lesson 5.2 transformations of sinusoidal...
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Lesson 5.2 Transformations of Sinusoidal Functions
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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 Sinusoidal Functions
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Part A: Reflections on the x and yaxis
Graph the functions Example 1:
Lesson 5.2 Transformations of Sinusoidal Functions
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Graph the functions Example 2:
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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
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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 cosine curve (0,0)
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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
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Graph the functions Example 5:
Can you identify other equations that would produce the graph of ?
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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.
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Example 7: Sketch the graph of over two cycles.
Method 1: Use Mapping Rule
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Example 7 cont'd
Method 2: Use Inequality
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Example 8: Sketch the graph of over two cycles.
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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 9 cont'd
Method 2: Use Inequality
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Example 10: Sketch the graph of over two cycles.
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Your Turn
Sketch the following functions over two cycles. Identify the vertical displacement, amplitude, period, phase shift, domain and range.
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Example 1: 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 =
From graph there are __ cyles in
period = b =
phase shift
Sine Function Cosine Function phase shift locate start of cycle locate start of cycle
Equation: Equation:
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Example 2: Write the equation of the function in the form and
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Example 3: Write the equation of the function in the form and
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Example 4: Word Problems
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p.253 #1516p.253 #67, 1314 p.253 #19
Questions
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Lesson 5.2 Transformations of Sinusoidal Functions
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Assign p.254255 #20, 24Word Problems
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