# exponential behavior from pattern of data

In the image below from this video lesson, the teacher shows how to get an exponential function from a pattern of data, also copied below. You can see that her solution using the formula (a)(b) to the power of x is f(x) = 4(3) to the power of x. She explains that she is multiplying 4 by 3 because the values in the y column increase by 3, but she doesn't explain why she is using 4 for the a value?

Question: for the set of data below, if the b value of (a)(b) to the power of x is 3, why is the a value 4?

x   y
-1   4/3
0   4
1   12
2   36
3  108


If you know, that the data are the result of an exponential function, then you take two data points and write down two equations. The general function is $f(x)=a\ \cdot b^x$. For example you can take the first two data points $(-1/\frac{4}{3})$ and $(0/4)$. Now you can insert the values in the equation:
$a\cdot b^{-1}=\frac{4}{3}$
$a\cdot b^{0}=4$
Solve this little equation system for a and b. By looking at the second equation it can be easily evaluated, that $a=4$.