1. 5.5 a&b Graphs of Sine and Cosine
HW p odd, odd

2. Graphical Analysis: graph the function in the given window
Domain:
Range:
Continuous
Alternately increasing and decreasing
in periodic waves by
Symmetry: Origin (odd function)
Bounded
Absolute Max. of 1
Absolute Min. of –1
No Horizontal Asymptotes
No Vertical Asymptotes
End Behavior:
and
do not exist

3. The “Do Now” – first, graph the function in the given window
Other notes:
This function is periodic, with
period
By definition, sin(t) is the
y-coordinate of the point P on the
unit circle to which the real number
t gets wrapped by
So now let’s “explore” where this wavy graph comes from…

4. Now, a Complete Analysis of the Cosine Function
Domain:
Range:
Continuous
Alternately increasing and decreasing
in periodic waves by
Symmetry: y-axis (even function)
Bounded
Absolute Max. of 1
Absolute Min. of –1
No Horizontal Asymptotes
No Vertical Asymptotes
End Behavior:
and
do not exist

5. Definition: Sinusoid
A function is a sinusoid if it can be written in the form
where a, b, c, and d are constants and neither a nor b is 0.
In general, any transformation of a sine function (or the graph
of such a function – such as cosine) is a sinusoid.
This is the format that we are used to seeing, thus it is OK to
continue using this format…I use this format.

6. Transformations Horizontal stretches and shrinks affect the period
There is a special vocabulary for describing our traditional
graphical transformations when applied to sinusoids…
Horizontal stretches and shrinks affect the period
and the frequency.
Vertical stretches and shrinks affect the amplitude.
Horizontal translations bring about phase shifts.

7. Definition: Amplitude of a Sinusoid
The amplitude of the sinusoid is
Similarly, the amplitude of is
Graphically, the amplitude is half the height of the wave.

8. Transformations Confirm these answers graphically!!!
Find the amplitude of each function and use the language of
transformations to describe how the graphs are related.
(a)
(b)
(c)
Amplitudes: (a) 1, (b) 1/2, (c) |–3| = 3
The graph of y is a vertical shrink of the graph of y by 1/2. 2 1
The graph of y is a vertical stretch of the graph of y by 3,
and a reflection across the x-axis, performed in either order. 3 1
Confirm these answers graphically!!!

9. Definition: Period of a Sinusoid
The period of the sinusoid is
Similarly, the period of is
Graphically, the period is the length of one full cycle of
the wave.

10. Transformations
Find the period of each function and use the language of
transformations to describe how the graphs are related.
Periods
(a)
(b)
(c)

11. Transformations
Find the period of each function and use the language of
transformations to describe how the graphs are related.
(a)
The graph of y is a horizontal
stretch of the graph of y by 3, a
vertical stretch by 2, and a
reflection across the x-axis,
performed in any order. 2 1
(b)
(c)

12. Transformations Confirm these answers graphically!!!
Find the period of each function and use the language of
transformations to describe how the graphs are related.
(a)
The graph of y is a horizontal
shrink of the graph of y by 1/2, a
vertical stretch by 3, and a
reflection across the y-axis,
performed in any order. 3 1
(b)
Confirm these answers
graphically!!!
(c)

13. Transformations New Terminology: When applied to sinusoids, we
How does the graph of differ from the
graph of ?
 A translation to the left by c units when c > 0
New Terminology: When applied to sinusoids, we
say that the wave undergoes a phase shift of –c.

14. Transformations Confirm these answers graphically!!!
Write the cosine function as a phase shift of the sine function.
Write the sine function as a phase shift of the cosine function.
Confirm these answers graphically!!!

15. Reminder: Graphs of Sinusoids
The graphs of these functions have the following characteristics:
Amplitude =
Period =
A phase shift of
A vertical translation of

16. Guided Practice Graph one period of the given function by hand.
Amplitude =

17. Guided Practice Graph one period of the given function by hand.
Amplitude =

18. Guided Practice
Identify the maximum and minimum values and the zeros of the
given function in the interval  no calculator!
Maximum: At
Minimum: At
Zeros:

19. Finally, a couple of whiteboard problems
Find the amplitude of the function and use the language of
transformations to describe how the graph of the function is
related to the graph of the sine function. 1.
Amplitude 2; Vertical stretch by 2
Amplitude 4; Vertical stretch by 4,
Reflect across x-axis 3.

20. More whiteboard… Period 7. Horizontal shrink by 1/3 9. Period
Find the period of the function and use the language of
transformations to describe how the graph of the function is
related to the graph of the cosine function. 7.
Period
Horizontal shrink by 1/3 9.
Period
Horizontal shrink by 1/7,
Reflect across y-axis

21. Whiteboard Graph three periods of the given function by hand.
Amplitude =
Period =

22. Whiteboard Graph three periods of the given function by hand.
Amplitude =
Period =

23. Whiteboard Graph three periods of the given function by hand.
Amplitude =
Period =

24. Whiteboard State the amplitude and period of the given sinusoid, and
(relative to the basic function) the phase shift and vertical
translation.
Amplitude:
Period:
Phase Shift:
Vertical Translation: 7 units up

25. Whiteboard
Identify the maximum and minimum values and the zeros of the
given function in the interval  no calculator!
Maximum: At
Minimum: At
and
Zeros:

26. Whiteboard State the amplitude and period of the given sinusoid, and
(relative to the basic function) the phase shift and vertical
translation.
Amplitude:
Period:
Phase Shift:
Vertical Translation: 1 unit down