Lesson 5.2 Transformations of Sinusoidal...

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Lesson 5.2 Transformations of Sinusoidal Functions 1 Lesson 5.2: Transformations of Sinusoidal Functions (Sine and Cosine) Reflections Horizontal Translation (c) Vertical Translation (d) Remember: vertical stretch horizontal stretch

Transcript of 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

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Part A: Reflections on the x and y­axis

Graph the functions                  Example 1:

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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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Assign p.254­255 #20, 24Word Problems

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