Finding an exponential equation with 2 points

Question

If you have two points, for example, (2, 6) and (3, 18), how do you find the equation if you know its exponential?

I've heard about y=ab^x but I'm not sure what those variables represent.

Answer 1

You have two points on the curve to determine the two parameters $a,b$ so plug them in, giving $6=ab^2, 18=ab^3$ Can you find $a,b?$

Answer 2

A linear function is one which increases (or decreases) by the same amount every time the input is increased by a unit. For example, if $f$ is linear and $f(3)=5$ and $f(4)=10$, then you know that $f(5)=15$ and $f(6)=20$. More generally, the slope of this function would be 5, and you could write the equation $f(x)=mx+b$ where $m=5$ and $b=-10$.

An exponential function is similar, except that the function increases or decreases by a fixed ratio for each increment. If $g$ is exponential, and $g(3)=5$ and $g(4)=10$, then you would know that $g(5)=20$ and $g(6)=40$. The general form for an exponential function is $g(x)=ab^x$, where $a$ is the intercept (note that $g(0)=a$) and $b$ is the ratio term. In this case, we have $g(x)=\frac{5}{8}\cdot 2^x$.

In your example, the function increases by a factor of $\frac{18}{6}=3$ as the input goes from $x=2$ to $x=3$. So the ratio term is $3$, and the function has the form $g(x)=a\cdot 3^x$. You should be able to find $a$ from here.