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Midline — Definition, Formula & Examples

Midline is the horizontal line halfway between the maximum and minimum values of a periodic function. It represents the vertical center of the wave, acting as the baseline around which the graph oscillates.

For a sinusoidal function of the form y=asin⁡(bx+c)+dy = a\sin(bx + c) + d or y=acos⁡(bx+c)+dy = a\cos(bx + c) + d, the midline is the horizontal line y=dy = d, equal to the average of the function's maximum value (d+∣a∣d + |a|) and minimum value (d−∣a∣d - |a|).

Key Formula

midline=y=d=ymax+ymin2\text{midline} = y = d = \frac{y_{\text{max}} + y_{\text{min}}}{2}
Where:
  • dd = Vertical shift (the constant added outside the trig function)
  • ymaxy_{\text{max}} = Maximum output value of the function
  • yminy_{\text{min}} = Minimum output value of the function

How It Works

To find the midline from an equation like y=3sin⁡(2x)+5y = 3\sin(2x) + 5, identify the vertical shift dd. Here d=5d = 5, so the midline is y=5y = 5. To find the midline from a graph, locate the highest and lowest points and average them: midline =max+min2= \frac{\text{max} + \text{min}}{2}. The graph oscillates equally above and below this line, with the distance from the midline to either extreme equal to the amplitude.

Worked Example

Problem: A sinusoidal function has a maximum value of 8 and a minimum value of 2. Find the midline and the amplitude.
Find the midline: Average the maximum and minimum values.
midline=8+22=102=5\text{midline} = \frac{8 + 2}{2} = \frac{10}{2} = 5
Find the amplitude: The amplitude is the distance from the midline to the maximum.
amplitude=8−5=3\text{amplitude} = 8 - 5 = 3
State the midline: The midline is the horizontal line at y=5y = 5, and the function oscillates 3 units above and below it.
y=5y = 5
Answer: The midline is y=5y = 5 and the amplitude is 3.

Why It Matters

The midline appears on the SAT and in Algebra 2 / Precalculus whenever you write or interpret sinusoidal models. In real-world applications like modeling tides or temperatures, the midline represents the average value — for instance, the average sea level around which tides rise and fall.

Common Mistakes

Mistake: Confusing the midline with the x-axis (y=0y = 0) when there is a vertical shift.
Correction: The midline is y=dy = d, not y=0y = 0. Always check for a constant term added to the trig function, or average the max and min from the graph.

Related Terms