5.5. Frequency and Period of Sinusoidal Functions http://www.ck12.org
The ability to measure the period of a function in multiple ways allows different equations to model an identical
graph. In the image above, the top red line would represent a regular cosine wave. The center red line would
represent a regular sine wave with a horizontal shift. The bottom red line would represent a negative cosine wave
with a horizontal shift. This flexibility in perspective means that many of the examples, guided practice and practice
problems may have multiple solutions. For now, try to always choose the function that has a period starting atx=0.
Frequency is a different way of measuring horizontal stretch. For sound, frequency is known as pitch. With
sinusoidal functions, frequency is the number of cycles that occur in 2π. A shorter period means more cycles can
fit in 2πand thus a higher frequency. Period and frequency are inversely related by the equation:
period=frequency^2 π
The equation of a basic sine function isf(x) =sinx. In this caseb, the frequency, is equal to 1 which means one
cycle occurs in 2π. This relationship is a common mistake in graphing sinusoidal functions. Students findb=^12 and
then mistakenly conclude that the period is^12 when it is in fact stretched to 4π.
Example A
Rank each wave by period from shortest to longest.
Solution:
The red wave has the shortest period.
The green and black waves have equal periods. A common mistake is to see that the green wave has greater
amplitude and confuse that with greater periods.
The blue wave has the longest period.
Example B
Identify the amplitude, vertical shift, period and frequency of the following function. Then graph the function.
f(x) =2 sin(x 3 )+ 1
Solution: a= 2 ,b=^13 ,d=1. Sinceb=^13 (frequency), then the period must be 6π.