5.6. Phase Shift of Sinusoidal Functions http://www.ck12.org
Find an equation that predicts the height based on the time. Choose whent=0 carefully.
- Use the equation from Guided Practice #1 to predict the height of the tide at 6:05 AM.
- Use the equation from Guided Practice #1 to find out when the tide will be at exactly 8 ft on September 19th.
Answers: - There are two logical places to sett=0. The first is at midnight the night before and the second is at 10:15
AM. The first option illustrates a phase shift that is the focus of this concept, but the second option produces a
simpler equation. Sett=0 to be at midnight and choose units to be in minutes.
TABLE5.10:
Time (hours : minutes) Time (minutes) Tide (feet)
10:15 615 9
16:15 975 1
22:15 1335 9
615 + 2975 = 795 513352 + (^975) = 1155 5
These numbers seem to indicate a positive cosine curve. The amplitude is four and the vertical shift is 5. The
horizontal shift is 615 and the period is 720.
720 =^2 bπ→b= 360 π
Thus one equation is:
f(x) = 4 ·cos( 360 π (x− 615 ))+ 5
- The height at 6:05 AM or 365 minutes is:f( 365 )≈ 2 .7057 feet.
- This problem is slightly different from question 2 because instead of givingxand using the equation to find they,
this problem gives theyand asks you to find thex. Later you will learn how to solve this algebraically, but for now
use the power of the intersect button on your calculator to intersect the function with the liney=8. Remember to
find all thexvalues between 0 and 1440 to account for the entire 24 hours.
There are four times within the 24 hours when the height is exactly 8 feet. You can convert these times to hours and
minutes if you prefer.
t≈532.18 (8:52), 697.82 (11:34), 1252.18 (20 : 52), 1417.82 (23:38)