Chapter 1Chapter 1

Section 1.7Section 1.7

Symmetry & TransformationsSymmetry & Transformations

Points and SymmetryPoints and Symmetry

Types of SymmetryTypes of SymmetrySymmetry with respect to the Symmetry with respect to the x-axisx-axis

(x, y) & (x, -y) are reflections across the x-(x, y) & (x, -y) are reflections across the x-axisaxis

y-axisy-axis

(x, y) & (-x, y) are reflections across the y-axis(x, y) & (-x, y) are reflections across the y-axis

OriginOrigin

(x, y) & (-x, -y) are reflections across the (x, y) & (-x, -y) are reflections across the originorigin

Even and Odd FunctionsEven and Odd Functions

Even Function: graph is symmetric to Even Function: graph is symmetric to the y-axisthe y-axis

Odd Function: graph is symmetric to Odd Function: graph is symmetric to the originthe origin

Note: Except for the function f(x) = 0, Note: Except for the function f(x) = 0, a function can not be both even and a function can not be both even and odd.odd.

Algebraic Tests of Algebraic Tests of Symmetry/Tests for Even & Symmetry/Tests for Even &

Odd FunctionsOdd Functions f(x) = - f(x) f(x) = - f(x) symmetric to x-axissymmetric to x-axis

neither even nor oddneither even nor odd(replace y with –y)(replace y with –y)

f(x) = f(-x)f(x) = f(-x) symmetric to y-axissymmetric to y-axiseven functioneven function

(replace x with –x)(replace x with –x)

- f(x) = f(-x)- f(x) = f(-x) symmetric to originsymmetric to originodd functionodd function

(replace x with –x and y (replace x with –x and y with –y)with –y)

Basic FunctionsBasic Functions

Basic FunctionsBasic Functions

Basic FunctionsBasic Functions

Basic FunctionsBasic Functions

Basic FunctionsBasic Functions

Basic FunctionsBasic Functions

Basic FunctionsBasic Functions

Transformations Transformations withwith

the Squaring the Squaring FunctionFunction

v x = 12

x2

t x = 3x2

s x = -x2

r x = x+4 2q x = x-2 2

h x = x2-2

g x = x2+3

f x = x2

Transformations with Transformations with thethe

Absolute Value Absolute Value FunctionFunction

v x = 1

2 x

t x = 3x

s x = - x

r x = x+4

q x = x-2

h x = x -2

g x = x +3

f x = x

Transformation RulesTransformation Rules EquationEquationHow to obtain the graphHow to obtain the graph

For (c > 0)For (c > 0)

y = f(x) + c y = f(x) + c Shift graph y = f(x) up Shift graph y = f(x) up cc units units

y = f(x) - c y = f(x) - c Shift graph y = f(x) down Shift graph y = f(x) down cc unitsunits

y = f(x – c) y = f(x – c) Shift graph y = f(x) right Shift graph y = f(x) right cc unitsunits

y = f(x + c) y = f(x + c) Shift graph y = f(x) left Shift graph y = f(x) left cc

units units

Transformation RulesTransformation Rules EquationEquation How to obtain the graphHow to obtain the graph

y = -f(x) (c > 0)y = -f(x) (c > 0) Reflect graph y = f(x) over Reflect graph y = f(x) over x-axisx-axis

y = f(-x) (c > 0)y = f(-x) (c > 0) Reflect graph y = f(x) over Reflect graph y = f(x) over y-axisy-axis

y = af(x) (a > 1)y = af(x) (a > 1) Stretch graph y = f(x) Stretch graph y = f(x) vertically by vertically by

factor of factor of aa

y = af(x) (0 < a < 1)y = af(x) (0 < a < 1) Shrink graph y = f(x) Shrink graph y = f(x) vertically by vertically by

factor of factor of aaMultiply y-coordinates of y = f(x) by aMultiply y-coordinates of y = f(x) by a