The graph of
is rotated counterclockwise about the origin through an acute angle .
What is the largest value of 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 , which
is , and the minimal slope which is .
Thus the maximal acute angle through which the graph can be rotated counterclockwise
is degrees.
Thus the maximal acute angle through which the graph can be rotated clockwise
is degrees. (Your answer should be negative to indicate
the clockwise direction.)
Note that a line makes angle with the horizontal,
where .
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 of a fixed slope intersect a graph of in at most one point, what can you say about rotating the graph of ?
2. If some line intersects the graph of in two or more points, what can you say about rotating the graph of ?
3. If some line intersects the graph of in two or more points, what does the Mean Value Theorem tell us about ?