Section 5.2: Transformations of Sinusoidal Functions (Sine and...
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Section 5.2 Tansformations of Sinusoidal Functionssoln.notebook 1 February 03, 2017 Section 5.2: Transformations of Sinusoidal Functions (Sine and Cosine)
Transcript of Section 5.2: Transformations of Sinusoidal Functions (Sine and...
Section 5.2 Tansformations of Sinusoidal Functionssoln.notebook
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Section 5.2: Transformations of Sinusoidal Functions (Sine and Cosine)
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Remember: VS =
HS =
Reflection in xaxis if a < 0Reflection in yaxis if b < 0
New:Horizontal Translation (c)Vertical Translation (d)
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Investigation Part C and D
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Part C: How does changing the value of d affect the graph?
"d" is the Vertical Translation or Vertical Displacement
1. If d > 0, the graph is shifted upward d units If d < 0, the graph is shifted downward d units.
2. SA, midline, or central axis: y = d
3. Range is affected: Minimum Point = midline (VT) – amplitude = d – a Maximum Point = midline(VT) + amplitude = d + a
Range =
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‐ Cosine function the start of the first cycle of the cosine curve is (0, 1) and it goes down
‐ Sine function the start of the first cycle of the sine curve is (0, 0) and it goes up
How does changing c affect the graph?
c is the Horizontal Translation or Phase Shift
i) (x + c) Horizontal Translation: HT = c (the base graph is shifted c units left)
ii) (x – c) Horizontal Translation: HT = c (the base graph is shifted c units right)
Part D: Horizontal Translation (c) (Phase Shift)
Keep in mind
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The value that is chosen for the phase shift will determine whether the graphs are perceived as having a reflection in the x axis or not.
If you choose the xvalue of the maximum point no reflectionIf you choose the x value of the minimum point reflection in xaxis
Example 5: Graph the functions y = cos x and y = cos (x π).
Can you identify other equations that would produce the same graph as y = cos (x π)?
Mapping Rule:
Be careful when describing phase shift!
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Example 6: Graph the functions: and
Think about other equations:
The value that is chosen for the phase shift will determine whether the graph is perceived as having a reflection on the xaxis or not.
(0, 0) on the SA and graph is going from a minimum to a maximum no reflection(0, 0) on the SA and graph is going from a maximum to a minimum reflection in xaxis
Mapping Rule:
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Example 9: Sketch the graph of
Mapping Rule: (x, y)
A) Identify the vertical displacement, amplitude, period, phase shift, domain and range for the function.
Vertical displacement:
Amplitude:
Period:
Phase Shift:
Domain:
Range:
B) How would you modify the equation so it would have no xintercepts?
C) Suppose we wanted to write the equation in the form
What values could be used for a ,b ,c, and d ?
Questions: p.250252 # 1ce (no graph), 2cf (no graph), 3a, 4, 5, , 14, p.253 #19
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Write the Equation of the Sinusoidal Function Given the Graph
Identify the key characteristics of the graph and then link them to the parameters in the equation.
maximum value = minimum value = period =
Sinusoidal Axis = amplitude = b =
Horizontal translation:
Sine Function Cosine Function
Phase shift locate start of cycle phase shift locate start of cycle
Equation: Equation:
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a
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Page 250253, Questions 6 – 7, 13 – 16Worksheet "Trig Graphs Worksheet"
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Applications of Sine and Cosine
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Example 14:
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Example 15: The depth d, in metres, of the water in the harbor is approximated
by the equation , where t is the time in hours,
after the first high tide.
A) Graph the function for two cycles starting at t = 0.
B) What is the period of the tide?
Use period to find scale of graph
C) If a boat requires a minimum of 3.5m of water to launch safely, for how many hours per cycle can the boat safely launch?
D) What is the depth of the water at 7h? At what other times is the water level at this depth?
max = d+a
min = d+a
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