Watch the video from 5:35:
In the description, the narrator says that "demand depends on price".
However, in the plot, he is keeping price "P" as the dependent variable.
How does that make sense?
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As the other answer already stated, this is mostly due to a convention that goes back to Marshall. However, I think there is an additional reason for this.
Optimizing economists often think "on the margin" and a central equation in many economic models is "marginal benefit = marginal cost." In a competitive market the marginal revenue is simply the price that firms take as given. However, firms with market power can influence the price. A monopolist takes the demand as given and calculates "marginal revenue" to equalize it to "marginal cost." It is unintuitive to think about marginal revenue in terms of price, but much more intuitive to think about marginal revenue in terms of quantity. With quantity on the horizontal axis, you can easily draw the graph of marginal revenue as you intuitively think about it into your p-q diagram, where you already plotted your inverse demand function.