4.1 and 4.2 Sine Graph

1
4.1 and 4.2 Sine Graph
Sine Cosine are periodic functions, repeating
every period of 2? radians
y
y sin (x)

• x y
• 0 0
• ?/2 1
• 0
• 3?/2 -1
• 2? 0

2
1
x
0
-1
-2
Does the graph always start at zero? Try y
sin (x - ?/2)
What is the amplitude? (How High is
it?) What about the graph of y 2sin(x)
Does the period ever change? What about y
sin (2x)
2
Equation of a Sine Curve
y A sin (Bx C) amplitude A
period 2?
B phase shift C (right/left) B

• Graphing Guidelines
• Calculate period
• 2. Calculate interval
• (period / 4)
• Calculate phase
• shift-start-value
• 4. Create X/Y chart w/
• 5 key points from
• start-value

y sin (x) y 2 sin (x) y sin (2x) y sin (x
- ?/2)
y
2
1
x
0
-1
Does the graph ever shift vertically? YES y A
sin (Bx C) D D is the vertical shift What
would y sin (x - ?/2) 1 look like?
-2
3
Equation of a Cosine Curve
y A cos (Bx C) amplitude A
period 2?
B phase shift C (right/left) B

• x ycos(x)
• 0 1
• ?/2 0
• -1
• 3?/2 0
• 2? 1

y cos (x) y 2 cos (x) y cos (2x) y cos (x
- ?/2)
y
2
1
x
0
-1
-2
4
4.3 Equation of a Tangent Function
y A tan (Bx C) 1. Find draw two
asymptotes Set Bx C ? and - ? (solve for
x) 2 2 2.
Find plot the x-intercept (Midway between
asymptotes) 3. Plot points x ¼ from 1st
asymptote, y -A x ¾ before 2nd
asymptote, y A
y tan(x)
1
0
x
? 2
?
-? 2
3? 2
2?
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Graphing a Cotangent Function
y A cot (Bx C) 1. Find and draw two
asymptotes Set Bx C 0 and ? (solve for x)
2. Find
plot the x-intercept (Midway between
asymptotes) 3. Plot points x ¼ from 1st
asymptote, y A x ¾ before 2nd
asymptote, y -A
y cot(x)
1
0
x
? 2
?
-? 2
3? 2
2?
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4.4 Graphing variations of ycsc(x)
csc(x) 1/sin(x). To graph the cosecant
1) Draw the corresponding sine graph 2)
Draw asymptotes at x-intercepts 3) Draw
csc minimum at sin maximum 4) Draw csc
maximum at sin minimum
y sin (x) y csc (x)
y
2
1
x
0
-1
-2
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Graphing variations of ysec(x)
sec(x) 1/cos(x). To graph the secant
1) Draw the corresponding cosine graph 2)
Draw asymptotes at x-intercepts 3) Draw
sec minimum at cos maximum 4) Draw sec
maximum at cos minimum
y cos (x) y sec (x)
y
2
1
x
0
-1
-2
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4.5 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 ? are constants with ? gt
0. Amplitude of motion is a Period is 2?/
? Frequency is ?/2 ? oscillations per unit of time

• Example An object attached to a coiled spring is
  pulled down 5 in
• from equilibrium position and released. The time
  for 1 complete oscillation is 4 sec.
• Give an equation that models the position of the
  object at time t
• Time of object release is t 0 and
  distance is then 5 in below equilibrium
• so, S(0) -5 and we use S(t) cos ?t.
  Period 4 sec ? 2?/ ? 4 ? ? ?/ 2
• So, S(t) -5 cos (?/ 2)t
• (b) Determine the position at t 1.5 sec
  S(1.5) -5 cos (?/ 2)(1.5) 3.54 in.