Symmetry

1
Chapter 1

• Section 1.7
• Symmetry Transformations

2
Points and Symmetry
3
Types of Symmetry

• Symmetry with respect to the
• x-axis
• (x, y) (x, -y) are reflections across the
  x-axis
• y-axis
• (x, y) (-x, y) are reflections across the
  y-axis
• Origin
• (x, y) (-x, -y) are reflections across the
  origin

4
Even and Odd Functions

• Even Function graph is symmetric to the y-axis
• Odd Function graph is symmetric to the origin
• Note Except for the function f(x) 0, a
  function can not be both even and odd.

5
Algebraic Tests of Symmetry/Tests for Even Odd
Functions

• f(x) - f(x) symmetric to x-axis
• neither even nor odd
• (replace y with y)
• f(x) f(-x) symmetric to y-axis
• even function
• (replace x with x)
• - f(x) f(-x) symmetric to origin
• odd function
• (replace x with x and y with y)

6
Basic Functions
7
Basic Functions
8
Basic Functions
9
Basic Functions
10
Basic Functions
11
Basic Functions
12
Basic Functions
13
Transformations withthe Squaring Function
14
Transformations with theAbsolute Value Function
15
Transformation Rules

• Equation How to obtain the graph
• For (c gt 0)
• y f(x) c Shift graph y f(x) up c units
• y f(x) - c Shift graph y f(x) down c
  units
• y f(x c) Shift graph y f(x) right c units
• y f(x c) Shift graph y f(x) left c units

16
Transformation Rules

• Equation How to obtain the graph
• y -f(x) (c gt 0) Reflect graph y f(x) over
  x-axis
• y f(-x) (c gt 0) Reflect graph y f(x) over
  y-axis
• y af(x) (a gt 1) Stretch graph y f(x)
  vertically by
• factor
  of a
• y af(x) (0 lt a lt 1) Shrink graph y f(x)
  vertically by
• factor
  of a
• Multiply y-coordinates of y f(x) by a