We are given a table that shows the water temperature D each month t at Miami Beach. We want to write a model that gives us the average monthly high temperature D as a of the time t.
| Water Temperatures at Miami Beach, FL
|
| t
|
1
|
2
|
3
|
4
|
5
|
6
|
| D
|
71
|
73
|
75
|
78
|
81
|
85
|
| t
|
7
|
8
|
9
|
10
|
11
|
12
|
| D
|
86
|
85
|
84
|
81
|
76
|
73
|
To do so, we will use a graphing calculator to create the function model. Then, we can find and interpret the of the function. Let's do this one at time!
Creating the Function Model
Let's begin by drawing a . We will do it by pushing the STAT button and choosing the first option Edit.
Then we can enter the values. The first column represents the months and the second one represents the temperature.
Having entered the values, we can plot them by pushing the 2nd button and the Y= button. Then, we will choose one of the plot types from the list. Let's turn the plot ON, choose the type to be a scatter plot, and assign L1
and L2
as XList
and Ylist.
Now let's go to the ZOOM button. Here, we will choose the ninth option ZoomStat.
The scatter plot of the data appears.
Looking at the plot, the curve appears . Therefore, we can perform a sinusoidal regression. To do so, we need to push the STAT button. Later, we will go to CALC
and choose option C, SinReg.
Before we perform the regression, we will store the function so that we can draw it later. Let's go to the fifth row, StoreRegEQ,
and press the VARS button. After that, we will go to Y-VARS,
choose the first option, Function,
and choose one of the functions to store our equation.
Now we can perform the regression.
As we drew the scatter plot, we can also draw the function.
The model seems to be a good approximation. Before we summarize our findings, let's round the SinReg
values to the nearest hundredth.
D=7.38sin(0.497t-2.05)+78.64
Finding and Interpreting the Period
Since the sinusoidal regression found a model of the form y= a sin ( bx + c) + d, we can factor out b so that this model matches the . This will help us find the period of the function.
|
|
General Form
|
Our Function
|
| Sinusoidal Regression Form
|
y= a sin ( bx + c) + d
|
D= 7.38sin( 0.497t+( - 2.05))+ 78.64
|
| Standard Form
|
y= a sin b (x +c/b) + d
|
D= 7.38sin 0.497(t+( - 2.05)/0.497)+ 78.64
|
The period of any sine function is found using the expression 2 π b. Let's substitute the value of b from our function into this expression to find the period.
2 π/b
2 π/0.497
12.642223...
≈ 12.6
