Recall what pattern equally-spaced data has if they fit an exponential function.
See solution.
Practice makes perfect
If the table of values contains equally spaced inputs, then we can look for a pattern in the outputs. But what if our data does not have equally spaced inputs? We will address each case, one at a time.
Table of Values Containing Equally Spaced Inputs
If the x-values are equally spaced, the quotients of the consecutive corresponding y-values of an exponential function are constant. For example, consider the following table of values for the function y = 2^x.
Notice that the constant quotient is not necessarily equal to the base. This depends on the spacing between the inputs. Nevertheless, once we know that the data fits an exponential function we can find the corresponding function using two of these points in the general form of an exponential function.
y = ab^x
Table of Values Not Containing Equally Spaced Inputs
Consider the general form of an exponential function y=ab^x. If we take the logarithm on both sides, we can rewrite it as a linear function.
Notice that log a and log b are constants. The equation has the format of a linear equation in slope-intersect form, where log y is our dependent variable and x is the dependent variable.
y = mx+ b
log y = (
log b)x+ log a
Hence, if the points (x,log y) lie in a line or close to it, an exponential model is appropriate. This is useful, because linear models are easier to recognize.
