The app below shows the graph of the exponential function [math]f\left(x\right)=3\cdot2^x[/math] for [math]x\ge0[/math].[br]The [i]y[/i]-coordinate of the red dot currently shown in the graph is [math]f\left(0\right)[/math].[br]Use the slider to show some points with integer [i]x[/i]-coordinates and[i] y[/i]-coordinates equal to [math]f\left(x\right)[/math] in the graph, and view their values in the table below the graph (values are approximated to 2 decimal places).[br][br]Drag the slider to its maximum value, and observe the table of values.[br]What do you notice?[br][i]Each term is the product of the previous term by[/i] 2.[br]If we want to be formal, we will then say that [math]f\left(1\right)=2\cdot f\left(0\right)[/math], [math]f\left(2\right)=2\cdot f\left(1\right)[/math], and so on.[br][br][b]Note[/b]: You can modify the function at any time, by dragging the two black handles on it. Explore the relationship between two consecutive terms of the sequence of values related to the new function.[br]
Consider the exponential function [math]f\left(x\right)=3\cdot2^x[/math] and let [math]n\in\mathbb{N}[/math].[br]Show that [math]f\left(n+1\right)=2\cdot f\left(n\right)[/math].
[math]f\left(n+1\right)=3\cdot2^{n+1}=3\cdot2^n\cdot2^1=\left(3\cdot2^n\right)\cdot2[/math][br]Since [math]f\left(n\right)=3\cdot2^n[/math], we can conclude that [math]f\left(n+1\right)=2\cdot f\left(n\right)[/math]
We have now a recursive formula that shows the relationship between the values of the function at two points that are distant 1 unit from each other:[br][center][math]f\left(n+1\right)=2\cdot f\left(n\right)[/math].[br][/center]We can read the formula as "[i]the given exponential function grows multiplicatively by[/i] 2 [i]units over any interval of length[/i] 1 [i]unit[/i]".
The example with which we started reasoning about the growth rate of an exponential function was the restriction of an exponential function with base [math]b>1[/math]over [math]x\ge0[/math], and we considered integer values of [i]x[/i] to make reasoning simpler, but the same reasoning holds for any [math]x\in\mathbb{R}[/math].[br][br]Generalizing the formula obtained above for any exponential function of the form [math]f\left(x\right)=ab^x[/math] with [math]b>1[/math], we can say that these functions grow multiplicatively by [math]b[/math] units over any interval of length 1 unit.[br][br]Moreover, they grow multiplicatively by [math]b^{\ell}[/math] over any interval of length [math]\ell[/math].[br][i]Proof[/i]: [math]f\left(x+l\right)=ab^{x+\ell}=ab^x\cdot b^{\ell}=f\left(x\right)\cdot b^{\ell}[/math][br][br]If 0<[i]b[/i]<1, we have an exponential decay and the graph of the function is decreasing, but the general properties remain the same. Use the applet above to create graphs of exponential decay functions, and observe the related tables of values.[br][br][center][b][i]Exponential functions[/i] → [i]Multiplicative growth[/i][/b][/center]