5.3. Amplitude of Sinusoidal Functions http://www.ck12.org
Concept Problem Revisited
The most common mistake is doubling or halving the amplitude unnecessarily. Many people forget that the number
ain the equation, like the 3 inf(x) =3 sinx, is the distance from thexaxis to both the peak and the valley. It is not
the total distance from the peak to the valley.
Vocabulary
Theamplitudeof the sine or a cosine function is the shortest vertical distance between the sinusoidal axis and the
maximum or minimum value.
Thesinusoidal axisis the neutral horizontal line that lies between the crests and the troughs of the graph of the
function.
Guided Practice
- Identify the amplitudes of the following four functions:
- Graph the following function:f(x) =−8 sinx.
- Find the amplitude of the functionf(x) =−3 cosxand use the language of transformations to describe how the
graph is related to the parent functiony=cosx.
Answers: - The red function has amplitude 3. The blue function has amplitude 2. The green function has amplitude^12.
- First identify where the function starts and ends. In this case, one cycle (period) is from 0 to 2π. Usually sine
functions start at the sinusoidal axis and have one bump up and then one bump down, but the negative sign swaps
directions. Lastly, instead of going up and down only one unit, this function has amplitude of 8. Thus the pattern
is:
Starts at height 0
Then down to -8.
Then back to 0.
Then up to 8
Then back to 0.