6. The population of a certain city grows exponentially. When the city was founded, it only had 1000 residents. After 1 year the city had 1050 residents. Write an equation that models the population $y$ of the city after $x$ years

Step 1 ：The population of the city grows exponentially, which means it follows the formula: \(y = a \cdot b^x\) where \(y\) is the final amount (population in this case), \(a\) is the initial amount (initial population), \(b\) is the growth factor, and \(x\) is the time (in years).

Step 2 ：Given that the initial population (\(a\)) is 1000 and the population after 1 year (\(y\)) is 1050, we can substitute these values into the equation to find the growth factor (\(b\)).

Step 3 ：Solving the equation \(1050 = 1000 \cdot b^1\) gives us the value of \(b\), which is 1.05.

Step 4 ：Now that we have the growth factor, we can substitute \(a\) and \(b\) into the exponential growth formula to get the equation that models the population of the city.

Step 5 ：\(\boxed{y = 1000 \cdot 1.05^x}\) is the equation that models the population \(y\) of the city after \(x\) years.