1. Sinusoids and Transformations
Sec. 4.4b

2. 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.

3. 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.

4. 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.

5. 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!!!

6. 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.

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

8. 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)

9. 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)

10. Definition: Frequency of a Sinusoid
The frequency of the sinusoid is
Reciprocal of the period!!!
Similarly, the frequency of is
Graphically, the frequency is the number of complete cycles
the wave completes in a unit interval.

11. Transformations
Find the frequency of the given function, and interpret its
meaning graphically.
Sketch the graph of the function in by
Interpretation:
Frequency:
The graph completes 1 full cycle
per interval of length
Period:

12. 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.

13. 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!!!

14. 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.

15. More whiteboard… Period ; Horizontal shrink 7. by 1/3 9.
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

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

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

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

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

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

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

22. 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:

23. 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:

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: 1 unit down

25. 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

26. Practice Problems
Construct a sinusoid with period and amplitude 6 that goes
through (2,0).
First, solve for b:
Find the amplitude:
Let’s just take the positive value again.
To pass through (2,0), we need a phase
shift of 2  h = –2
Either will work!!!

27. Practice Problems
Construct a sinusoid y = f(x) that rises from a minimum value of
y = 5 at x = 0 to a maximum value of y = 25 at x = 32.
First, sketch a graph of this sinusoid…
Amplitude is half the height:
The period is 64:
We need a function whose minimum is at x = 0. We could shift
the sine function horizontally, but it’s easier to simply reflect the
cosine function……by letting a = –10

28. Practice Problems Support this answer graphically???
Construct a sinusoid y = f(x) that rises from a minimum value of
y = 5 at x = 0 to a maximum value of y = 25 at x = 32.
But since cosine is an
even function:
This function ranges from –10 to 10, but we need a function
that ranges from 5 to 25………vertical translation by 15:
Support this answer
graphically???

29. Practice Problems One possibility:
Construct a sinusoid with the given information.
Amplitude: 2, Period: , Point (0,0)
One possibility:

30. Practice Problems One possibility:
Construct a sinusoid with the given information.
Amplitude: 3.2, Period: , Point (5,0)
One possibility: