Math Analysis--Trig. Applications Notes Name ______
Writing Trig. Equations from Periodic Behavior
Example 1: Breathing Cycles
Our normal breathing cycle takes place every 5 seconds and is measured in liters per second. The velocity of our breath is positive as we inhale and negative as we exhale. If y represents our velocity of air flow after x seconds, find a function that models air flow in a normal breathing cycle assuming our inhale and exhale velocity is approximately 0.6 liters per second each.
1.) Sketch a curve that represents this cycle.
Ask, would a sine or cosine curve best represent this cycle? A sine curve is best in this situation. Sketch a sine curve below and label the information given:
2.) Use the model to write the equation:
a.) Find the amplitude:______Is there a flip in the sine curve? _____ A= ______
b.) Find the period: One cycle is ____ seconds. So, . Solving for B we get: B = ______
c.) Is there a phase shift? _____ Is there a vertical shift? ______
d.) Write the equation:
You can check by graphing this on your calculator! (switch to radians first!)
Example 2: Tidal Cycles
The depth of water at a boat dock varies with the tides. The depth is 5 feet at low tide and 13 feet at high tide. On a certain day, the low tide occurs at 4 A.M. and high tide at 10 A.M. If y represents the depth of the water, in feet, x hours after midnight, write a function that models the water’s depth.
1.) Sketch a curve that represents this cycle.
Ask, would a sine or cosine curve best represent this cycle? We can use either curve (the amount of phase shift is the only difference) but let’s use cosine since low tide occurs first. Sketch a cosine curve below and label the information given: (Watch! The curve does not start at 0, it starts at 4.)
2.) Use the model to write the equation:
a.) Find the vertical shift:Add the low tide depth and the high tide depth, then divide by 2: (_____(lt)+____(ht)/2 = ______So, our D value= ____ This is also our new equilibrium line.
b.) Find the amplitude: How much is above and below the equilibrium line along the y-axis of the curve? ______This is our amplitude. Is there a flip in the curve? _____
So, our A value = ______
b.) Find the period: One cycle is ____ hours. So, . Solving for B we get:
B value = ______
c.) Is there a phase shift? _____ (If the curve of a complete cycle starts somewhere other than 0, there will be a phase shift.) How far away from 0 along the x-axis does the curve start? The curve starts at an x value of _____. So, =_____ Plug in the B value we found to find C. C value = ______.
d.) Write the equation: Check on the calculator!
You try!
A region that is 30 degrees north of the Equator averages a minimum of 10 hours of daylight in December. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year (1=January, 2=February, etc.). If y represents the number of hours of daylight in month x, write a function to model the hours of daylight.
1.) Sketch a curve that represents this cycle. Would a sine or cosine curve best represent this cycle?
2.) Determine A, B, C, and D values of the model:
a.) D Value (vertical shift) = ______
b.) A Value (amplitude+flip?) = ______
c.) B Value (use ) = ______
d.) C Value (use) = ______
e.) Write the final equation: ______