Review the basic features of sinusoidal functions: midline, amplitude, and period.

What are midline, amplitude, and period?

Midline, amplitude, and period are three features of sinusoidal graphs.
Midline\maroonC{\text{Midline}} is the horizontal line that passes exactly in the middle between the graph's maximum and minimum points.
Amplitude\greenD{\text{Amplitude}} is the vertical distance between the midline and one of the extremum points.
Period\purpleC{\text{Period}} is the distance between two consecutive maximum points, or two consecutive minimum points (these distances must be equal).
Want to learn more about midline, amplitude, and period? Check out this video.

Finding features from graph

Given the graph of a sinusoidal function, we can analyze it to find the midline, amplitude, and period. Consider, for example, the following graph.
It has a maximum point at (1,7)(1,7), then a minimum point at (3,3)(3,3), then another maximum point at (5,7)(5,7).
The horizontal line that passes exactly between y=7y=7 (the maximum value) and y=3y=3 (the minimum value) is y=5\maroonC{y=5}, so that's the midline.
The vertical distance between the midline and any of the extremum points is 2\greenD2, so that's the amplitude.
The distance between the two consecutive maximum points is 4\purpleC4, so that's the period.
Problem 1
What is the midline equation?
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Want to try more problems like this? Check out these exercises: