Section Section 4-34-3Reflecting Graphs; Reflecting Graphs;

SymmetrySymmetryObjective:

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

Introduction• In this section we will see the relationship

between a function’s equation and its graph.– When a function’s equation is altered, its graph will

predictably change

• We will start with the reflection of a graph.– What does the word reflection mean?

• mirror image

– Example:Definition:

Line of Reflection – located halfway between a point and its reflection

• Acts like a mirror

Where is the line of reflection?

• The Line of Symmetry (also called the Mirror Line) does not have to be up-down or left-right, it can be in any direction.

Introduction

These are the four most common lines of symmetry

IntroductionRecall:– When a function’s equation is altered, its graph will predictably change

• Let’s try some examples by graphing the following.– KEY - Look for a relationship between the function’s equation and its

graph.y = x2

y = - x2

y = x2 - 1

1 - x y 2

Reflection over the x-axis

Partial Reflection over the x-axis

y = 2x - 1

y = 2(-x) - 1

Reflection over the y-axis

x y

x- y

Reflection over the y-axis

Line of Reflection

Line of Reflection

Line of Reflection

Line of Reflection

Moral of the story – Small changes in an equation greatly change the graph

IntroductionRecall:– When a function’s equation is altered, its graph will predictably change

• Let’s try some examples by graphing the following.– KEY - Look for a relationship between the function’s equation and its

graph.y = x2

y = - x2

y = x2 - 1

1 - x y 2

Reflection over the x-axis

Partial Reflection over the x-axis

y = 2x - 1

y = 2(-x) - 1

Reflection over the y-axis

x y

x- y

Reflection over the y-axis

Line of Reflection

Line of Reflection

Line of Reflection

Line of Reflection

Moral of the story – Small changes in an equation greatly change the graph

Reflection in the x-axis• The graph of y = -f(x) is obtained by reflecting the graph of

y = f(x) in the x-axis.

Notice: the point (x,y) from f(x) (the original graph) becomes the point (x,-y) on –f(x) (the reflected graph)

y = f(x)

y = -f(x)

y = x2 - 3

y = -(x2 - 3)

Note: The graph of is identical to the graph of y = f(x) when f(x) ≥ 0 and is identical to the graph of y = -f(x) when f(x) < 0. We will see several examples:

)(xfy

y = f(x)

f(x)y

y = x2 - 3

3 - xy 2

Recall:

Recall:

• The graph y = f(-x) is obtained by reflecting the graph of y = f(x) in the y-axis.

Reflection in the y-axis

y = 1.5x

y = f(x)y = 1.5-x

y = f(-x)

Notice: the point (x,y) from f(x) (the original graph) becomes the point (-x,y) on f(-x) (the reflected graph)

y = (x + 3)2

y = f(x)y = (-x + 3)2

y = f(-x)

Recall:

Reflection in the Line y = x• Reflecting the graph of an equation in the line y = x is

equivalent to interchanging x and y in the equation.

y = x2

Original graph and equation Reflection in the line y = x

x = y2

Reflected graph and altered equation

(switched x and y)

Reflection in the Line y = x• Reflecting the graph of an equation in the line y = x is

equivalent to interchanging x and y in the equation.

y = x2

Original graph and equation Reflection in the line y = x

x = y2

Reflected graph and altered equation

(switched x and y)

SymmetrySymmetry• A line l is called an axis of symmetry of a graph if it is possible

to pair the points of a graph in such a way that l is the perpendicular bisector of the segment and the joining pair.Ex)

l = axis of symmetry

Recall:

Let’s try a few:

1. The graph of y = f(x) is shown at the right. Sketch the graph of each of the following equations. y = f(x)

a) y = -f(x)

y = -f(x)

y = f(x)

b) y = f(x)

y = f(x)

y = f(x)c) y = f(-x)

y = f(x)

y = f(-x)

More Examples

• Page 135 #2 and #3

Homework• p136-137: 1-4 (all), 6-30 (multiples of three),

31, 38 (increasing and decreasing functions)

• Extra Credit: 32, 33, 35