Stephen, I started this question by drawing simple wheel diagram. The center is at 38 cm, which is half the diameter of 76 cm. Since the center is 38 cm off the ground, and the reflector is 15 cm off the ground, the reflector is 38-15=23 cm. from the center of the wheel.
We can use sine or cosine to graph this; both work. I'll be referring to A,B,C, and D throughout this, as they are the coefficients and constant in: y=A cos (Bx+C)+D.
I chose cosine because it doesn't start at 0, so I could just make A negative to have it start with negative y values. Either one will need a horizontal translation, or "phase shift".
A=23 because that's how far the reflector goes up and down from the center, or midline of the function. D is the midline, so D=38 is the location of the wheel's center.
The period is how often a trig graph repeats a full cycle. Normally, a sine or cosine graph goes through one full cycle in 2π radians, or 360 degrees. The formula for changing the period of a function is simple: Period = 2π/B. This bike wheel rotates fully every .75 seconds. (or every 3/4 seconds)
Setting up my period formula, I get:
3/4 = 2π/B. Cross multiply to get 3B = 8π. Divide by 3 to get B for this function: B=8π/3.
We've got the right amplitude, period, and vertical shift. Now we need the horizontal translation, or "phase shift". The formula for phase shift is -C/B. Because the function needs a minimum value at t =.5, (remember t is the variable used for x here), we need to shift the function right 0.5= 1/2. So -C/B = 1/2. Since B=8π/3,
-C/(8π/3)=1/2. Solve for C to get C=-4π/3.
Plug A,B,C, and D into our original equation to get
f(t) = -23 cos ((8π/3)x + 4π/3) +38
I hope this helps, and have fun mathing!
Liz Z
