Graphs of sine and cosine functions
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Graphs of Sine and Cosine Functions. Section 4.5. Fast Wave, Slow Wave. Who cares about waves??. Create the sine and cosine Graphs. Creating the Graphs. sine and cosine graphs. Sine graph plotted Cosine graph plotted. What about negative values?. Does the graph repeat? When?.

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Fast wave slow wave
Fast Wave, Slow Wave

Who cares about waves??



Sine and cosine graphs
sine and cosine graphs

Sine graph plotted

Cosine graph plotted



Does the graph repeat when
Does the graph repeat?When?

One complete cycle of the cosine and sine graphs is called the ___period___


Key points
Key Points

What did you find???

Key Points

What are key points? What do they mean?


Review graphing sine and cosine
Review – Graphing sine and cosine

  • Where do they graphs come from?!?

  • Graphing Sine and Cosine – Review

  • graphing sine and cosine - radians


Listing the key points
Listing the Key Points

Sinx:(0°,0) (90°, 1) (180°,0) (270°, -1) (360°, 0)

Cosx:(0°,1) (90°,0) (180°, -1) (270°, 0) (360°, -1)


Transformations to the parent graphs sine and cosine
Transformations to the parent graphs – sine and cosine

  • We will see how transformations affect the basic sine and cosine parent graphs


General equations
General Equations:

  • y = asin(bx – c) + d

  • y = acos(bx – c) + d


Vertical translations
Vertical Translations

  • A vertical shift is the vertical distance between the midline of the graph and the x –axis.

  • For y = sinx +d and y = cosx +d, the constant d causes a vertical shift in the graph

  • Which value (x or y) is influenced by the change in the “d” value?


What s happening
What’s Happening??

Transformations of Sine and Cosine


Vertical translations1
Vertical Translations

  • The shift is d units upward for d >0

  • The shift is d units downward for d< 0

  • The graph oscillates about line y = d instead of x-axis

  • Does the period change when “d” changes?



Amplitude
Amplitude

  • The amplitude of y = asinx and y = acosx represents the vertical distance between the midline and the maximum or minimum

  • Amplitude = |a|

  • The constant factor “a” is a scaling factor - a vertical stretch or shrink of the basic sine and cosine curves


What s happening1
What’s Happening??

Transformations of Sine and Cosine


Amplitude what does it do
Amplitude…What Does It Do??

  • If a ≥ 1, the basic sine curve is stretched vertically

  • If a ≤ -1, the basic sine curve is reflected across the x-axis and vertically stretched

  • The graph of y = asinx ranges between -a and a instead of between -1 and 1


Amplitude what does it do1
Amplitude…What Does It Do??

  • If 0 <a < 1, the basic sine curve is vertically shrunk

  • If -1< a < 0, the basic sine curve is reflected across the x-axis and vertically shrunk

  • The graph of y = asinx ranges between -a and a instead of between -1 and 1