The app below shows the graph of the linear function [math]f\left(x\right)=0.5x+1[/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 of [math]f\left(x\right)[/math] in the graph, and view their values in the table below it.[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 sum of the previous term and[/i] 0.5![br]If we want to be formal, we will then say that [math]f\left(1\right)=f\left(0\right)+0.5[/math], [math]f\left(2\right)=f\left(1\right)+0.5[/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 linear function [math]f\left(x\right)=0.5x+1[/math] and let [math]n\in\mathbb{N}[/math].[br]Show that [math]f\left(n+1\right)=f\left(n\right)+0.5[/math].
[math]f\left(n+1\right)=0.5\left(n+1\right)+1=0.5n+0.5+1=\left(0.5n+1\right)+0.5[/math][br]Since [math]f\left(n\right)=0.5n+1[/math], we can conclude that [math]f\left(n+1\right)=f\left(n\right)+0.5[/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)=f\left(n\right)+0.5[/math].[br][/center]We can read the formula as "[i]the given linear function grows additively by[/i] 0.5 [i]units over any interval of length[/i] 1 [i]unit[/i]".
The example with which we started reasoning about the growth rate of a linear function was the restriction of a linear function 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 linear function of the form [math]f\left(x\right)=mx+b[/math] , we can say that these functions grow additively by [math]m[/math] units over any interval of length 1 unit.[br][br]Moreover, they grow additively by [math]m\ell[/math] units over any interval of length [math]\ell[/math] units.[br][i]Proof[/i]: [br][math]f\left(x+\ell\right)=m\left(x+\ell\right)+b=mx+m\ell+b=\left(mx+b\right)+m\ell=f\left(x\right)+m\ell[/math][br][br][center][b][i]Linear functions[/i] → [i]Additive growth[/i][/b][/center]