The graph of [math] is rotated counterclockwise about the origin through an acute angle [math]. What is the largest value of [math] for which the rotated graph is still the graph of a function? What about if the graph is rotated clockwise?

To answer this question we need to find the maximal slope of [math], which is , and the minimal slope which is .

Thus the maximal acute angle through which the graph can be rotated counterclockwise is [math] degrees.

Thus the maximal acute angle through which the graph can be rotated clockwise is [math] degrees. (Your answer should be negative to indicate the clockwise direction.)

Note that a line [math] makes angle [math] with the horizontal, where [math].

Hints: Recall that a graph of a function is characterized by the property that every vertical line intersects the graph in at most one point. In view of this:
1. If ALL lines [math] of a fixed slope [math] intersect a graph of [math] in at most one point, what can you say about rotating the graph of [math]?
2. If some line [math] intersects the graph of [math] in two or more points, what can you say about rotating the graph of [math]?
3. If some line [math] intersects the graph of [math] in two or more points, what does the Mean Value Theorem tell us about [math]?