# Exponential Growth

**Problem**

On her computer, Alex has a screen saver that models exponential growth. The screen saver shows colored dots whole number can be modeled but the equation where *t* stands for the number of seconds since the screen saver’s cycle started. One second after the cycle’s start, there were 6 dots on the screen; three seconds after the cycle’s start there were 24 dots. Find the value of *a*.

**Solution**

What do you know? You’re told the equation is . And you know two data points for that equation: and What can you do with that?

Plug the known data points into the equation:

That’s a system of two equations with two unknowns. You can solve it by substitution. Note that if you multiply the *N*(1) equation by 4, you get 24, the same value that equals *N*(3).

Replace the 24 in *N*(3) with

Divide by :

Exponential bases are always positive, so . And that’s the answer.

To check, go back to one of the original equations and solve for then plug both *a* and into the other original equation and see if you get a true statement:

and

Check.