REFLECTING GRAPHSAND

SYMMETRY

Section 4.3

REFLECTING GRAPHS

x-axisy-axis

Line y = xAbsolute value of a graph

Around a point

GRAPH USING CALCULATOR2

2

y x

y x

3 2

3 2

2

( 2 )

y x x

y x x

How are the graphs of y=f(x) and y = - f(x) related?

Reflected over the x-axis!!

GRAPH

( 1)( 3)

( 1)( 3)

y x x x

y x x x

Any negative y values are reflected over the x-axis!!

GRAPH USING CALCULATOR

y x

y x

How are the graphs of y = f(x)

and y = f(-x) related?

Reflected over the y-axis!!

GRAPH2

2

y x

x y

How is the graph of an equation affected

when you interchange the variables in the equation?

The graph is reflected over the line y = x!!

4

2

-2

-4

-5 5

Sketch the graphs of:( )

( )

( )

y f x

y f x

y f x

Sketch the graphs of:( )

( )

( )

y f x

y f x

y f x

3

2

1

-1

-2

-3

-4 -2 2 4

Sketch the graph and the reflection in the line y = x. Also, give the equation of the reflected graph.

2 4y x x 4

2

-2

-4

-5 5

Bellwork

Given an isosceles right triangle shown below. Express the Area, A, of the triangle as a function of the hypotenuse, c.

ac

Summary of Symmetry X-axis y = -f(x) (make y negative)

Y-axis y = f(-x) (make x negative)

Line y = x x = f(y) (switch x and y)

Origin y = -f(-x) (both x and y are negative)

Is there any symmetry in the graph of the functions?

4 4 1x y 3 1x y

Use symmetry to sketch the graph of

15

10

5

-5

-10

-15

-10 10

3 1x y

Use symmetry to sketch the graph of 12xy

15

10

5

-5

-10

-15

-10 10

Cubic Functions and Symmetry:

What type of symmetry is in a cubic?

What is the point of symmetry?

3

bx

a

Given the polynomial below, what is the point of symmetry?

2 3( ) 9 6 2f x x x x

The graph of a cubic function has a local max at (-5, 6) and a point of symmetry a (-1, 2). What is the local minimum?