Question

determining the regression equation
mr. valdez puts $10,000 in a retirement account and does not make any deposits or withdrawals. the table below shows the amount of money in the account after x years.

years	amount ($)
10	16,000
20	27,000
30	43,000
40	70,000

what values, rounded to the nearest hundredth, complete the exponential regression equation that models the data?
$f(x) = \square (\square)^x$

Explanation:

Step1: Recall the exponential regression formula

The general form of an exponential regression equation is \(y = ab^{x}\), where \(a\) is the initial value when \(x = 0\).
From the table, when \(x=0\), \(y = 10000\), so \(a = 10000\).

Step2: Calculate the value of \(b\)

We can use another point \((x,y)\) to find \(b\). Let's use the point \((10,16000)\).
Substitute \(a = 10000\), \(x = 10\) and \(y=16000\) into \(y=ab^{x}\):
\(16000=10000\times b^{10}\)
First, divide both sides by \(10000\):
\(\frac{16000}{10000}=b^{10}\)
\(1.6 = b^{10}\)
Then take the \(10^{th}\) root of both sides:
\(b=1.6^{\frac{1}{10}}\)
Using a calculator, \(b\approx1.05\)

Answer:

\(f(x)=10000(1.05)^{x}\)