Question

What is the relationship among proportional relationships lines, rates of change, and slope?

Answer 1

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:

$y=kx \\ where \ k \ is \ the \ proportionality \ constant$

This constant is also the slope of the straight line.

On the other hand, the average rate of change between any two points $(x_{1},f(x_{1}) \ and \ (x_{2},f(x_{2})$ 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 $m_{sec}$, so:

$ARC=\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}} \\ \\ \therefore ARC=\frac{Change \ in \ y}{Change \ in \ x}=m_{sec}$

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:

$Slope=\frac{change \ in \ y}{change \ in \ x}=\frac{rise}{run}$

So, we can say that the relationship of these three concepts is that they all talk about the slope of a curve.

Answer 2

It introduces the relationship between two variables and is called correlation. Proportionality or variation is state of relationship or correlation between two variables It has two types: direct variation or proportion which states both variables are positively correlation. It is when both the variables increase or decrease together. On the contrary, indirect variation or proportion indicates negative relationship or correlation. Elaborately, the opposite of what happens to direct variation. One increases with the other variables, you got it, decreases. This correlations are important to consider because you can determine and identify how two variables relates with one another. Notice x = y (direct), y=1/x (indirect)