What is the shape of a sine curve?

[phEight]

URGENT: Sinusoid graph problem!

Hello

Here is the problem:

Naturalists find that the poplulations of some kinds of predatory animals vary periodically. Assume that the population of foxes in a certain forest varies sinusoidally with time. Records started being kept when time t = 0. A minimum number, 200 foxes, occurred when t = 2.9 years. The next maximum, 800 foxes, occurred at t = 5.1 years.

a.) Sketch a graph of this sinuoid.
b.) Write an equation expressing the number of foxes as a function of time,
t.
c.) Predict the population when t = 7
d.) Foxes are declared to be an endangered species when their population drops below 300. Between what two non-negative values of t were foxes first endangered?

I do not know how this graph should look like... as in whether it should be sine or cosine, and how I would properly show the minimums and maximums. Any help would be appreciated.

 

[jtbell]

Assume that the population of foxes in a certain forest varies sinusoidally with time. Records started being kept when time t = 0. A minimum number, 200 foxes, occurred when t = 2.9 years. The next maximum, 800 foxes, occurred at t = 5.1 years.

a.) Sketch a graph of this sinuoid.

In a sinusoidal curve, the maxima and minima occur at equal intervals. Your given minimum and maximum are 2.2 years apart. The next minimum occurs 2.2 years after the given maximum, the next maximum 2.2 years after that, etc. You can also work your way backward from the given minimum.

This gives you enough information to plot the maximum and minimum points on a graph, and you can connect them with at least a rough sketch of the curve between them.

 

[BobG]

A cosine curve is the same as the sine curve, except shifted over by \pi / 2 radians. That's why only one term, "sinusoidal" is used.

Two data points is hardly enough to determine the curve (it's rare for the data to match a sine curve perfectly). Since the problem states that you are to "assume that the population of foxes in a certain forest varies sinusoidally with time", they probably mean to use a perfect sine curve.

In a sine curve, how far apart are your max and min? They are \pi radians apart, while the period of a sine curve is 2 \pi radians. That concept allows you to find your period.

It also brings your phase shift into play, since the sine curve won't be at zero at t=0.

In a sine curve, your maximum and minimum should be equidistant from your baseline, or reference. That means you'll need to add in a vertical shift to set your baseline midway between the max and min.

Your amplitude is the difference between the max and your vertical shift - how far above or below the baseline your graph goes.