The formula for price elasticity of demand uses price as independent variable.
But I wonder what the elasticity graph looks like on an ordinary microeconomics system of coordinates, where demand is on the X axis.
After some reflection, it seems to me that my problem amounts to : how to convert the elasticity formula ( in which the input is price) into an equivalent formula in which the input ( indep. variable) is demand. For it seems to me that only the transformed formula would be suitable for ordinary microeconomics representation ( with demand on the X axis).
My goal is to visualize the graph of the elasticity function for a linear demand curve .
The problem I face is that the elasticity function graph I came up with looks unfamiliar.
I suppose my formula for the elasticity function contains a mistake, but I can't locate it.
Here is what I've done ( and I add a Desmos image below).
(1) I start with a demand function ( with price as independent variable) :
$$D(x)= a-bx$$.
(2) I transform this function into a price function ( with demand as independent variable), in order to obtain the traditonal demand curve ( with demand on the X axis and price on the Y axis) :
$$P(x)= - \frac 1b x +\frac ab$$.
(3) I use the calculus version of the elasticity function, namely :
$$ \epsilon_{\small P}= \frac {\mathit dD(P)} {\mathit dP} \times \frac {\mathit P} {\mathit D}$$
wich ( so it seems) should yield
$$ \Large\epsilon(x) = D'(x) \frac {P(x)}{Q(x)}$$
and finally ( since $D'(x)=-b$ here)
$$ \Large\epsilon(x) = -b \frac {P(x)}{Q(x)}$$.
But, as I said above, the graph of my alledged elasticity function looks unfamiliar.
In particular, it seems to me that the elasticity should be equal to $1$ for the X-value of the middle point on the demand curve.
Desmos (https://www.desmos.com/calculator/fp7elscgtq) :