3.1 Symmetry & Coordinate
Graphs
I. Symmetry

Point symmetry – two distinct points P and P’ are
symmetric with respect to point M if and only is M is
the midpoint of
PP '

When the definition is extended to a set of points,
such as a graph of a function, then each point P in
the set must have an image point P’ that is also in
the set. A figure that is symmetric with respect to a
given point can be rotated 180 degrees about that
point and appear unchanged.
A. Symmetry with Respect to the
Origin
Symmetry with respect to the origin




A function has a graph that is symmetric
with respect to the origin if and only if f(-x)
= -f(x) for all x in the domain of f.
A graph will have symmetry about the
origin if we get an equivalent equation
when all the y’s are replaced with -y and
all the x’s are replaced with -x.
So for every point (x, y) on the graph, the
point (-x, -y) is also on the graph.
It is a reflection about both the x- and yaxis.
Ex 1 Is each graph symmetric with respect
to the origin? How do you know?
The graph does not appear to be symmetric with
respect to the origin. We can verify this algebraically by
the following two-step method:
Step 1: find f(-x) and –f(x)
Step 2: if f(-x) = -f(x), then the graph has symmetry
about the origin. If not, then it is not.
If you have an equation instead of a function, you can:
Step 1: Replace all x’s with –x and all y’s with –y.
Step 2: if you get the same equation, then it is
equivalent about the origin.
−𝑓 𝑥 = −𝑥 6
≠
𝑓 −𝑥 = 𝑥 6
No, f(x) = x6 is not symmetric with
respect to the origin.
Ex 1 Is each graph symmetric with respect
to the origin? How do you know?
The graph appears to be
symmetric about the origin, but
lets check algebraically.
Remember the two steps:
Step 1: find f(-x) and –f(x)
Step 2: if f(-x) = -f(x), then the graph has
symmetry about the origin. If not, then it is not.
−𝑓 𝑥 = 3𝑥 3 − 5𝑥
=
−𝑓 𝑥 = 3𝑥 3 − 5𝑥
Yes, 𝑓 𝑥 = −3𝑥 3 + 5𝑥 is symmetric
with respect to the origin.
B. Line symmetry

Two points P and P’ are symmetric with respect to a
line l if and only if l is the perpendicular bisector of PP '
A point P is symmetric to itself with respect to line l if
and only if P is on l.

Graphs that have line symmetry can be folded along
the line of symmetry so that the two halves match
exactly. Some graphs, such as the graph of an
ellipse, have more than one line of symmetry.

Common lines of symmetry: x-axis, y-axis, y = x
and
y = -x.
Ex 2: Determine whether the graph of x2 + y = 3 is symmetric with respect to
the x-axis, y-axis, the line y = x, the line y = -x, or none of these. Answer: y-axis
You can figure this out without actually graphing the equation. Here is how:
Symmetry with
respect to the line:
Test
Results
x-axis
(a, b) and (a, -b) should produce
equivalent equations.
x2 + y = 3
x2 + y = 3
a2 + b = 3
a2 - b = 3
No, these are not equivalent
equations, so it is not
symmetric with respect to the
x-axis.
y-axis
(a, b) and (-a, b) should produce
equivalent equations.
x2 + y = 3
x2 + y = 3
a2 + b = 3
a2 + b = 3
Yes, these are equivalent
equations, so it is symmetric
with respect to the y-axis.
y=x
(a, b) and (b, a) should produce
equivalent equations.
x2 + y = 3
x2 + y = 3
a2 + b = 3
b2 + a = 3
No, these are not equivalent
equations, so it is not
symmetric with respect to the
line y = x.
y = -x
(a, b) and (-b, -a) should produce
equivalent equations.
x2 + y = 3
x2 + y = 3
a2 + b = 3
b2 - a = 3
No, these are not equivalent
equations, so it is not
symmetric with respect to the
line y = -x.
Ex 3: Determine whether the graph 𝑦 = 𝑥 + 1 is symmetric
with respect to the x-axis, y-axis, both or neither.
Test both:
Symmetry with
respect to the line:
Test
Results
x-axis
(a, b) and (a, -b) should produce
equivalent equations.
𝑦 = 𝑥 +1
𝑦 = 𝑥 +1
𝑎 = 𝑏 +1
𝑎 = 𝑏 +1
Yes, these are
equivalent equations, so
it is symmetric with
respect to the x-axis.
y-axis
(a, b) and (-a, b) should produce
equivalent equations.
𝑦 = 𝑥 +1
𝑦 = 𝑥 +1
𝑎 = 𝑏 +1
𝑎 = 𝑏 +1
Yes, this are equivalent
equations, so it is
symmetric with respect
to the y-axis.
Answer: Both
II. Even, Odd, or Neither Functions
Not to be confused with End Behavior
 To determine End Behavior, we
check to see if the leading degree is
even or odd
 With Functions, we are determining
SYMMETRY (if the entire function is
even, odd, or neither)

A.
Symmetric with respect to the y-axis
Symmetric with respect to the origin
To determine whether a function is even, odd, or
neither, determine whether f(-x) = f(x) (even), f(-x) =
-f(x) (odd), or neither.
Ex. 1
Even, Odd or Neither?
Graphically
f ( x)  x  x
3
Algebraically
𝑓 −𝑥 = (−𝑥)3 −(−𝑥)
𝑓 −𝑥 = −𝑥 3 + 𝑥
Ex. 2Even, Odd or Neither?
Graphically
f ( x)  x  1
2
Algebraically
f ( x)  x  1
2
f(-x)=(-x)2+1
f(-x)=x2+1
Ex. 3
Even, Odd or Neither?
Graphically
f ( x)  x 1
3
Algebraically
f ( x)  x 1
3
f(-x) = (-x)3-1
f(-x) = -x3-1
Ex. 4
Even, Odd or Neither?
f ( x)  2 x  3
4
f ( x)  x  x
3
B. Copy and complete the graph so that it is an even
function and then an odd function.
Even: symmetric
about the y-axis
Odd: Symmetric
about the origin