4.3 Reflecting Graphs; Symmetry4.3 Reflecting Graphs; Symmetry

In this section and the next we will see how the graph of an equation is transformed when the equation is altered.

This will allow us to graph a simple equation and – by reflecting it, stretching or shrinking it, or sliding it – to obtain the graph of a related, more complicated equation.

We begin by considering the reflection of a graph in a line.

The line of reflection acts like a mirror and is located halfway between a point and its reflection.

Objective To reflect graphs and to use symmetry to sketch graphs.

1. The graph of y = –f(x) can be obtained by reflecting the graph of y = f(x) in the x-axis. Algebraically, to obtain a reflecting graph of y = f(x), we only need to multiply (–1) on the original function.

Reflection in the Reflection in the xx-axis-axis

2. The graph of y = | f(x)| is keeping the graph of y = f(x) when f(x) ≥ 0 and reflecting the graph of y = f(x) when f(x) < 0 . The graph of y = | f(x)| has no dip below the x-axis. So graph of y = | f(x)| only flips the negative portion of graph of y = f(x).

Reflection in the Reflection in the yy-axis-axis3. The reflection graph of y = f(x) about the y-axis can be obtained

algebraically by the graph of y = f(–x).

Reflection in the Line Reflection in the Line yy = = xx4. Reflecting the graph of an equation in the line y = x is equivalent to interchange the variables x and y in the equation.

x = y2 – 3y

SymmetrySymmetryA line l is called an axis of symmetry of a graph if it is possible to pair the points of the graph in such a way that l is the perpendicular bisector of the segment joining each pair.

A point O is called point of symmetry of a graph if it is possible to pair the points of the graph in such a way that O is the midpoint of the segment joining each pair

l is the axis of symmetry O is the point of symmetry

2

Type of Symmetry Example

1

:

Symmetry in the

-axis y x

Meani g

x

n

2

, is on the graph whenever , is. 1

:

In the equation, leave alone and

substitute for . Does an equivalent

equivalent

x y x y y x

Testing an equation of a graph

x

y y

equation result?

Type of Symmetry Example

Special Tests for the Symmetry of a Graph

Both points (x, y) and (x, –y) on the graph

You only need to graph one branch. The other branch can be obtained by reflecting the graphed branch about the x-axis.

2

Type of Symmetry Example

:

Symmetry in

the -ax

is y x

Meaning

y

2

, is on the graph whenever , is.

:

In the equation, substitute for and

leave alone. Does an equivalent

eq

equivalent

x y x y y x

Testing an equation of a graph

x x

y

uation result?

Special Tests for the Symmetry of a Graph

Type of Symmetry Example

p. 134

Both points (x, y) and (–x, y) on the graph

You only need to graph one branch. The other branch can be obtained by reflecting the graphed branch about the y-axis.

3 3 1

: equivale

Symmetry in the lin

, is on the graph whenever , is.

e

t

n

y x x y

Meaning

y x x y

3 3 1

:

In the equation, interchange and .

Does an equivalent equation result?

y x

Testing an equation of a graph

x y

Type of Symmetry Example

p. 134

Special Tests for the Symmetry of a Graph

Both points (x, y) and (y, x) on the graph

You only need to graph one branch. The other branch can be obtained by reflecting the graphed branch about the line y = x.

3

: e

Symmetry in the or

quivalent

, is

i

o

gi

n the graph when r

n

eve

y x

Meaning

x y

3 , is.

:

In the equation, substitute for and

for . Does an equivalent equation

result?

x y y x

Testing an equation of a graph

x x

y y

Type of Symmetry Example

p. 134

Special Tests for the Symmetry of a Graph

Both points (x, y) and (–x, –y) on the graph

You only need to graph one branch. The other branch can be obtained by reflecting the graphed branch about the origin.

Even and Odd FunctionsEven and Odd FunctionsA function f is an even function if and only if

1. The domain of f is symmetric about the origin; and 2. f(– x) = f(x) for all x in the domain.

The graph of an even function is symmetry with respect to y-axis.

A function f is an odd function if and only if

1. The domain of f is symmetric about the origin; and2. f(– x) = –f(x) for all x in the domain.

The graph of an even function is symmetry with respect to origin.

More on Graphs of Even and Odd FunctionsMore on Graphs of Even and Odd Functions

Even and Odd FunctionsEven and Odd FunctionsExample 1. Given function f (x) = 3x2 – 2 defined on [2, 10]. Is

function f (x) an even function?

Since the domain of f is NOT symmetric about the origin, eventhough f(– x) = f(x) is algebraically true in the expression. This function is NOT an even function.

Challenge Question:

How do you change the condition of the above function f so that it can become an even function?

Since the condition #2 for even function is always true, we only need

to modify the domain of function f so that it is symmetric about the origin. One of the solutions is changing domain to [-10, 10].

More on Even and Odd FunctionsMore on Even and Odd FunctionsExample 2. Given part of the graph of an odd function. Find the complete graph of this function and state the domain and range.

-3-12 3 12

8

-8

5

-5

The complete graph as shown at the right. The domain of the this function is

[-12, -3] [3, 12]

The range of the this function is

[-8, 8]

Assignment

P. 128 #2 – 10 (even), 18 – 26 (even)

P. 136 #1 – 15 (odd)