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### Lesson 3-2 Families of Graphs

Warm up

Determine whether the graph of each equation is symmetric wrt the x-axis, the y-axis, the line y = x the line y = -x or none.

1.

2.

Objective: Identify transformations of simple graphs and sketch graphs of related functions.

Family of graphs

A family of graphs is a group of graphs that displays 1 or more similar characteristics.

Parent graph – the anchor graph from which the other graphs in the family are derived.

Identity Functions

- f(x) =x y always = whatever x is

Polynomial Functions

f(x) = x2 The graph is a parabola.

Square Root Function

- f(x)=

Absolute Value Function

- f(x) =|x|

Greatest Integer Function (Step)

y=[[x]]

Rational Function

y=x-1 or 1/x

Reflections

- A reflection is a “flip” of the parent graph.
- If y = f(x) is the parent graph:
- y = -f(x) is a reflection over the x-axis
- y =f(-x) is a reflection over the y-axis

Translations

y=f(x)+c moves the parent graph up c units

y=f(x) - c moves the parent graph down c units

Translations f(x) +c

6

y = f(x)

4

2

x

4

-6

2

8

6

-4

-2

-2

-4

-6

Vertical Translations

f(x) +2= x2 + 2

f(x) = x2

0

f(x) - 5 = x2 - 5

Translations

y=f(x+c) moves the parent graph to the left c units

y=f(x – c) moves the parent graph to the right c units

y = f(x)

4

2

x

4

-6

2

8

6

-4

-2

-2

-4

-6

Translations y =f(x+c)Horizontal Translations

5

2

f(x - 5)

f(x)

f(x + 2)

In other words, ‘+’ inside the brackets means move to the LEFT

Translations

y=c •f(x); c>1 expands the parent graph vertically (narrows)

y=c •f(x); 0<c<1 compresses the parent graph vertically (widens)

y co-ordinates doubled

Points located on the x axis remain fixed.

Translations y=cf(x)30

y = f(x)

Stretches in the y direction

3f(x)

20

2f(x)

f(x)

10

x

4

-6

2

8

0

6

-4

-2

0

-10

The graph of cf(x) gives a stretch of f(x) by scale factor cin the y direction.

-20

-30

y co-ordinates scaled by 1/3

Translations y = cf(x);0<c<130

y = f(x)

1/3f(x)

20

½f(x)

f(x)

10

x

4

-6

2

8

0

6

-4

-2

-10

The graph of cf(x) gives a stretch of f(x) by scale factor cin the y direction.

-20

-30

Translations

y=f(cx); c>1 compresses the parent graph horizontally (narrows)

y=f(cx); 0<c<1 expands the parent graph horizontally (widens)

y = f(x)

4

2

x

4

-6

2

8

6

-4

-2

-2

½ the x co-ordinate

1/3 the x co-ordinate

-4

-6

Translations y=f(cx)f(2x)

f(3x)

f(x)

Stretches in x

0

The graph of f(cx) gives a stretch of f(x) by scale factor 1/cin the x direction.

y = f(x)

4

2

x

4

-6

2

8

6

-4

-2

-2

-4

-6

Translations y=f(cx)Stretches in x

f(1/3x)

f(x)

f(1/2x)

0

The graph of f(cx) gives a stretch of f(x) by scale factor 1/cin the x direction.

All x co-ordinates x 2

All x co-ordinates x 3

Sources

http://mrstevensonmaths.wordpress.com/2011/02/07/transformation-of-graphs-2/; August 9,2013

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