If you have a graphing calculator (and know how to use it), you could graph the function and trace
the graph to find the maximum. But it’s really a lot easier if you conceptualize the situation. The
expression 3 − (x − 2)^2 will be at its maximum when the part being subtracted from the 3 is as small
as it can be. That part after the minus sign, (x − 2)^2 , is the square of something, so it can be no
smaller than 0. When x = 2, (x − 2)^2 = 0, and the whole expression 3 − (x − 2)^2 = 3 − 0 = 3. For any other
value of x, the part after the minus sign will be greater than 0, and the whole expression will be less
than 3. So 3 is the maximum value, and the answer is (D).
MAXIMUMS AND MINIMUMS
To find a maximum or minimum value of a function, look for parts of the expression—
especially squares—that have upper or lower limits.GRAPHING FUNCTIONS
Like almost everything else with functions, graphing is no big deal once you understand the
conventions. Example 4 provides a very good illustration.
Example 41. If which of the following could be the graph of y = p(x)?