1C-2 Why do I have to use f(x) - f(a) over x - a on the difference quotient ?
OCW Scholar - Single Variable Calculus
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The difference quotient is about change in height(y) over the change in width(x). If x-a is the difference in width of a function, the difference in height would be f(x)-(a).
If you're using \[\Delta x\] as the change in width then you want\[f(x+\Delta x) - f(x)\] to represent the change in height.
If you're h as the change in width then you would f(x+h)-f(x) as the change in height, all 3 forms are the same way of saying the change in height over the change in width, which is called the slope.
The difference quotient is about finding the instantaneous slope, that is when the limit of the change in width approaches zero.
Ok I get it now, thank you
This is just another way of finding the slope: the change in y divided by the change in x: