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. 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.

Warm up

3694 22 yx

43

2 y

x

Lesson 3-2 Families of Graphs

Objective: Identify transformations of simple graphs and sketch graphs of

related functions.

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.

Family of graphs

Identity Functions

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

Constant Function

f(x) = c In this graph the domain is all real numbers but the range is c.

c

Polynomial Functions

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

Square Root Function

f(x)= x

Absolute Value Function

f(x) =|x|

Greatest Integer Function (Step)

y=[[x]]

y=x-1 or 1/x

Rational Function

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

Reflections

Reflections

Parent Graph y =x3

y=-f(x)

y=f(-x)

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

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

Translations

Translations f(x) +c

0 2 4 6 8

x

y = f(x)

-2-4-6

2

4

6

-2

-4

-6

f(x) = x2

f(x) +2= x2 + 2

f(x) - 5 = x2 - 5

Vertical Translations

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

Translations y =f(x+c)

2 4 6 8x

y = f(x)

-2-4-6

2

4

6

-2

-4

-6

Horizontal Translations

f(x)

f(x - 5)

f(x + 2)

52

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)

Translations y=cf(x)

2 4 6 8

x

y = f(x)

-2-4-6

10

20

30

-10

-20

-30

0

The graph of cf(x) gives a

stretch of f(x) by scale factor

c in the y direction.

f(x)

2f(x)

3f(x)

0Points located on the x axis remain

fixed.

Stretches in the y

directiony co-ordinates

doubled

y co-ordinates tripled

2 4 6 8x

y = f(x)

-2-4-6

10

20

30

-10

-20

-30

0

The graph of cf(x) gives a

stretch of f(x) by scale factor

c in the y direction.

f(x)

½f(x)

1/3f(x)

y co-ordinates

halvedy co-

ordinates scaled by 1/3

Translations y = cf(x);0<c<1

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

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

Translations

Translations y=f(cx)

½ the x co-ordinate

02 4 6 8

x

y = f(x)

-2-4-6

2

4

6

-2

-4

-6

f(x)

f(2x)

f(3x)

The graph of f(cx) gives a

stretch of f(x) by scale factor 1/c

in the x direction.

1/3 the x co-ordinate

Stretches in x

0

2 4 6 8x

y = f(x)

-2-4-6

2

4

6

-2

-4

-6

f(x)

f(1/2x)f(1/3x)

The graph of f(cx) gives a stretch of

f(x) by scale factor 1/c in the x

direction.

All x co-ordinates x 3

All x co-ordinates x 2

Stretches in x

Translations y=f(cx)

Even Functions

Consider all functions with a domain and range in the element of reals. Some of these have an interesting property. Namely, they make no distinction between negative and positive numbers.

For example, consider f(x) = x2. How does this function treat 3 vs. -3?

We can prove it makes no distinction between positive and negative numbers.

F(x) = (x)2 = x2 While F(-x) = (-x)2 = x2

Even Functions also have symmetry about the y-axis.

To say that f(−x) = f(x) for all x in domain of f, is equivalent to saying that a point, (x,y), is on the graph of f if and only if (−x,y) is also on the graph, which is also equivalent to saying that the graph is symmetric about the y axis.

ODD Functions

While even functions, by definition, map every x and −x to the same number, odd functions are defined to be those functions that map −x to the opposite of where x gets mapped to.

That is, f(−x) = −f(x) for all x in the set of real numbers.

Now consider how f treats positive and negative numbers. For example, how does f treat 5 as compared with −5?

f(5) = (5)3 = 125 while f(−5) = (−5)3 = −125

Yet another way to put it is that one can always ’factor’ out the negative thru the function as in the above example,f(−5) = −f(5)

Odd functions are symmetric in respect to the origin.

Algebraic Test - Evens

We can determine if the graph is even without actually graphing the equation by Substituting (-x) in to the original equation. If we can manipulate it back to the Original expression, it is an even function.

Example: |y| = 2 - |2x||y| = 2 - |2(-x)| (substitute (-x) in for x.)|y| =2- |2x| since |-2x| = |2x| - This is an Even Function

Example: xy = -2(-x)y = -2 (substitute (-x) in for x.)-xy = -2xy = 2 – THIS IS NOT AN EVEN FUNCTION.

Algebraic Test - Odds

We can determine if the graph is even without actually graphing the equation by Substituting (-x) in to the original equation. If we can manipulate it back to the Original expression, it is an even function.

Example: |y| = 2 - |2x||-y| = 2 - |2x| (substitute (-y) in for y.)|y| =2- |2x| since |-y| = |y| This is also an odd function

Example: xy = -2x(-y) = -2 (substitute (-y) in for y.)-xy = -2xy = 2 – THIS IS NOT AN ODD FUNCTION.