4.1 Nonlinear Functions and Their Graphs
Increasing, decreasing, maximum, and minimum
Here’s the graph of a function:
y
local
maximum




absolute
minimum
   







x

local
minimum








it is increasing on the intervals [-3.75, -2] and [1, )
it is decreasing on the interval [-2,1]
it has no absolute maximum
it has an absolute minimum of -3
it has a local maximum of 3.5
it has a local minimum of -1






a function could have several local minima
a function could have several local maxima
it could have no local max or min
it can have only one each of absolute max and min
it could have no absolute max or min
an “end point” can never be a local max or min
4.1-1
Symmetry
The graph of y = x2 looks like:

y



   





x





 the pieces of it on either side of the y-axis are reverse
(mirror) images of each other
 we say that it is symmetric with respect to the y axis
The graph of y = x3 looks like:

y



   





x





 if we rotate the image by ½ turn, we get the same image
 we say that it is symmetric with respect to the origin
4.1-2
Tests for symmetries; even/odd functions
 if f(-x) = f(x)
o the graph of f is symmetric with respect to the y-axis
o we say that f is an even function
 if f(-x) = -f(x)
o the graph of f is symmetric with respect to the origin
o we say f is an odd function
Examples (you will be required to use this method for
demonstrating whether a function is even, odd or neither):
f(x) = 3x2 + 3
g(x) = x3 + x
h(x) = x2 + x
f(-x) = 3(-x)2 + 3 = 3x2 + 3 = f(x)
 f is even
g(-x) = (-x)3 + (-x) = -x3 - x = - (x3 + x) = - g(x)  g is
odd
h(-x) = (-x)2 + (-x) = x2 - x  h is neither even nor odd
x
-2
-1
0
f(x)
-2
-.5
0
f(-x)
2
.5
0
for each input, f(-x) = -f(x)  f is odd
1
.5
-.5
2
2
-2
4.1-3