Sec. 3.1 – Symmetry
We’re going to be doing a lot of graphing in this chapter. Let’s start by
looking at the graphs of some polynomial functions.
You can use a graphing calculator to graph a polynomial function and
approximate the zeros of the function. The zeros are important points in
the graph because that is where the graph crosses the x-axis. Other
important points are the y-intercepts, maximum, and minimum points.
To find the zeros with the calculator, graph the function. Then, on the
Casio, go to G-Solv, ROOT. If there is more than one zero, push the right
arrow key to get the other zeros. On the TI-83, go to Calc, ZERO.
Symmetry
Some common lines of symmetry are the x-axis, the y-axis, the line y = x,
the line y = -x, and the origin. The following “test points” will help you to
determine if a graph is symmetric.
Symmetry with
respect to
x-axis
Definition
Assume that (x, y) is
on the graph.
(x, -y) is on the graph.
Ex: (-3, 1) and (-3, -1)
are on the graph.
y-axis
6
x
(2,  12) are on
the graph.
6
x  y2  4
-6
6
y
x
-6
6
y   x2  4
-6
(-x, -y) is on the graph.
Ex: (2, 12) and
y
-6
(-x, y) is on the graph.
Ex: (1, 3) and (-1, 3)
are on the graph.
the origin
Sample Graph
6
y
x
-6
6
-6
x 2  y 2  16
the line y = x
(y, x) is on the graph.
Ex: (2, 3) and (3, 2)
are on the graph.
the line y = -x
6
x
-6
6
xy  6
-6
(-y, -x) is on the graph.
Ex: (2, -3) and (3, -2)
are on the graph.
y
6
y
x
-6
6
-6
xy  6
Example 1: Determine whether the graph is symmetric with respect to the
x-axis, the y-axis, the origin, the line y = x, and the line y = -x.
y  x 4  3x 2  8
x-axis: Does substituting (x, -y) produce the same equation?
 y  x 4  3x 2  8
y   x 4  3x 2  8
NO! The graph is not symmetric wrt the x-axis.
y-axis: Does substituting (-x, y) produce the same equation?
y  ( x)4  3( x) 2  8
y  x 4  3x 2  8
YES! The graph is symmetric wrt the y-axis.
origin: Does substituting (-x, -y) produce the same equation?
 y  ( x)4  3( x)2  8
y   x 4  3x 2  8
NO! The graph is not symmetric wrt the origin.
y = x : Does substituting (y, x) produce the same equation?
x  y4  3y2  8
NO! The graph is not symmetric wrt the line y = x.
y = -x: Does substituting (-y, -x) produce the same equation?
 x  ( y)4  3( y)2  8
NO! The graph is not symmetric wrt the line y = -x.
A look at the graph confirms that the graph is only symmetric
y
14
with respect to the y-axis.
x
- 10
-2
10
Example 2: Determine whether the graph is symmetric with respect to the
x-axis, the y-axis, the origin, the line y = x, and the line y = -x.
xy  6
x-axis: Does substituting (x, -y) produce the same equation?
x( y )  6
 xy  6
xy  6
NO! The graph is not symmetric wrt the x-axis.
y-axis: Does substituting (-x, y) produce the same equation?
( x) y  6
 xy  6
xy  6
NO! The graph is not symmetric wrt the y-axis.
origin: Does substituting (-x, -y) produce the same equation?
( x)( y )  6
xy  6
YES! The graph is symmetric wrt the origin.
y = x : Does substituting (y, x) produce the same equation?
( y )( x)  6
xy  6
YES! The graph is symmetric wrt the line y = x.
y = -x: Does substituting (-y, -x) produce the same equation?
( y )( x)  6
xy  6
YES! The graph is symmetric wrt the line y = -x.
A look at the graph confirms that the graph is symmetric
with respect to the origin, the line y = x, and the line y = -x.
6
y
x
-6
6
-6
Even and Odd Functions
A function is an even function if its graph is symmetric with respect to the
y-axis. It has exponents that are all even.
A function is an odd function if its graph is symmetric with respect to the
origin. It has exponents that are all odd.
Example 3: Determine if the following functions are even, odd, or neither.
a) y  x 2  2
6
y
x
-6
6
-6
The graph is symmetric wrt the y-axis, so the function
is even.
b) y  ( x  2)2
6
y
x
-6
6
-6
The graph is not symmetric wrt either the y-axis or the
origin, so the function is neither even nor odd.
c) y  x3
6
y
x
-6
6
-6
The graph is symmetric wrt the origin, so the function
is odd.
Example 4: Given point P, determine the coordinates of the point that is
symmetric with respect to the x-axis, the y-axis, the origin,
the line y = x, and the line y = -x.
a) P(3, 4)
b) (4, -8)
x-axis: (3, -4)
x-axis: (4, 8)
y-axis: (-3, 4)
y-axis: (-4, -8)
origin: (-3, -4)
origin: (-4, 8)
y = x : (4, 3)
y = x : (-8, 4)
y = -x: (-4, -3)
y = -x: (8, -4)
Example 5: The graph below is a portion of a complete graph. Sketch a
complete graph for each of the following symmetries:
a) the x-axis, b) the y-axis, c) the origin, d) the line y = x,
and e) the line y = -x.
6
y
x
-6
6
(The graph contains the
point (3, -1).
a) the x-axis. The graph needs to contain the point (3, 1).
-6
6
y
x
-6
6
-6
b) the y-axis. The graph needs to contain the point (-3, -1).
6
y
x
-6
6
-6
c) the origin. The graph needs to contain the point (-3, 1).
6
y
x
-6
6
-6
d) the line y = x. The graph needs to contain the point (-1, 3).
6
y
x
-6
6
-6
e) the line y = -x. The graph needs to contain the point (1, -3).
6
y
x
-6
6
-6