Step 1 - Modeling with trig function.sagews

%md

Here we have some data:

Syracuse weather data (month, max temp):  (1, 32),(2,34),(3,43),(4,57),(5,69),(6,78),(7,82),(8,80),(9,72),(10,60),(11,48),(12,36)

We'll make a scatter plot of the data first and call the plot 'p'.

Syracuse weather data (month, max temp): (1, 32),(2,34),(3,43),(4,57),(5,69),(6,78),(7,82),(8,80),(9,72),(10,60),(11,48),(12,36)

data=[(1, 32),(2,34),(3,43),(4,57),(5,69),(6,78),(7,82),(8,80),(9,72),(10,60),(11,48),(12,36)]
p=scatter_plot(data)
p

%md
Next, we determine the parameters for amplitude, period, phase shift and midline.  This gets a little tedious.  Take careful note of the lines that you would actually edit!!!!!!!!

data=[(1, 32),(2,34),(3,43),(4,57),(5,69),(6,78),(7,82),(8,80),(9,72),(10,60),(11,48),(12,36)]#you will edit this with new data
a,b,c,d=var('a,b,c,d')
model(x)=a * sin(b*(x - c))+d
values=find_fit(data,model,initial_guess=(25,pi/6,7,60))#you will edit the guess values based on the scatterplot
params=[a,b,c,d]
for i in range(4):
    params[i]=values[i].rhs()

model(x)=params[0] * sin(params[1]*(x - params[2]))+params[3]
model(x)
p=scatter_plot(data,xmin=-2,xmax=14,ymin=-10,ymax=100)
p+=plot(model(x),xmin=-2,xmax=14,ymin=-10,ymax=100)
p

25.61099628075631*sin(0.5089522520223427*x - 2.068485038219847) + 56.87966137474419

find_fit(data,model,initial_guess=(25,pi/6,7,60))

[a == 25.61099628075631, b == 0.5089522520223427, c == 4.064202545524923, d == 56.87966137474419]

a = 25.61099628075631; b = 0.5089522520223427; c = 4.064202545524923; d = 56.87966137474419

y=a * sin(b*(x - c))+d
show(y)

$\displaystyle 25.6109962807563 \, \sin\left(0.508952252022343 \, x - 2.06848503821985\right) + 56.8796613747442$

p=plot(y,0,12)
p+s

Your turn!

1. Follow the link http://mathbits.com/MathBits/TISection/Statistics2/sinusoidal.html to find data for the two other cities. Type in the data and create separate scatter plots. Then find the sinusoidal model for each.

2. Plot the function: f(x)=sin(2x-π)-2, g(x)=-3sec(πx); h(x)=tan(x-π/2) in a single graphing window from [-2π,2π]. Use different colors for each graph!