Just look at the difference between x = 7 and x = 8, the value of the function increases by more than 38 MILLION!

How does this compare to other graphs/functions?

As the graph below shows, exponential growth

at first, has a lower rate of growth than the linear equation f(x) =50x

at first, has a slower rate of growth than a cubic function like f(x) = x^{3}, but eventually the growth rate of an exponential functionf(x) = 2^{ x}, increases more and more -- until the exponential growth function has the greatest value and rate of growth!

Moral of the story: Exponential growth eventually grows at massive rates, even though it starts out growing slowly. This 'trend' is true and, when compared to other graphs, exponential growth eventually outpaces most other functions's rate of increase.

You can get $1 for the first year, $2 for the second, $4 for the 3rd, doubling the amount very year -- for twenty years.

Explanation

Option 1 $1000 a year for twenty years

This would give you $20,000.

Option 2 the exponential growth function y = 2^{x}

2^{20} = $1,048,576

This question demonstrates the trait that we keep coming back to: exponential growth might not start out with much (1 and 2 dollars in the first years), but eventually its rate of growth is humongous!

Below is a picture of the two graphs in question. Option 1 can be modeled of the graph of $$ y = 2x $$, a linear equation. On the other hand, the graph of $$ y = 2^x $$, an exponential growth equation, quickly surpasses the line.

Closer Look at Graph/Equation

Below is a picture of the most commonly studied example of exponential growth, the equation $$ y = 2^x $$