1. 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.
x-axis, y-axis, origin
none of these

2. Lesson 3-2 Families of Graphs
Objective: Identify transformations of simple graphs and sketch graphs of related functions.

3. 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.

4. Identity Functions
f(x) =x y always = whatever x is

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

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

7. Square Root Function
f(x)=

8. Absolute Value Function
f(x) =|x|

9. Greatest Integer Function (Step)
y=[[x]]

10. Rational Function
y=x-1 or 1/x

11. 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

12. Reflections
Parent Graph
y =x3
y=f(-x)
y=-f(x)

13. Translations y=f(x)+c moves the parent graph up c units
y=f(x) - c moves the parent graph down c units

14. Vertical Translations
Translations f(x) +c 2 4 6 8 x
y = f(x) -2 -4 -6
Vertical Translations
f(x) +2= x2 + 2
f(x) = x2
f(x) - 5 = x2 - 5

15. 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

16. Translations y =f(x+c)2 4 6 8 x
y = f(x) -2 -4 -6
Horizontal Translations 2 5
f(x - 5)
f(x)
f(x + 2)
In other words, ‘+’ inside the brackets means move to the LEFT

17. 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)

18. Translations y=cf(x) 3f(x) 2f(x) f(x)30
y = f(x)
y co-ordinates tripled
Stretches in the y direction
3f(x) 20
y co-ordinates doubled
2f(x)
f(x) 10
Points located on the x axis remain fixed. x -6 -4 -2 2 4 6 8
-10
The graph of cf(x) gives a stretch of f(x) by scale factor c in the y direction.
-20
-30

19. Translations y = cf(x);0<c<12 4 6 8 x
y = f(x) -2 -4 -6 10 20 30
-10
-20
-30
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 halved
y co-ordinates scaled by 1/3

20. Translations
y=f(cx); c>1 compresses the parent graph horizontally (narrows)
y=f(cx); 0<c<1 expands the parent graph horizontally (widens)

21. Translations y=f(cx) f(3x) f(2x) f(x)4 6 8 x
y = f(x) -2 -4 -6
f(3x)
f(2x)
f(x)
Stretches in x
1/3 the x co-ordinate
½ the x co-ordinate
The graph of f(cx) gives a stretch of f(x) by scale factor 1/c in the x direction.

22. Translations y=f(cx) f(1/3x) f(1/2x) f(x)4 6 8 x
y = f(x) -2 -4 -6
Stretches in x
f(1/3x)
f(1/2x)
f(x)
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 2
All x co-ordinates x 3

23. 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 = x 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.

24. 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.

25. 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.

26. 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.