1. Unit 4: Transformations and piecewise functions
Final Exam Review

2. Topics to cover Domain and Range Transformations of Functions
Transformations of Points
Toolkit Functions
Piecewise Functions

3. Domain and range
Domain and Range are important to be able to recognize in the graph of a function
Domain
X values
LEFT to RIGHT
Range
Y values
BOTTOM to TOP
Use Interval Notation
If there is an OPEN CIRCLE, use a PARENTHESIS
If there is a CLOSED CIRCLE, use a BRACKET
If there is an ARROW, use −∞ or ∞ and a PARENTHESIS

4. Domain and range
Example:
Domain: [-7, 5)
Range: [-3, 1)

5. Domain and range
Now you try:

6. Transformations of functions
You can move a function anywhere on a graph using transformations
You can go:
UP AND DOWN
LEFT AND RIGHT
STRETCH AND SHRINK
REFLECT OVER THE X AXIS AND Y AXIS

7. Transformations UP AND DOWN
Use these rules when translating a function up and down
Up:
f(x) + c
Example: y = f(x) + 4 will move a function Up 4
Down:
f(x) – c
Example: y = f(x) – 5 will move a function Down 5

8. Transformations left and right
Use these rules when translating a function left and right
Left:
f(x + c)
Example: y = f(x + 2) will move a function Left 4
Right:
f(x – c)
Example: y = f(x – 6) will move a function Right 6

9. Stretch and Shrink Use these rules when stretching and shrinking
c ∙ f(x) when c > 1
Example: y = 3 f(x) will stretch a function by 3
Shrink:
c ∙ f(x) when 0 < c < 1
Example: y = 𝟐 𝟑 f(x) will shrink a function by 𝟐 𝟑

10. Reflections over the x axis and y axis
Use these rules when reflecting a functions over the x axis or the y axis
X axis:
-f(x)
The negative is OUTSIDE of the function
Y axis:
f(-x)
The negative is INSIDE of the parenthesis with the x

11. Given the function, write the transformations
Example
y = 5 f(x – 3)
y = -f(x) + 2
y = 1 3 f(−x+7) - 8
Answers
Stretch 5, Right 3
Reflect over the X axis, Up 2
Shrink 𝟏 𝟑 , Reflect over the y axis, Left 7, Down 8

12. Given the function, write the transformations
Now you try:
y = f(x – 4) – 3
y = 6 f(x + 2)
y = − 1 3 f(x) + 6

13. Given the transformations, write the equation
Example
Left 1, Up 9
Shrink , Reflect over the x axis
Stretch 10, Right 4, Down 2
Answers
y = f(x + 1) + 9
y = - 𝟑 𝟒 f(x)
y = 10 f(x – 4) – 2

14. Given the transformations, write the equation
Now you try:
Right 7, Reflect over the y axis
Stretch 4, Left 2, Down 5
Reflect over the x axis, Shrink 4 5 , Up 8

15. Transforming points When transforming points, you must figure out:
If the transformation is affecting the X VALUE or the Y VALUE
Then ADD, SUBTRACT, or MULTIPLY the point by the correct value
UP AND DOWN will affect the y value
LEFT AND RIGHT will affect the x value
STRETCHING AND SHRINKING will affect the y value

16. Transforming points Example
Move the point (5, -2) using the transformations in the function y = f(x – 4)
Answer: (9, -2)
*This transformation moves the function right 4, so the x value is moved right 4 spaces

17. Transforming points Now you try:
Move the point (-4, 2) using the functions:
y = f(x) – 5
y = f(x + 1)
y = 3 f(x)
y = f(x – 2) + 9

18. Toolkit functions
Toolkit functions are the basic functions that can be transformed.
They are:
LINEAR FUNCTION y = x
QUADRATIC FUNCTION y = x2
CUBIC FUNCTION y = x3
SQUARE ROOT FUNCTION y = 𝑥
CUBE ROOT FUNCTION y = 3 𝑥
RECIPROCAL FUNCTION y = 1 𝑥
ABSOLUTE VALUE FUNCTION y = 𝑥
EXPONENTIAL FUNCTION y = ex

19. Toolkit functions Example:
Write the correct toolkit function and the transformations that happened
y = 4 (x + 3)2
Toolkit Function: y = x2
Transformations: Stretch 4, Left 3

20. Toolkit functions Now you try:
Write the correct toolkit function and the transformations that happened to it.
y = - (x + 1)3
y = 1 𝑥+4 −3
y = 6 𝑥 −2 +5

21. Piecewise functions
Piecewise Functions are functions that are in SEVERAL pieces
Each function has a restriction on its DOMAIN
Use this restriction in order to EVALUATE a function at certain points
Example: 𝑓 𝑥 = 2𝑥+4, 𝑥<−3 &𝑥 −5, 𝑥≥−3

22. Piecewise functions Example: 𝑓 𝑥 = 9𝑥−4, 𝑥<1 &𝑥+2, 𝑥≥1
Evaluate f(4)
Since 4 is greater than or equal to 1, you should plug 4 into the SECOND equation
Answer: f(4) = = 6

23. Piecewise functions Now you try
Example: 𝑓 𝑥 = 5𝑥+3, 𝑥<3 &2𝑥 −7, 𝑥≥3
Evaluate 1. f(-4)
2. f(6)
3. f(3)

24. ALL DONE