Write a function to predict the amount of sales after 10 years with the chart x:0,1,2,3 y:25000,30000,36000,43200

Question

We are learning about linear, quadratic, and exponential models

Answer 1

Well, the only option for a linear model is the least squares regression line (a.k.a. Best Fit Trend Line) because the slopes are not the same.

 

Any least squared calculator will find it for you.

It is y = 6060* X + 24460.   So based on this model, after 10 years, the sales are

6060*10 + 24460 =  60600 + 24460 = 85060

 

As you have given four points of data, An exponential model is possible. So is a cubic.

 

The exponential model is y = c*E^kx = c*exp(kx).

Since f(0) = 25000, then C = 25000.

Since f(1) = 30000, 25000* exp(k) = 30000

                             exp(k) = 6/5

                             so k = ln(6/5)

The exponential model is then y = f(x) = 25000*exp( (ln(6/5)X)

At 10 years, the sales based on this model is 154793.41                                                    

 

 

The cubic model is Ax^3+Bx^2+C^x+25000 , since f(0)=25000

Plugging in the other points of data produces the 3 x 3 system of equations:

        A +B +C = 5000

       8A + 4B + 2C = 11000

      27A + 9B + 3C = 18200

                                                                                         

The solution of this system produces the following cubic function:

    (100/3)x^3 + 400X^2 + 13700/3X + 25000

The sales after 10 years based on this model is 110666.67

 

Finally, a recursive model lies hidden inside the 4 points of data itself.....

Observe that the difference of sales between each year increases by 20%

 

year  sales     difference

0       25000 

1       30000     5000

2       36000     6000     <-- 20% from year 1

3       43200     7200     <-- 20% from year 2

 

 

So a recursive function can be written as:

  f(x) =   {  25000 if x=1,

                  30000 if x=2,

                  

                  f(x-1) +  1.2* (f(x-1)-f(x-2)) if x>1

 

the recursive expression in the function can be written as:

           2.2* f(x-1) - 1.2*f(x-2) for any year x>1

 

The following table shows the sales after 10 years based on

this recursive model:

year     sales

0          25000
1          30000
2          36000
3          43200
4          51840
5          62208
6          74649.6
7          89579.52
8          107495.424
9          128994.5088
10        154793.4106

               

              

Notice this recursive model agrees with the exponential.

 

 

 

Answer 2

We know that the y values do not have a common difference.  So we know it will not be a linear model. 

We know that as the x values increase, the y values increase.  This represents an exponent model.  So we want a exponential regression.

 

Go to STATS on graphing calculator, then go to EDIT.  You will see a chart labeled L1 to L3.  In column L1, enter all the x values given.  In column L2, enter all the y values that correspond to their x values.  Then go to STATS, go to CALC, then scroll down to ExpReg.  Select that option to get the equation.