From MathMotivation.com – Permission Granted For Use and Modification For NonProfit Purposes
Function Graphing Rules
Are you tired of calculating points?
The rules in this section will allow you to graph
something like y = 2(x –4)
2
– 3 without calculating a single point! Also, you will learn
rules that will give you insight into the symmetry of graphs like y = x
4
– 2x
2
without
calculating a point!
Even Functions
A function is defined as
even
if opposite real values of x result in the same yvalue. In
other words, a function is even if f(a) = f(a) for any real value of “a”. For example,
f(x) = x
2
is an even function since f(a) = a
2
and f(a) = a
2
.
Graphs of Even Functions
Since opposite values of x result in the same yvalue, the
graph of an even function
will always have symmetry with respect to the yaxis
.
For example, the graph of
y=x
2
is shown below.
Example:
Plot the graph of y = x
4
– x
2
.
Then determine whether or not this is an even
function and note any symmetry of the graph.
Letting x=0 results in y=0, so our yintercept is (0,0).
We can find the xintercepts by letting y=0 and solving 0 = x
4
– x
2
.
Factor 0 = x
4
– x
2
with the Distributive Property to get
0 = x
2
(x
2
– 1) = x
2
(x + 1)(x  1).
Apply the Zero Product Law to get solutions
x
2
= 0, x + 1 = 0, and x  1 = 0.
Extract square roots and apply the Addition Property of Equality to solve and get
x = 0, x = 1, x = 1.
This results in xintercepts of (0,0), (1,0), and (1,0).
Plot these points and find more
points on each side of each xintercept. Draw a smooth curve through as shown on the
following page.
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From MathMotivation.com – Permission Granted For Use and Modification For NonProfit Purposes
It becomes clear that this is an even function as x=2 and x=2 both resulted in y=12 and
both x=1/2 and x=1/2 both resulted in y = 3/16.
Also,
both terms of y = x
4
– x
2
involve even powers of x, making opposite input values of x result in the same y
value
.
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 Fall '09
 Crandall
 Anthropology, Even and odd functions

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