Key takeaway: The midline of a periodic graph is the horizontal line y = D exactly halfway between its maximum and minimum values, the center line of the wave.
Grade 11 – Algebra 2F-TF.B.5Trigonometry
Definition of Midline
The midline is the invisible axis around which a wave oscillates. Crests sit as far above it as troughs sit below it, so it splits the graph into mirror-sized halves vertically.
In the equation \(y=A\sin(Bx)+D\) or its cosine twin, the constant \(D\) is what lifts or lowers the whole graph, so the line is simply \(y=D\). From a picture you can find it without any formula: add the highest and lowest heights and divide by two. Real data such as average daily temperature or sea level often oscillates around a line like this.
Formula and notation
For a wave with extreme values \(y_{\max}\) and \(y_{\min}\):
\[y=D, \qquad D=\frac{y_{\max}+y_{\min}}{2}.\]
In the form \(y=A\cos(Bx-C)+D\) the line is still \(y=D\); the other constants do not move it.
Key facts
- It is horizontal, so its equation always has the form y = a number.
- The wave crosses this line twice in every cycle.
- Maximum equals D + |A| and minimum equals D − |A|.
- With no vertical shift, D = 0 and the center line is the x-axis.
Midline example
The water level in a canal lock cycles between a low of 3.2 feet and a high of 11.8 feet. Write the midline and then the high and low in terms of the equation \(y=A\sin(Bx)+D\).
-
Average the extremes: \(D=\frac{11.8+3.2}{2}=\frac{15}{2}=7.5\).
-
The distance from there to either extreme is \(11.8-7.5=4.3\), so \(|A|=4.3\).
-
Verify: \(7.5+4.3=11.8\) and \(7.5-4.3=3.2\).
The center line is \(y=7.5\), with \(|A|=4.3\) feet.
Common mistake
Watch out
For \(y=4\sin x+9\), Ravi says the center line is \(y=4\).
Why it is wrong
The number multiplying the sine is the height of the swing, not the position of the line. The vertical shift is the added constant, so the line is \(y=9\). The wave then oscillates between 5 and 13.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 2 vocabulary · Trigonometry