Question

after door - to - door canvassing, census workers determine the number of unoccupied homes. the table shown to the right lists the number of homes, in millions, that were unoccupied in four census years. answer questions a through c.

a) use a graphing calculator to fit an exponential function to the data, where x is the number of years after 1980.

( y=square(square)^{x} )
(type integers or decimals rounded to six decimal places as needed.)

Explanation:

Step1: Input data into graphing calculator

Input the data points \((x,y)\) where \(x = 0,10,20,30\) and \(y=5.8,7.6,9.7,14.7\) into the graphing calculator's list - editor (usually \(L_1\) for \(x\) - values and \(L_2\) for \(y\) - values).

Step2: Use exponential regression

On the graphing calculator, access the regression menu (e.g., on TI - 84: STAT → CALC → ExpReg). Select \(L_1\) as the \(x\) - list and \(L_2\) as the \(y\) - list.
The exponential regression formula is of the form \(y = ab^x\).
When we perform the exponential regression on the data \((0,5.8)\), \((10,7.6)\), \((20,9.7)\), \((30,14.7)\):
The calculator will calculate the values of \(a\) and \(b\).
We know that when \(x = 0\), \(y=a\times b^0=a\). From the point \((0,5.8)\), \(a = 5.8\).
Using the regression formula \(y=ab^x\) and the other data points, the value of \(b\) is calculated as follows:

$$y = 5.8b^x$$

Substitute \(x = 10\) and \(y = 7.6\):

$$7.6=5.8b^{10}$$

$$b^{10}=\frac{7.6}{5.8}\approx1.310345$$

$$b=\sqrt[10]{1.310345}\approx1.027797$$

Answer:

\(y = 5.8(1.027797)^x\)