
• maximum when x0

• sine curve
• middle of increasing side is where x0

cos shift
sin shift
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Vertical Phase Shift

• Midline

midline
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Graph Makingwith cosine
Y 2 cos (2(x (p/6))) 5
Go to graph
d ?
d 5
a ?
a 2
Go to graph
Go to graph
c ?
c -p/6
Go to graph
b ?
b 2
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Cosine Curvewith vertical phase shift
Back
X scale p/3 Y scale 1
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Cosine Curve with amplitude
Back
X scale p/3 Y scale 1
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Cosine Curvehorizontal phase shift
Back
X scale p/3 Y scale 1
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Cosine Curvewith period
Distance p
Distance p
Back
X scale p/3 Y scale 1
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Graph of Y 2 cos (2(x (p/6))) 5Y 2 sin
( 2 (x (p/12))) 5
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Graph Making with sine

• Y 2 sin ( 2 (x (p/12))) 5

Go to graph
d ?
d 5
Go to graph
a ?
a 2
Go to graph
c ?
c p/12
Go to graph
b ?
b 2
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Sine Curvewith vertical phase shift
Back
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Sine Curvewith amplitude
Back
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Sine Curvewith horizontal phase shift
Back
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Sine Curvewith period
Back
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Graph of Y 2 cos (2(x (p/6))) 5Y 2 sin
( 2 (x (p/12))) 5
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Equation from Graph
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Equation from Graph
a ?
a 5
b 3
b ?
cos c -10 sin c 20
c ?
d ?
d -3
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Cosine Equation from Graph
f(x) a cos (b(xc)) d
f(x)
5 cos (
3(x
-10))
-3
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Sine Equation from Graph
f(x) a sin (b(xc)) d
f(x)
5 sin (
3(x
20))
-3
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Example

• Give the cosine and sine equations for the
  following

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Answer
Y 4 cos (3(x(p/3)))3
Y 4 sin (3(x-(p/6)))3