How Does Rotating the x-Axis Create a New Polynomial Graph?

[batballbat]

To show the addition of the term mx to a polynomial graph, my book takes an example with y=x^3. To produce a function y=x^3 - 3x it draws the graph of y= x^3. Then the line y= - 3x is drawn in the same graph. quote" If we think of the ordinates of y=x^3 as attached to the x-axis and constrained to remain vertical, the graph of y=x^3 will become the graph of y=x^3- 3x if the x-axis is rotated about the origin until it coincides with the line y= -3x.

Can somebody explain me in detail how rotating in such a way will produce the new graph?
It mentions such transformation is shear. what is such motion? Is it applicable to all functions?

 

[Dr. Seafood]

That's interesting, I haven't seen that interpretation before. A "shear" of a set (graph) is when all the points in the set (on the graph) are translated parallel to the vertical axis, such that points with a greater vertical distance from the axis are translated further horizontally. More specifically, points are translated parallel to the axis by a distance proportional to their perpendicular distance from the axis.

So, to the x-coordinate, we add a proportion of the y-coordinate (height above the axis), i.e. the point (x, y) is changed into the point (x + my, y).

Umm ... brb on the "rotation" interpretation of this. I'll edit later.

 

[batballbat]

i am not clear with the translation you mentioned. The points of graph y=x^3 are translated along the vertical axis but with distance proportional to the perpendicular distance from which axis??

also please try to explain the rotation method