> This is nonsense. Kids are not expected to look at polynomial equations and be able to deduce the shape of the graph without a graphing calculator

It is not nonsense. I'll draft an example.

Any second degree polynomial is a parabola that is either pointing up (positive a term), or down (negative a term). That term is an indication of how curved it is.

-b/2a is the X coordinate of the parabola's inflection point.

Plug that value into the equation and it'll give you the Y coordinate.

You now know the inflection point of the parabola, you know which way it points, and how steep it is, and exactly where the polynomial's roots should live (and whether or not it has any real ones!). If you remember what the squares of 0.5, 1, and 2 are, you can now connect the dots on a 'pretty good' plot.

This took yuo longer to read than it takes to do.

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Similar transformations can be applied to sine waves, root functions, exponentials, logarithms, and reciprocals.

If you can't do this, or don't understand how to do this, you have not learned and understood the material. If all you've learnt is how to plug the formula into a magic $160 box to look at the pretty picture, and how to ask it to solve for roots, you and your teachers have wasted your time. The point of all this isn't looking at plots, the point is understanding how you can manipulate these equations, and what these manipulations do to them. This should all be drilled to the point of being intuitive.

Anything so complicated that basic algebraic manipulations won't get you the rough shape in seconds of work... Is more complicated than a high schooler is taught to solve.