Building an Exponential Function from a Situation

A table is exponential when its outputs multiply by the same number every time the input increases by 1, not when they add the same number. Check a table by dividing each output by the one before it. If that ratio is constant, you have exponential growth or decay. If instead subtracting consecutive outputs gives the same constant difference, the table is linear, no matter how big that difference is or how steep the table looks. Steepness is not evidence of exponential behavior; a constant ratio is the only proof.

To build the function, read a straight off the table: it is the output when x = 0, the starting amount before any growth happens. Then find b, the growth factor, by looking at how the output changes over one step of x. If you only have two points that are n steps apart, the output multiplies by some ratio over those n steps, so b is the nth root of that total ratio. Once you have a and b, you write y = a·b^x, and every other value in the table should check out by repeated multiplication.

This skill is the bridge between spotting a pattern and actually solving exponential equations later: once you can write the function, you can plug in a given output and solve for x using logarithms, or plug in x and predict the output. Getting a and b right here is what makes every later log law and equation-solving step point at the correct function.