Finding a Formula for the Ferris Wheel Function
Name_________________________
Over the course of a year, the length of the day—that is, the number of hours of daylight—changes
every day. Here is a table giving the length of day, rounded to the nearest tenth of an hour, for Boston.
Note: January 1st corresponds to t = 0.
Day
0 10 20 30 40
50
60
70
80
90 100 110 120 130 140 150 160 170 180
Daylight Hours 9.1 9.3 9.6 9.9 10.3 10.8 11.2 11.7 12.2 12.7 13.1 13.6 14 14.4 14.8 15.2 15.3 15.3 15.3
Day
190 200 210 220 230 240 250 260 270 280 290 300 310 320
Daylight Hours 15.1 14.9 14.6 14.2 13.8 13.3 12.9 12.4 11.9 11.5 11 10.6 10.1 9.8
330
9.5
340
9.2
350 360
9.1 9.1
1.
Plot these points on the graph paper provided. Connect the points using appropriate concavity.
2.
What term could be used to describe the shape of the graph?
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3.
Approximate the equation of the midline.
Sketch this as a dotted line on your graph.
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4.
What is the period?
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5.
What is the amplitude?
6.
For what values of t does the graph appear to
intersect the midline?
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7.
What are these special points called?
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8.
Over what intervals does the function appear to be increasing?
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9.
Over what intervals does the function appear to be decreasing?
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10.
Over what intervals is the function concave up?
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11.
Over what intervals is the function concave down?
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12.
Which of the special values in problem #6 occurs when the
function is increasing?
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Now, we are going to graph the function on your calculator. To do this, we need an equation. We
will use the model
f ( x)  A sin B( x  C )  D
where
A = amplitude
2
B=
period
C = phase shift
D = midline
13.
Calculate the B value by dividing 2 by the period.
Leave your answer in terms of .
14.
Write the equation using the model above. You found the value
of C in problem #12 for the sine equation.
15.
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Graph this equation on your calculator using an appropriate window.
Make sure you are in radian mode.
Now we can answer some questions about the daylight hours in Boston.
16.
17.
18.
Approximate the number of daylight hours in Boston
on May 15th.
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When does the longest day of the year occur?
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the shortest?
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On what date(s) will the length of daylight be 13.7 hours?
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Now here’s a couple more to practice!
Find the amplitude, midline, period, and phase shift. Then, write the equation using sine. Check your
solution using your calculator.
1.
2.