Lesson 5.2: Transformations of Sinusoidal Functions (Sine...
Transcript of Lesson 5.2: Transformations of Sinusoidal Functions (Sine...
Lesson 5.2 Transformations of sine and cosine function
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Lesson 5.2: Transformations of Sinusoidal Functions (Sine and Cosine)
Reflections
Horizontal Translation (c)
Vertical Translation (d)
Remember: vertical stretch
horizontal stretch
Lesson 5.2 Transformations of sine and cosine function
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Part A: Reflections on the x and yaxis
Graph the functions Example 1:
Lesson 5.2 Transformations of sine and cosine function
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Part B: Vertical Translation (vertical displacement)
Graph the functions Example 3:
vertical translation:
midline:
maximum:
minimum:
How does changing the value of affect the graph?
vertical translation:
midline:
maximum:
minimum:
vertical translation:
midline:
maximum:
minimum:
Use amplitude and vertical translation to determine the maximum and minimum value
Lesson 5.2 Transformations of sine and cosine function
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Part C: Horizontal Translation (Phase Shift)
Graph the functions
Example 4:
Locate the start of the first cycle of the cosine curve (0,1)
How does changing the value of affect the graph?
Cosine Function
Sine Function
the start of the first cycle of the cosine curve (0,1)
the start of the first cycle of the sine curve (0,0)
Lesson 5.2 Transformations of sine and cosine function
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Think about the equations:
Since the function is periodic, there are several equations that can correspond to a given graph where the phase shift is different.
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.
The xvalue of the maximum point no reflection
The xvalue of the minimum point reflection on xaxis
Lesson 5.2 Transformations of sine and cosine function
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Graph the functions Example 5:
Can you identify other equations that would produce the graph of ?
Lesson 5.2 Transformations of sine and cosine function
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Graph the functions Example 6:
Locate the start of the first cycle of the sine curve (0,0)
(i) where the xcoordinate hits the sinusoidal axis going from a minimum to a maximum no reflection
(ii) where the xcoordinate hits the sinusoidal axis going from a maximum to a minimum reflection on xaxis
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.
Lesson 5.2 Transformations of sine and cosine function
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Example 7: Sketch the graph of over two cycles.
Method 1: Use Mapping Rule
Lesson 5.2 Transformations of sine and cosine function
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Example 8: Sketch the graph of over two cycles.
Lesson 5.2 Transformations of sine and cosine function
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Example 9: Sketch the graph of over two cycles. Identify the vertical displacement, amplitude, period, phase shift, domain and range for the function.
Method 1: Use Mapping Rule
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Example 10: Sketch the graph of over two cycles.
Lesson 5.2 Transformations of sine and cosine function
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Worksheet: Sketch the graphs of cosine and sine functions
Worksheet
Sketch the following functions over two cycles. Identify the vertical displacement, amplitude, period, phase shift, domain and range.
p.250252 # 1ce (no graph), 2cf (no graph), 3a, 4, 5, 67, 1314Questions:p.253 #19
Lesson 5.2 Transformations of sine and cosine function
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Example 11: Write the equation of the function in the form
Identify the key characteristics of the graph and then link them to the parameters in the equation.
and
Write the Equation of the Sinusoidal Function Given the Graph.
maximum value =
minimum value =
Sinusoidal Axis =
amplitude =
period =
b =
phase shift
Sine Function Cosine Function phase shift locate start of cycle locate start of cycle
Equation: Equation:
Horizontal Translation:
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Example 12: Write the equation of the function in the form and
Example 13: Write the equation of the function in the form and
Questions: p.253 #1516
Lesson 5.2 Transformations of sine and cosine function
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Example 14: 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 tide.
Word Problems
(a) Graph the function for two cycles starting at .
(b) What is the period of the tide?
(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?
Inequality
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Example 15:
A mechanic rolls a wheel of radius 42cm and the wheel as it rolls makes 1 revolution every two seconds. The diagram below illustrates the vertical motion of a point on the wheel over time. Find an equation of the function shown as a transformation of sine or cosine.
Question: p.254 #24