First of all, let's define some concepts. In mathematics, proportional relationships happen when two values always change by the same multiple. That is, when one doubles, the other doubles as well. So, you can always reduce a proportional relationship to the same equation as follows:
[tex]y=kx \\ where \ k \ is \ the \ proportionality \ constant[/tex]
This constant is also the slope of the straight line.
On the other hand, the average rate of change between any two points [tex](x_{1},f(x_{1}) \ and \ (x_{2},f(x_{2})[/tex] is the slope of the line through the two points. The line through the two points is called the secant line and its slope is denoted as [tex]m_{sec}[/tex], so:
[tex]ARC=\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}} \\ \\ \therefore ARC=\frac{Change \ in \ y}{Change \ in \ x}=m_{sec}[/tex]
Finally, The slope of a nonvertical line is the number of units the line rises (or falls) vertically for each unit of horizontal change from left to right, so:
[tex] Slope=\frac{change \ in \ y}{change \ in \ x}=\frac{rise}{run} [/tex]
So, we can say that the relationship of these three concepts is that they all talk about the slope of a curve.
