Chapter 3 Section 2: General Sinusoidal Graphs

Main Concepts:
Period: the number of degrees per cycle
Frequency: the number of cycles per degree

General Sinusoidal Equation:
y= C+ A cos B(θ-D)

or

y= C+ A sin B(θ-D) where...

·│A│ is the amplitude (A is the vertical dilation, which can be positive or negative)
The amplitude is the distance from the sinusoidal axis to a max. point or a min. point

· B is the reciprocal of the horizontal dilation
When you multiply by θ it dilates by a factor of 1/B

· C is the location of the sinusoidal axis

· D is the phase displacement (horizontal translation)

You can calculate the period from the value of B by:

Period= 360/B

or

B= 360/ period

PROPERTY: Period and Frequency of a Sinusoid:
For the general equations y= C+ A cos B(θ-D) or y= C+ A sin B(θ-D)

period= 1/│B│(360)

and

frequency= 1/period = │B│/(360)

Graphing Sinusoids and finding their equations:

Concave: hollowed out side (holds water)

Convex: bulging side (spills water)

Point of Inflection: occurs where a graph stops being concave one way and starts being concave the other way.

Upper Bound: the line through the "high points" of the graph

Lower Bound: the line through the "low points" of the graph

Critical Points: the "high points" and "low points" of the graph (max. and min.)

Example Problem:

You should be able to determine the amplitude, the sinusoidal axis, the period, and the phase displacement from an equation.

Equation: y= 3+ 2 cos 4 (x-20)

To figure out these values refer to the equation y= C+ A cos B(θ-D)

Amplitude (A)=2

Sinusoidal axis (C)=3

Period (360/B)=90

Phase displacement (D)= 20

Graphed:

New max on graph becomes 20 then add the period to equal the next max at 110. 65 is the middle point between 20 and 110. The sinusoidal axis is at 3. To get the amplitude go up the amplitude (2) up and down from the sinusoidal axis and those points become the max. and min.

Additional resources:
http://www.math.duke.edu/education/ccp/materials/precalc/singraph/contents.html


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