Midline

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Midline: meaning, method and solved example (Algebra 2) – Maths PDF

In short: The midline is the horizontal line that runs through the exact middle of a periodic graph, halfway between its maximum and minimum, with equation y = (max + min)/2.

Algebra 2TrigonometryF-TF.B.5

What is Midline?

The midline is the resting level of a wave. Picture a buoy bobbing on the sea: it rises and falls, but its average height is the surface when the water is calm, and that level is the resting line.

For \(y = a\sin(bx + c) + d\) or the cosine version, it is the horizontal line \(y = d\). From data, \(d = \frac{\text{maximum} + \text{minimum}}{2}\). The curve is symmetric around it, and the amplitude is the distance from this line to a peak.

How to spot it

  • A wave equation ends with a constant added or subtracted, like + 7 or – 5, outside the trig function.
  • A graph oscillates and the question asks for the horizontal line around which it is centered.
  • Daily high and low values, such as 78°F and 52°F, are given and a typical level is wanted.
  • You need to write a sinusoidal model from a table of maximum and minimum readings.

The method

  1. If the equation is given, read the constant \(d\) outside the trig function; the center line is \(y = d\).
  2. If the graph or data is given, add the maximum and minimum values.
  3. Divide that sum by 2 to get the height of the center line.
  4. Draw it as a dashed horizontal line and check that the peaks are equally far above and below.
13°13°high 78 (noon)low 52 (midnight)midline y = 65hours 0 to 24
Midline = (78 + 52)/2 = 65, with 13° of swing on each side.

Midline: solved example

Problem

A town’s temperature over one day peaks at \(78^{\circ}\text{F}\) and bottoms out at \(52^{\circ}\text{F}\). Find the midline and the amplitude of a sinusoidal model.

Solution

The center line is the average of the extremes: \(\frac{78 + 52}{2} = 65\), so the line is \(y = 65\).

The amplitude is \(\frac{78 – 52}{2} = 13\). The model oscillates 13 degrees around \(y = 65\).

Try it

Question 1

Name the midline of \(y = 3\cos(2x) + 8\).

Show the answer

The constant added is 8, so the line is \(y = 8\).

Question 2

A wave has maximum 3.5 and minimum -1.5. Find its midline.

Show the answer

\(\frac{3.5 + (-1.5)}{2} = 1\), so \(y = 1\).

Midline vs. Amplitude

This line is a location: a horizontal line at a certain height. Amplitude is a distance: how far the wave leaves that line. Two waves can share the same center yet have different amplitudes, and two waves with equal amplitudes can be centered at different heights.

Amplitude

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