Algebra III
Lesson 31
Symmetry – Reflections - Translations
Symmetry
The graph looks the same on either side of the – point, line,
plane – of symmetry.
Sample:
Not symmetric
about origin
0
Symmetric
about origin
0
Not symmetric
about line
Symmetric
about line
Visualize the point or
line as a mirror. What
would be on the other
side of the mirror is
what is really on the
other side of the
mirror.
Symmetry about the y-axis
Yes
Yes
No
Grab the y-axis and flip the graph
over. The new graph would be the
same as the original.
Test for symmetry about the y-axis
In the equation of the graph, replace the ‘x’ with ‘-x’. If after doing any
algebra that can be done, the equation is the same as the original,
then there is symmetry about the y-axis.
y = x2
y = (-x)2
symmetric
y = x3
y = (-x)3
not symmetric
Symmetry about the x-axis
Test for symmetry about the x-axis
Replace ‘y’ with ‘-y’. If the equation
remains equivalent, then there is
symmetry about the x-axis.
Symmetry about the origin
Test for symmetry about the origin
Replace ‘y’ with ‘-y’ and ‘x’ with ‘-x’. If
the equation remains equivalent, then
there is symmetry about the origin.
Note: symmetry about the x and y
axes automatically means symmetry
about the origin, but not always the
other way around.
Other symmetries
Symmetry can also happen about other lines or points.
There is no good test for these situations, except visually.
Reflections
Reflections are new mirror images of a graph with respect
to something.
By contrast, symmetry is a mirror image onto itself.
Reflection
about the y-axis
Reflection
about the x-axis
Spin the graph by the object of reflection to get the reflection.
To find the new equation for a reflected graph.
For reflection about the y-axis, replace ‘x’ with ‘-x’.
For reflection about the x-axis, replace ‘y’ with ‘-y’.
Given y = f(x)
Reflection about the y-axis: y = f(-x)
Reflection about the x-axis: -y = f(x)
Note: f(-x) is not the same as –f(x).
y = -f(x)
Translation
Translation is where the entire graph is moved to another spot.
The shape of the graph remains the same.
Translation on the x-axis
y = x2
y = (x-2)2
x
y
x
y
-2
4
0
4
-1
1
1
1
0
0
2
0
1
1
3
1
2
4
4
4
y = (x+2)2
x
y
-4
4
-3
1
-2
0
-1
1
0
4
y = (x-4)2
y = (x+4)2
Translation on the y-axis
y = x2
y = x2-2
x
y
x
y
-2
4
-2
2
-1
1
-1
-1
0
0
0
-2
1
1
1
-1
2
4
2
2
y = x2+2
x
y
-2
6
-1
3
0
2
1
3
2
6
y = x2-4
y = x2+4
What is the equation of this graph?
General form for translation
(y-a)=(x-b)2
y=(x-b)2+a
Where would this graph be?
y = (x+2)2 - 3
y = (x-4)2+2
Example 31.1
The graph of the equation y = x is shown on the left below. The
graph on the right is the same graph reflected in the y-axis. Write
the equation of the graph on the right.
What kind of reflection is this?
(What is it reflected about?)
y-axis
for this replace ‘x’ with ‘-x’
y= x
y= −x
Example 31.2
The graph of the equation y = x is shown on the left below. The
graph on the right is the same graph reflected in the x-axis and
translated 3 units to the left. Write the equation of the graph on the
right.
What was done?
Reflected on x-axis: ‘y’ with ‘-y’.
y= x
y=− x
Moved 3 to left (in ‘-’ direction)
y=− x
y = − ( x + 3)
Example 31.3
Graph the equation of y = 1 x . Then write the equation of the
graph that has the same shape but is translated 3 units to the left
and 4 units down.
Note: x≠0
x
y
-2
-1/2
-1
-1
-1/2
-2
1/2
2
1
1
2
1/2
Move 3 left
y= 1
x
y=
1
( x + 3)
Move 4 down
y=
1
( x + 3)
y=
( y + 4) =
1
−4
( x + 3)
1
(x + 3)
Practice
a) Determine whether the graph of each equation is symmetric about
the x-axis, the y-axis, and the origin.
1)
2)
y = x2
y-axis
y = (-x)2
x2 + y2 = 9
yes
(-x)2 + y2 = 9
yes
x-axis
(-y) = x2
no
x2 + (-y)2 = 9
yes
origin
(-y) = (-x)2
no
(-x)2 + (-y)2 = 9
yes
b) The graph of the function f(x) = |x| is shown. Sketch the graph of the
function g(x) = |x-1|.
f(x)
g(x)
Z
c) Givens: ∠Z is a right angle.
∠ZXV ≅ ∠ZYU
U
V
XV ≅ YU
Prove: ∆XZV ≅ ∆YZV
Statements
1) ∠Z is a right angle.
Y
X
Reasons
Z
Z
1) Given
∠ZXV ≅ ∠ZYU
XV ≅ YU
2) ∠Z ≅ ∠Z
2) Reflexive
3) ∠ZXV ≅ ∠ZYU
3) AAÆAAA
4) ∆XZV ≅ ∆YZU
4) ASA
V
X
U
Y
d) Evaluate
7!
without using a calculator.
4!3!
=
7 ⋅ 6 ⋅ 5 ⋅ 4!
4!3!
=
7 ⋅6⋅5
3 ⋅ 2 ⋅1
=7•5
= 35