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I'm now reading about the formal definition of line as the graph of the function $f(x)=f(x_0)+m(x-x_0)$ for an arbitrary $x_0$ through which the graph passes, where the constant $m$ is called slope. If we denote $a-a_0$ by $\Delta a$, it can be easily proved that $m$ for some $x_0$ is $\dfrac{\Delta f(x)}{\Delta x}$.

What makes the equation define straight lines is that if and only if the graph is a straight line does $m$ stay constant for any $x_0$.

I don't understand the intuition behind this definition.

Why does $\dfrac{\Delta y}{\Delta x}=const.$? And why only in them?

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    $\begingroup$ Not sure I understand your question but the ratio is constant because of the properties of similar triangles. $\endgroup$
    – Karl
    Mar 2, 2017 at 18:29

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The difference quotient $m=\frac{f(x)-f(x_0)}{x-x_0}$ denotes the rate of change of your function.

If the rate of change is constant for any input $x$, then the function has to be linear, since a function can only "curve", if the rate of change differs between two arbitrary points $x_i, x_j$.

You can compare that with driving a car: Imagine you are driving a constant speed of $20 km/h$. You would then cover $20 km$ in an hour, $40 km$ in two hours and so on (in a linear manner).

To cover more or less kilometers, you would have to adjust speed, hence adjust the rate of change with respect to the kilometers covered. Therefore, your function would not be linear anymore.

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From your first equation you see that $$ m=\frac{f(x)-f(x_0)}{x-x_0} $$ so, in general, the value of $m$ is not a constant, but depends from $x$ and $x_0$.

If it is the same for any couple $x$ and $x_0$, we can chose an $x_1$ (different from $x$ and $x_0$) and we have $$ m=\frac{f(x)-f(x_0)}{x-x_0}=\frac{f(x_1)-f(x_0)}{x_1-x_0} $$

and this is the equation of a straight line passing through the points $(x_0,f(x_0))$ and $(x_1,f(x_1))$.

So, any three points of $y=f(x)$ stay on the same straight line.

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  • $\begingroup$ I know this definition, I want to see how it defines the straight line passing through these points - I see now thanks to Karl's answer that in any straight line the value of $m$ is constant, but I don't see why the converse is true - if we take any $f:\mathbb R \rightarrow \mathbb R$ such that $\dfrac{\Delta x}{\Delta y}=const.$ the graph of it is a straight line too. $\endgroup$
    – asdasdfsss
    Mar 2, 2017 at 19:35
  • $\begingroup$ You can work with the concept of derivatives: If the function is not linear, the derivative will not evaluate to a constant for any input $x$.. $\endgroup$ Mar 2, 2017 at 19:46
  • $\begingroup$ Now I understand your question !. I've modified my answer and I hope it's better. $\endgroup$ Mar 2, 2017 at 19:49

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