The general form of transformed sine and cosine functions are y=asin b(x-l)+k and y=a cos b(x-l)+k. Determine whether the graph is modeled by a sine or a cosine function, and then find the values of a, b, l, and k.
h = - 36.25 cos (Ď€/12 t) + 34.25
Practice makes perfect
We are told that the highest point of a bucket on the wheel is 70.5 feet above the viewing platform and the lowest point is - 2. We want to write a model for the height of the bucket as a function of time t. To do so, we will consider the general forms of transformed sine and cosine functions.
Sine Function:& y= asin b(x- l)+ k
Cosine Function:& y= acos b(x- l)+ k
In these functions | a| is the amplitude, 2Ď€ b is the period ( b>0), l is the horizontal shift, and k is the vertical shift. To write a function of a sinusoid, we will follow four steps.
Determine whether the model should be modeled by a sine or a cosine function, and identify any horizontal shift l.
Find the period and the value of b.
Find the amplitude and the value of a.
Let's follow these steps one at a time!
Step 1
The vertical shift k is the mean of the maximum and minimum values.
k=maximum value+ minimum value/2
We know that the maximum value is 70.5 and the minimum value is - 2.
k=70.5+( -2)/2
Let's solve this formula to find k.
Since the minimum point occurs at t=0, the function passes through y-axis at its minimum point. Then, the height can be modeled as a cosine function reflected across the x-axis and with no horizontal shift. This means that l= 0.
Step 3
The wheel makes a complete turn every 24 seconds. This means that the period is 24. Remember that the formula for a period of sine and cosine functions is 2Ď€ b. With this information, we can write an equation in terms of b.
24=2Ď€/b
Let's solve this equation!
Next, let's find the amplitude | a| of the graph. This is half the difference between the maximum and minimum values.
| a| =maximum value- minimum value/2
In our case, the maximum value is 70.5 and minimum value is - 2.
| a|=70.5-( -2)/2
We will evaluate this formula to find the amplitude.
Since the graph is a reflection in the x-axis of cosine function, we know that a<0. Therefore, a= - 36.25.
Equation of the Model
Because of the location of the minimum point, we concluded that the height should be modeled by a cosine function. We also found that a= - 36.25, b= π12, l= 0, and k= 34.25. Finally, we can write the equation of the model.
h = - 36.25 cos π/12( t - 0)+ 34.25
⇕
h=- 36.25 cos π/12 t + 34.25
