![y= 19 cos [\frac{\pi}{5} (x-4)]+22](/tpl/images/0435/0789/71c43.png)
![y=19 sin[\frac{\pi}{5} (x-1.5)]+22](/tpl/images/0435/0789/b2769.png)
Explanation:
Assuming this problem: "The wheel is 3ft off of the ground and the diameter of it is 38ft. It takes you 4 seconds to reach the top. The wheel makes a revolution every 10 seconds. Draw a graph and write a funtion."
From the info given we know that the ferris wheel is already off the ground 3 feet when the person get on it.
We know that the diameter of the ferris wheel is D=38 feet.
Based on this the maximum point would be 38+3 = 41 ft
Based on this we have the maximum and the minimum point for the function and would be 41 the maximum and 3 the minimum. And we want to adjust a sinusoidal function.
We can find the amplitude like this:
And the moddle point for the fucntion would be 3+19 =22 ft or 41-19=22ft.
Now we need to take in count the info provided: We need 4 seconds to reach the top, so then the the point for the maximum would be(x=4,y=41).
The other info provided is that "it takes 10 seconds to reach the maximum". So then the next highest point would be (14,41).
In order to get the low point of the ferris wheel, we need to divide the distance from the first high point from the second high point by 2. Since we have 10 seconds, and we have a sinuosoidal function then the low point will be 5 seconds after the first point (x=4+5=9,y=3)
We can find also the midpoints between the maximum and the minimum. There is 5 seconds between two consecutive highest points, and if we divide that also by 2 we get the mid points. 5/2 is 2.5, and 4+2.5=6.5 and 9+2.5= 11.5. So then the middle points are (6.5, 22) and (11.5,22).
Now we need to create the equation based on the info given.
From the previous work we have some parameters:
Amplitude =19 ft
Period =10 sec
Sinc we are constructing a sinusoidal function then we can use a cosine or a sine equation. And we have two equivalent expressions:
And the plot obtained is attached on the figure.
