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Midline, amplitude, and period review

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.
A period of a trigonometric wave. A horizontal dashed line extends through the middle of the trigonometric wave and is labeled the midline. The distance from the midline to the highest point of the wave is the same distance as the lowest point is from the midline. The distance from the midline to the maximum or from the midline to the minimum is called the amplitude. The distance from the point on the midline before the maximum point and the point on the midline after the minimum point is labeled the period.
Midline is the horizontal line that passes exactly in the middle between the graph's maximum and minimum points.
Amplitude is the vertical distance between the midline and one of the extremum points.
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.
A graph of a trigonometric wave on an x y coordinate plane. The x and y axes scale by one. The midline is at y equals five. The highest points on the graph go up to seven on the y axis and the lowest points on the graph go to three on the y axis. A period spans an interval of four units on the x axis. No points are labeled.
It has a maximum point at (1,7), then a minimum point at (3,3), then another maximum point at (5,7).
A graph of a trigonometric wave on an x y coordinate plane. The x and y axes scale by one. The midline is at y equals five. The highest points on the graph go up to seven on the y axis and the lowest points on the graph go to three on the y axis. A period spans an interval of four units on the x axis. Maximum points are at (one, seven) and (five, seven). The minimum point between them is labeled (three, three).
The horizontal line that passes exactly between y=7 (the maximum value) and y=3 (the minimum value) is y=5, so that's the midline.
A graph of a trigonometric wave on an x y coordinate plane. The x and y axes scale by one. The midline is a dashed line at y equals five. The highest points on the graph go up to seven on the y axis and the lowest points on the graph go to three on the y axis. A period spans an interval of four units on the x axis. Maximum points are at (one, seven) and (five, seven). The minimum point between them is labeled (three, three).
The vertical distance between the midline and any of the extremum points is 2, so that's the amplitude.
A graph of a trigonometric wave on an x y coordinate plane. The x and y axes scale by one. The midline is a dashed line at y equals five. The highest points on the graph go up to seven on the y axis and the lowest points on the graph go to three on the y axis. A period spans an interval of four units on the x axis. Maximum points are at (one, seven) and (five, seven). A vertical dashed line connects from each maximum point to the midline to show the amplitude. The minimum point between them is labeled (three, three). A vertical dashed line connects from the minumum point to the midline to show the amplitude as well.
The distance between the two consecutive maximum points is 4, so that's the period.
A graph of a trigonometric wave on an x y coordinate plane. The x and y axes scale by one. The midline is a dashed line at y equals five. The highest points on the graph go up to seven on the y axis and the lowest points on the graph go to three on the y axis. A period spans an interval of four units on the x axis. Maximum points are at (one, seven) and (five, seven). A bracket shows the distance between the two maximum points. A vertical dashed line connects from each maximum point to the midline to show the amplitude. The minimum point between them is labeled (three, three). A vertical dashed line connects from the minumum point to the midline to show the amplitude as well.
Problem 1
What is the midline equation?
y=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

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

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