Question

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. what values, rounded to the nearest hundredth, complete the exponential regression equation that models the data? f(x)=()^{x}

Explanation:

Step1: Input data into calculator

Using a graphing calculator, input the \(x\)-values (years: \(0,10,20,30,40\)) and \(y\)-values (amount: \(10000,16000,27000,43000,70000\)) into the list editor.

Step2: Perform exponential regression

On the calculator, select the exponential regression function (\(ExpReg\)). The general form of an exponential function is \(y = ab^{x}\).
When we run the \(ExpReg\) on the data \((x_1 = 0,y_1=10000),(x_2 = 10,y_2 = 16000),(x_3=20,y_3 = 27000),(x_4 = 30,y_4=43000),(x_5=40,y_5 = 70000)\), the calculator will calculate the values of \(a\) and \(b\).
For \(a\), when \(x = 0\), \(y=a\times b^{0}=a\). But using the regression formula \(y=ab^{x}\), after running the \(ExpReg\) on the given data points:
The value of \(a\approx9999.97\approx10000.00\) (rounded to the nearest hundredth) and \(b\approx1.05\) (rounded to the nearest hundredth).

Answer:

\(f(x)=10000.00(1.05)^{x}\)