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. 4 x 2  9 y 2  36
 2. 2 x  y  4
3
Lesson 3-2 Families of Graphs
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
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]]
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
Reflections
y=f(-x)
Parent Graph
y =x3
y=-f(x)
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
Vertical
Translations
y = f(x)
f(x) +2= x2 + 2
4
f(x) = x2
-6
-4
2
0
-2
-2
f(x) - 5 =
x2
-5
-4
-6
2
4
6
x
8
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)
6 y = f(x)
Horizontal
Translations
4
2
2
5
f(x - 5)
f(x)
-6
-4
2
-2
f(x + 2)
In other words, ‘+’
inside the brackets
means move to the LEFT
-2
-4
-6
4
6
x
8
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)
30 y = f(x)
Stretches in the
y direction
3f(x)
20
2f(x)
-4
-2
y co-ordinates
doubled
f(x)
10
-6
y co-ordinates
tripled
2
0
4
6
0
-10
The graph of
cf(x) gives a
stretch of f(x) by
scale factor c in
the y direction.
-20
-30
Points located on the x
axis remain fixed.
x
8
Translations y = cf(x);0<c<1
30 y = f(x)
y co-ordinates
halved
20
½f(x)
y co-ordinates
scaled by 1/3
10
-6
-4
1/3f(x)
-2
The graph of cf(x)
gives a stretch of
f(x) by scale
factor c in the y
direction.
0
-10
-20
-30
f(x)
2
4
6
x
8
Translations
 y=f(cx); c>1 compresses the parent
graph horizontally (narrows)
 y=f(cx); 0<c<1 expands the parent
graph horizontally (widens)
Translations y=f(cx)
6 y = f(x)
f(3x)
Stretches in x
f(2x)
f(x)
4
2
0
-6
-4
2
-2
The graph of f(cx)
-2
gives a stretch of
f(x) by scale factor
1/c in the x direction.
-4
-6
4
6
½ the x co-ordinate
1/3 the x co-ordinate
x
8
Translations y=f(cx)
6 y = f(x)
Stretches in x
f(1/3x)
f(x)
f(1/2x)
4
2
0
-6
-4
2
-2
The graph of f(cx)
gives a stretch of f(x)
by scale factor 1/c in
the x direction.
-2
4
6
All x co-ordinates x 2
All x co-ordinates x 3
-4
-6
x
8
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 = -2
xy = 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 = -2
x(-y) = -2 (substitute (-y) in for y.)
-xy = -2
xy = 2 – THIS IS NOT AN ODD FUNCTION.