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What is the relationship among proportional relationship lines rate of change and slope?
The rate of change is the same as the slope.
How does the slope differ from average rate of change?
They are the same for a straight line but for any curve, the slope will change from point to point whereas the average rate of change (between two points) will remain the same.
Can a rate change and the slope of the line be different quantities?
The instantaneous rate change of the variable y with respect to x must be the slope of the line at the point represented by that instant. However, the rate of change of x, with respect to y will be different [it will be the x/y slope, not the y/x slope]. It will be the reciprocal of the slope of the line. Also, if you have a time-distance graph the slope is the rate of chage of distance, ie speed. But, there is also the rate of change of speed - the acceleration - which is not DIRECTLY related to the slope. It is the rate at which the slope changes! So the answer, in normal circumstances, is no: they are the same. But you can define situations where they can be different.
What is the rate of change between any two points on a line?
it is the same as the slope, which can be found either graphically (rise over run) or algebraically using the formula (y2-y1)/(x2-x1)
How does the slope of a line represent the rate of change?
Assuming the variables are x (horizontal) and y (vertical), the slope of a straight line is defined as "rise over run". That is, the change in y divided by a change in x. This is exactly what the rate of change in y with respect to x, is. If the line is a curve, the instantaneous slope is defined as the gradient of the tangent to the curve and is the limiting value (as dx tends to 0) of the same measure.
