5.5. Frequency and Period of Sinusoidal Functions http://www.ck12.org
The amplitude is 6 soa=6. There is no vertical reflection. Since the period is 12 you can determine the frequency
b:
12 =^2 bπ→b=π 6
The vertical shift is 12 sod=12. Thus you have all the pieces to make an algebraic model:
f(x) = + 6 ·cos(π 6 x)+ 12
Concept Problem Revisited
If a sine graph is horizontally stretched by a factor of 3 then the general equation hasb=^13. This is becausebis the
frequency and counts the number (or fraction) of a period that fits in a normal period of 2π. Graphically, the sine
wave will make a complete cycle in 6πunits like Example B.
Vocabulary
Periodis the distance it takes for a repeating function to make one complete cycle.
Frequencyis the number of cycles a function makes in a set amount of time or distance on thexaxis. For sine and
cosine graphs this distance is 2π.
Guided Practice
- A fish is caught in a water wheel by the side of a river. Initially the fish is 2 feet below the surface of the
water. Twenty seconds later the fish is 14 feet in the air at the top of the water wheel. Model the fish’s height in a
graph and an equation. - Graph the following function:g(x) =−cos( 8 x)+2.
- Given the following graph, identify the amplitude, period, and frequency and create an algebraic model.