Single-variable calculus from first principles
Sketching a curve by hand looks like art, but it's really a checklist. Run through the same ordered steps every time and the shape assembles itself, each step adding a new piece of information to the picture.
The order matters: each step builds on the last. By the time you reach the sketch, you already know where the curve lives, where it crosses, where it turns, and which way it bends, so drawing is just connecting the dots.
Before a plane leaves the gate, the pilot doesn't sketch the takeoff from intuition, they work through a pre-flight checklist, the same items in the same order every single flight. Sketching a curve works exactly that way. Run the steps in order, domain, intercepts, asymptotes, slope, concavity, and by the last line the shape is already cleared for drawing.