MATHEMATICS 11 ASSIGNMENT

Sinusoidal Functions

1. In your own words, define the following 4 terms: Amplitude, Sinusoidal Axis, Period, Phase Shift. Use a diagram (or several diagrams) to illustrate your definitions.

Amplitude ______
______
Sinusoidal Axis ______
______
Period ______
______
Phase Shift ______
______

2. We often represent equations in transformational form because we have developed, through repetitive practice, a systematic way to identify transformations from equations written in this form. What would be one (or two?) advantage(s) of having an equation written in standard form?

______

3. The following transformations are occurring on the graph of y = sin x

Vertical Translation = 4

Horizontal Translation = 90o

Vertical Stretch = 5

Horizontal Stretch = 2

a)  write a mapping rule
b)  What is the equation of the sinusoidal axis?
c)  What is the period?
d)  What is the amplitude?
e)  Write the equation in transformational form.
f)  Write the equation in standard form / g)  Graph the function:
.
From Equation to Graph …
Method 1: Equation à Mapping Rule à Find Key Points (using mapping rule) à Graph
Method 2: Equation à Identify Transformations (optional) à Identify Sinusoidal Axis, Phase Shift, Period & Amplitude (& possible reflection (Rx) ? à Graph ***
1. Draw in Sinusoidal Axis (use dotted line)
2. Put in “starting point” based on phase shift (this will be on sinusoidal axis for y=sin x and at max/min for y=cos x)
3. Draw another point exactly one period later at the same height (same y-value).
4. Use symmetry to draw in rest of points taking into account the amplitude and possibly a reflection (depending on where you chose to “start”).

Graph the following 3 equations: [Note: For full credit you MUST show each of the steps identified above. The steps shown will, obviously, be different for each Method.]

USE METHOD 1

.
/ USE METHOD 2

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USE ANY METHOD
(1 OR 2 – NOT with Graphing Tech!! J)
. / .

From Graph to Equation…
From Graph
1.  Identify Sinusoidal Axis, Phase Shift, Amplitude & Period.
2.  Write these as transformations of y=sin x or y=cos x (i.e. Write as VT/HT/VS/HS)
·  VT: This IS the value of your sinusoidal axis.
·  HT: This is your phase shift or “starting point” – for y=sin x, it is on the sinusoidal axis. For y=cos x, it is a max/min point. YOU choose J. Consider reflections…
·  VS: This IS your amplitude EXCEPT you have to consider reflections (Rx).
For y=sin x à If graph goes up from starting point, no reflection. Otherwise, Rx.
For y=cos x à If graph starts at max point, no reflection. If starts at min, Rx.
·  HS = period/360o
3.  Use transformations to write equation.
NOTES: We have been writing them in Transformational Form and changing them to Standard Form as needed. While the transformations all show up in the equation, they do not LOOK exactly the same. See notes & practice to help you complete this process.

Write an equation for each of the following graphs.

(You may use either a sine or a cosine equation.)

a) b)


WORD PROBLEMS

(Must include a graph/sketch and show all work for full value)

1

2.