Determining the elasticity of a supply/demand curve visually

Hi, I am self-studying microeconomics in preparation for a future course. I am confused about whether or not it is possible to visually determine the elasticity of a demand/supply curve by looking at its slope. Some sources I read tell you that the elasticity can be determined visually based on the steepness of the slope of the demand/supply curve.

However, I encountered other sources that say the opposite--that elasticity cannot be judged by the steepness of the slope. I read that the elasticity of demand changes at different points of a linear demand curve, but I can't find the same assertion repeated for linear supply curves.

Even more confusingly, I read from some sources that any supply curve that is upward sloping from the origin has a constant unitary elasticity regardless of the value of the slope of the supply curve. Doesn't this also contradict the assertion that the slope of the supply/demand curve indicates its relative elasticity, if a curve of ANY slope crossing the origin can be constant unitary elastic?

I am having trouble studying this from different sources. If someone can give an intuitive explanation for what is true (whether or not you can visually determine the elasticity of a demand/supply curve) I would be grateful.

Generally speaking it is not possible to tell if demand/supply is elastic/inelastic just by looking at steepness (although there are some exceptions).

Rigorously, the price elasticity of supply/demand is given by:

$$EL=\frac{\frac{\partial Q}{Q}}{\frac{\partial p}{p}}= \frac{\partial Q}{\partial p}\frac{p}{Q}$$

The slope of demand/supply curve will be given by $$\frac{\partial Q}{\partial p}$$ so looking at the slope can only tell you whether the first term is high/low but it does not tell you what the second term is $$p/Q$$. Consequently, generally speaking it is not possible to tell what the elasticity of supply/demand is just from the picture.

This is further complicated by the fact that what matters is the slope of the demand/supply function $$\frac{\partial Q}{\partial p}$$, but typically what you see on picture is not a demand/supply function but inverse demand/supply function, because by convention we plot price on $$y$$-axis, so what you see is not $$Q(p)$$ but $$p = f^{-1}(Q)$$.

This holds especially for linear supply/demand curves. For example general linear demand is given by:

$$Q = - a p + b$$

(where $$a$$ is the slope) hence general price elasticity for demand curve would be:

$$EL= -a \left(\frac{p}{-ap+b}\right)$$

(general formula for linear supply would be almost identical except $$a$$ would be positive) so as you can see above for linear function the price elasticity with change with price, and consequently knowing slope (steepness) will not help you to know whether the demand is elastic or not. For example, suppose that the slope is $$-10$$ (which is quite steep), then $$\frac{-10p}{-10p+b}$$ can still turn out to be less than 1. For example, for $$p=0$$ the elasticity here would also be zero.

There are some exceptions though. If the whole supply/demand is horizontal flat (as judged when looking at plot of $$p=f^{-1}(Q)$$ which means that $$Q(p)$$ is vertical) then we know that slope is infinity. Since, infinity multiplied by any scalar is still infinity we know that in this case the elasticity will be infinite (i.e. such supply/demand will be perfectly elastic) even just by looking at the picture.

Similarly, if the whole supply/demand is vertical (as judged when looking at plot of $$p=f^{-1}(Q)$$ which means that $$Q(p)$$ is horizontal) we know that the slope of demand/supply curve will be zero. Since anything multiplied by zero is still zero the whole elasticity will be zero in such case (i.e. such supply/demand will be perfectly inelastic).

Consequently, save for special cases it is not possible to know if a demand/supply is elastic or inelastic just by looking at a steepness of a slope. This is especially true for linear demand and supply where elasticity will range from $$0$$ to $$\infty$$ as price change along the demand.

Even more confusingly, I read from some sources that any supply curve that is upward sloping from the origin has a constant unitary elasticity regardless of the value of the slope of the supply curve.

This is true because a linear supply curve that goes through origin is generally given by $$Q_s= ap$$ which implies that elasticity is:

$$EL_{Q_s} = a \frac{p}{ap}=1$$

This actually holds also for linear demand function passing through origin since general linear demand that passes through origin would be given by $$Q_d=-ap$$ so the elasticity is:

$$EL_{Q_d} = -a \frac{p}{-ap} = 1$$

So this is another special case where you can exactly tell what a demand or supply elasticity is just by looking at it.