Two equations to describe relationship

Question

I have a question, so like I made a sinusoidal relationship, including 2 periods (using a bicycle wheel).

My table of values for the relationship is shown below:

So my question is : What can be the two equations that describe this relationship, one equation using sin() and one equation using cos() ?

x 0 15 30 45 60 75 90 105 120

f(x) 55 35 10 35 55 35 10 35 55

Answer 1

Hello Luhan,

First, to make a simple sine or cosine function, you need to change 35 to the number midway between 55 and 10 which is (55 + 10)/2 = 32.5. So change all your 35's to 32.5's.

The period is how long it takes for the pattern to repeat. How long does it take to go from 55 back to 55?

60 time units. The sine and cosine functions will have (2pi/60) times the angle.

The amplitude is the distance between the highest and lowest y-values divided by 2. So that would be (55-10)/2 - 22.5. That goes in front of your sine or cosine function. So now we could have y = 22.5 cos (2pi/60)t.

This graph would go from -22.5 to 22.5, but we want to go from 10 to 55. How do we do that? We add a number to everything (shift the graph up). What would make 22.5 increase to 55? Compute 55 - 22.5 = 32.5. So add 32.5. Also notice that -22.5 + 32.5 = 10 so the -22.5 would shift up to 10 just as want it to do. Also simplifying the 2pi/60, we have y = 22.5 cos(pi/30)t + 32.5

For sine, you would use something like y = sin(pi/30)t + 32.5, but you need to shift it to the left so that it starts at the high point (55 instead of the middle y-value as sine normally does). This is called a phase shift. You probably have a formula for this.

In the equation y = Asin(B(x + C)) + D, the horizontal shift is C to the left.

The high point of sine comes after 1/4 of a period which is pi/2 for the basic sine graph. For our data, 1/4 of a period is 1/4 of 60 = 15 so the C = 15. We want to shift the sine graph 15 to the left.

y = 22.5 sin[(pi/30)(t + 15)] + 32.5 would be our final equation.

But these answers will only work if you change the 35's to 32.5's.