SAT/ACT Practice
1
1. Simplify:
3
2
2
3
 
2
2. The sum of two positive consecutive integers is x. In
terms of x, what is the value of the smaller of these
two integers?
x 1
x
x 1
x
A.  1 B.
C.
D.
E. x  1
2
2
2
2
2
Objective: 3-1 Symmetry and Coordinate Graphs
Exercise
2

Draw your own function in Quadrant 1. Example:
Draw a reflection
of your function
over:
y
10
The x-axis
5
The y-axis
-10
-5
5
-5
-10
Objective: 3-1 Symmetry and Coordinate Graphs
10
x
The line y=x
The origin
(rotation of 180
degrees)
Point Symmetry
3
What is symmetry about the origin?
y
10
y
10
5
5
-10
-5
5
10
x
y
-10
-5
5
10
-5
-5
-10
5
-10
-10
-5
5
-5
-10
Objective: 3-1 Symmetry and Coordinate Graphs
10
x
10
x
Formal Definition
4


Two distinct points P and P’ are symmetric with
respect to point M if and only if M is the midpoint
of PP’. Point M is symmetric with respect to itself.
In other words, anything that is symmetric about the
origin will have the origin as the midpoint of any
point along the graph of the function.
Objective: 3-1 Symmetry and Coordinate Graphs
Determining Symmetry from a Graph
5
y
y
10
10
5
5
-10
-5
5
10
x
-10
-5
-5
-10
5
x
10
-5
-10
y
-10
y
10
10
5
5
-5
5
10
x
-5
-10
Objective: 3-1 Symmetry and Coordinate Graphs
-10
-5
5
-5
-10
10
x
6
Determining Point Symmetry
Algebraically
Symmetry about the origin.
If f(-x) = -f(x) then the function is symmetric about
the origin.
Example: Determine if f(x)=x5 is symmetric with
respect to the origin.

Objective: 3-1 Symmetry and Coordinate Graphs
Another Example
7

Determine whether the following function is symmetric with
respect to the origin:
x
g ( x) 
1 x
Objective: 3-1 Symmetry and Coordinate Graphs
Line Symmetry
8
As we have seen, sometimes functions are symmetric
with respect to a line (not just a single point).
Typical lines that are used:
x-axis
y-axis
y=x
y=-x
Each of these has a “test”

Objective: 3-1 Symmetry and Coordinate Graphs
X-Axis Symmetry Test
9
You have x-axis symmetry if you can substitute the
opposite of “y” and still get the same equation.
Example:
x = y2 + 2
If there is a point (2, 3) on the graph there will also
be a point at (2, -3)
Objective: 3-1 Symmetry and Coordinate Graphs
Y-Axis Symmetry Test
10
You have y-axis symmetry if you can substitute the
opposite of “x” and still get the same equation.
Example:
y = x6 + 4
If there is a point (2, 3) on the graph there will also
be a point at (-2, 3)
Objective: 3-1 Symmetry and Coordinate Graphs
Y=X Symmetry
11
You have y=x symmetry if you can switch the x and y
coordinates and still get the same equation.
Example:
xy = 6
If there is a point (2, 3) on the graph there will also
be a point at (3, 2)
Objective: 3-1 Symmetry and Coordinate Graphs
Y=-X Symmetry Test
12
You have y=-x symmetry if you can switch the x and y
coordinates and take the opposite of both terms
and still get the same equation. Example: x2 + y2
= 25
If there is a point (4, -1) on the graph there will also
be a point at (1, -4)
Objective: 3-1 Symmetry and Coordinate Graphs
A “Real” Example
13

Determine whether the graph of xy=-2 is symmetric
about the x-axis, y-axis, the line y=x, the line y = x, or none of these.
Objective: 3-1 Symmetry and Coordinate Graphs
You Try
14

Determine whether the graph of x2 + y = 3 is
symmetric with respect to the x-axis, the y-axis, the
line y = x, the line y = -x, or none of these.
Objective: 3-1 Symmetry and Coordinate Graphs
Using Symmetry to Graph
15

Determine whether the graph of:
|y| = 2 - |2x|
Is symmetric with respect to the x-axis, y-axis, or neither one of these.
Use this information to graph the function.
y
10
5
-10
-5
5
-5
-10
Objective: 3-1 Symmetry and Coordinate Graphs
10
x
You Try
16

Determine whether the graph of:
|y| = |x| + 1
Is symmetric with respect to the x-axis, y-axis, the line y=x, the line y=-x,
or none of these.
Use this information to graph the function.
Objective: 3-1 Symmetry and Coordinate Graphs
Odd and Even Functions
17


Functions whose graphs are symmetric with respect
to the y-axis are even functions.
Functions whose graphs are symmetric with respect
to the origin are odd functions.
Objective: 3-1 Symmetry and Coordinate Graphs
Homework
18

page 134, 14, 16, 22, 24, 26, 28-30 all, 32
Objective: 3-1 Symmetry and Coordinate Graphs