1. Transformations and Parent Graphs

2. Transformations Change a graph in some way
Move it
Reflect it
Stretch it or compress it
The shape of the graph does not change

3. The following table describes changes to the graph of f(x)
How it transforms
(x,y)
f(x) + b
f(x) – b
f(x + b)
f(x – b)
-f(x)
f(-x)
bf(x)
f(bx)
Moves graph up
(x, y+b)
Moves graph down
(x, y-b)
Moves graph left
(x-b, y)
Moves graph right
(x+b, y)
Reflects over x-axis
(x, -y)
Reflects over y-axis
(-x, y)
Vertical stretch of b
(x, by)
Horizontal compression of 1/b
(1/b x, y)

4. Observations
Notice that if b is with the x, it does the opposite of what you would expect
Think of it as if you were solving what was in the parenthesis
Usually these transformations will happen more than one at a time

5. Example x y -3 -2 2 1 4
Label each (x,y) point and make a table

6. Example cont. Now graph each of the following transformations
a. f(x+3)
Left 3
(x-3,y)
x-3 y -6 -2 -5 2 -3 1

7. Example cont. Now graph each of the following transformations
b. f(x)+3
up 3
(x,y+3) x
y+3 -3 1 -2 5 3 4

8. Example cont. Now graph each of the following transformations (x,-3y)
c. -3f(x)
Reflect in x axis, vertical stretch by 3 x
-3y -3 6 -2 -6 1 4

9. Example cont. Now graph each of the following transformations
(- ½ x,y)
d. f(-2x)
Reflect in y axis, hor. compression by 1/2
- ½ x y
1.5 -2 1 2
- ½

10. Example cont. Now graph each of the following transformations
(- x,y+3)
e. f(-x)+3
Reflect in y axis, up 3
- x
y+3 3 1 2 5
- 1 -4

11. Example cont. Now graph each of the following transformations
(1/3x-2,2y-4)
f. 2f(3x+6)-4
vertical stretch 2, horizontal compression 1/3, left 2, down 4
1/3x-2
2y-4 -3 -8
-2.7 -2 -4
- 1.7
- .7
Careful – this one is tricky!!

12. Example cont. Now graph each of the following transformations
(-x -2,-y-1)
g. -f(-x-2)-1
Reflect in x axis, reflect in y axis, left 2, down 1
-x-2
-y-1 1 -3 -2 -1
- 6
One more tricky one!!

13. Parent graphs
There are many common functions that we call parent graphs that we need to know
You will need to memorize all their shapes and which points make sense to plug in
Today we will graph all of them
Tomorrow we will apply the tranformations we know to these parent graphs

14. Quadratic x y -2 -1 1 2 4 1 1 4

15. Cubic x y -2 -1 1 2 -8 -1 1 8

16. Square Root x y 1 4 1 2

17. Absolute Value x y -2 -1 1 2 2 1 1 2

18. Rational x y -2 -1 1 2 -1/2 -1 error 1 1/2
Has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0 x y -2 -1 1 2
-1/2 -1
error 1
1/2

19. Has a horizontal asymptote at y = 0
Exponential Growth
Has a horizontal asymptote at y = 0
Can be any number greater than 1 x y 1 1 2
Will match the base in the equation

20. Has a horizontal asymptote at y = 0
Exponential Decay
Has a horizontal asymptote at y = 0 x y -1
Can be any number less than 1 1 2
Will match the base in the equation

21. Has a vertical asymptote at x = 0
Log
Has a vertical asymptote at x = 0 x y 1 6
Can be any number except 0 1
Will match the base in the equation

22. Has a vertical asymptote at x = 0
Log
Has a vertical asymptote at x = 0 x y 1 6
Can be any number except 0 1
Will match the base in the equation

23. Has a vertical asymptote at x = 0
Natural Log
Has a vertical asymptote at x = 0 x y 1 e 1
Equals approximately 2.7

24. sin x y 1 -1

25. cos x y 1 -1 1