Identify the general form of the exponential equation y = a(b^x) and the two points to be used. As an example, use the points (1,3), (2,9), presented in the form (x,y). Take the two points and substitute them into the equation y=a(b^x), giving 3 = a(b^1) and 9 = a(b^2) in this instance.
Rearrange the two equations to leave a on the right-hand side and try to solve the two simultaneous equations to find b: 3/(b^1) = a and 9/(b^2) = a. Since a = a, it can be stated that 3/(b^1) = 9/(b^2), which can be rearranged to yield 3(b^2) = 9(b^1) -> 3b^2 – 9b = 0 -> b(3b – 9) = 0. Therefore the solutions are either b = 0 or 3b – 9 = 0 -> 3b = 9 -> b = 3. Since the plotted curves of exponential functions never drop below the x-axis, ignore any values of b that are less than or equal to zero. Here, b must equal three.
Take this value of b and insert it into one of the rearranged equations to find the value of a: 3/(3^1) = a or 9/(3^2) = a. In both cases, a equals one.
Define the exponential equation, inserting the solutions for both a and b into the general form: y = 1(3^x), which can be simplified to y = 3^x. Therefore, the equation of the exponential curve that passes through the points (1,3), (2,9) is y = 3^x. For a more complete solution, draw a quick sketch of the exponential equation on a graph. Pick a range of values for x that will clearly demonstrate the exponential characteristics. A suitable range for this example would be from between minus one and three.