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## Algebra 2

### Unit 11: Lesson 6

Amplitude, midline and period

# 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.
start color #ed5fa6, start text, M, i, d, l, i, n, e, end text, end color #ed5fa6 is the horizontal line that passes exactly in the middle between the graph's maximum and minimum points.
start color #1fab54, start text, A, m, p, l, i, t, u, d, e, end text, end color #1fab54 is the vertical distance between the midline and one of the extremum points.
start color #aa87ff, start text, P, e, r, i, o, d, end text, end color #aa87ff is the distance between two consecutive maximum points, or two consecutive minimum points (these distances must be equal).

## 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 left parenthesis, 1, comma, 7, right parenthesis, then a minimum point at left parenthesis, 3, comma, 3, right parenthesis, then another maximum point at left parenthesis, 5, comma, 7, right parenthesis.
The horizontal line that passes exactly between y, equals, 7 (the maximum value) and y, equals, 3 (the minimum value) is start color #ed5fa6, y, equals, 5, end color #ed5fa6, so that's the midline.
The vertical distance between the midline and any of the extremum points is start color #1fab54, 2, end color #1fab54, so that's the amplitude.
The distance between the two consecutive maximum points is start color #aa87ff, 4, end color #aa87ff, so that's the period.
Problem 1
What is the midline equation?
y, equals

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

## Want to join the conversation?

• How to find tan or cot period?
• Same method as sin or cos except substitute pi for 2pi.

While Period of sin(Cx) = 2pi/C

Period of tan(Cx) = pi/C
Period of cot(Cx) = pi/C

Period of tan() and cot() occurs twice as frequently as sin() cos() because tan() is slope and when you travel halfway (pi radians) around the unit circle, you encounter another point on the same line (same slope).
• How do you find out the midline from just the equation (without the graph)?
• For f(x)=asin(bx+c)+d, d is the midline. Simple as that.
• How do you add all of parts (Midline, Amplitude, Period) to create the Final/ full equation?
• How do I find the equation of a sinusoidal wave given the frequency and amplitude of the wave?
• Awesome question! To write a sine function you simply need to use the following equation: f(x) = asin(bx + c) + d, where a is the amplitude, b is the period (you can find the period by dividing the absolute value b by 2pi; in your case, I believe the frequency and period are the same), c is the phase shift (or the shift along the x-axis), and d is the vertical shift (this value also represents the midline of the function). You can determine the vertical shift by dividing the maximum value plus the minimum value of the graph by 2. The period can be determined using the 2pi/|b| expression,but you might not always have to write it that way in the equation; I'm not quite sure how you would determine the phase shift given only the amplitude and frequency of the wave, but if you could graph the equation of the parent function and the function you have thus far, you might be able to determine the phase shift from that. Doing that will get you a solid equation. I hope this helps!
• what if it's a polynomial?
• that manipulated equation will help understand the graph
• How do you find the midline when you are only given one point (min or max) and the aplidtude?
• The amplitude is the vertical distance from the midline to the min or max. So if you are given the minimum, add the amplitude to the y-coordinate. If you are given the maximum, subtract the amplitude from the y-coordinate.
• how do you find the value of b in the equation?
• b represents the "y-intercept" which is the y-value of the function when it it touches the y-axis (in other words, when it has a x-value of zero). For example, the graph given as an example has a y-intercept of 5 as the graph touches the y-axis when its y value is 5.
• after determining the amplitude and period of a function, how do you determine the Sine/Cosine formulae
for them in a way that avoids the need to include a phase shift?
• I would say most of the time a phase shift is unavoidable. I think the simplest way is to just use what the proble tells you. if it describes the midline I would use sine and if it gives a max or min cosine. If it gives both use whichever comes first.

That is with word problems anyway, if you are given a picture then it's a little easier. If there is a point on the midline or a max or min that touches the y axis, then you use sine for a midline point and cosine for a max or min. In these cases then there will not be a phase shift. If there is no nice point that touches the y axis then there is definitely a phase shift. the easiest is to probably pick the closest "nice" point to the y axis then.

Does this make sense?
• I always seem to have a problem with finding the periodic function.. I just need more help with understanding what is plugged in as "K" in the formula 2pi/K?
y=1/2*tan theta-3