The general forms 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.
The amplitude and the vertical shift of the model are smaller.
Practice makes perfect
We know that a rope oscillates between 5 inches and 70 inches above the ground. We want to write a model for the height l of the rope 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 the minimum values.
k=maximum value+ minimum value/2We know that the maximum value is 70 and the minimum value is 5.
k=70+ 5/2
Let's evaluate this formula to find the value of k.
Since the minimum value 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
Since the rope makes 2 revolutions per second, we know that it makes a complete revolution every 0.5 seconds. This means that the period is 0.5. 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.
0.5= 2Ď€/b
Let's solve this equation!
Now, let's find the amplitude. This is half the difference between the maximum and the minimum values.
| a| =maximum value- minimum value/2
In our case, the maximum value is 70 and the minimum value is 5.
| a| = 70- 5/2
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= - 32.5.
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= - 32.5, b= 4Ď€, l= 0, and k= 37.5. With this information, we can write the equation of the model.
h(t) = - 32.5 cos 4Ď€ (t - 0)+ 37.5
⇕
h(t)= - 32.5 cos 4Ď€t + 37.5
Finally, we can compare it with the model given in the exercise.
Original Model
h(t) = - 36 cos 4Ď€t + 39 [0.8em]
New Model
h(t) = - 32.5 cos 4Ď€t + 37.5
Minimum and maximum values changed from 3 and 75 inches to 5 and 70 inches, which means that they are closer to each other. This makes the amplitude smaller. We can observe it in the equation of the model, the amplitude | a| = 32.5 in the new model is less than 36. Also, since the sum of minimum and maximum values decreased, the vertical shift k became smaller.
