Download Unit 4 - Husky Math

UNIT 4: TRANSFORMATIONS AND
PIECEWISE FUNCTIONS
Final Exam Review
TOPICS TO COVER
Domain and Range
Transformations of Functions
Transformations of Points
Toolkit Functions
Piecewise Functions
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
DOMAIN AND RANGE
Example:
Domain: [-7, 5)
Range: [-3, 1)
DOMAIN AND RANGE
Now you try:
1.
2.
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
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
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
STRETCH AND SHRINK
Use these rules when stretching and shrinking
Stretch:
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
f(x)
𝟑
will shrink a function by
𝟐
𝟑
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
GIVEN THE FUNCTION, WRITE THE
TRANSFORMATIONS
Example
Answers
1. y = 5 f(x – 3)
1. Stretch 5, Right 3
2. y = -f(x) + 2
2. Reflect over the X axis, Up 2
3. y =
1
f(−x
3
+ 7) - 8
𝟏
,
𝟑
3. Shrink Reflect over the y
axis, Left 7, Down 8
GIVEN THE FUNCTION, WRITE THE
TRANSFORMATIONS
Now you try:
1. y = f(x – 4) – 3
2. y = 6 f(x + 2)
3. y = −
1
f(x)
3
+6
GIVEN THE TRANSFORMATIONS, WRITE THE
EQUATION
Example
Answers
1. Left 1, Up 9
1. y = f(x + 1) + 9
3
4
𝟑
𝟒
2. Shrink , Reflect over the x axis 2. y = - f(x)
3. Stretch 10, Right 4, Down 2
3. y = 10 f(x – 4) – 2
GIVEN THE TRANSFORMATIONS, WRITE THE
EQUATION
Now you try:
1. Right 7, Reflect over the y axis
2. Stretch 4, Left 2, Down 5
3. Reflect over the x axis, Shrink
4
,
5
Up 8
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
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
TRANSFORMING POINTS
Now you try:
Move the point (-4, 2) using the functions:
1. y = f(x) – 5
2. y = f(x + 1)
3. y = 3 f(x)
4. y = f(x – 2) + 9
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
1
RECIPROCAL FUNCTION y =
𝑥
ABSOLUTE VALUE FUNCTION y = 𝑥
EXPONENTIAL FUNCTION y = ex
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
TOOLKIT FUNCTIONS
Now you try:
Write the correct toolkit function and the transformations that
happened to it.
1. y = - (x + 1)3
2. y =
1
𝑥+4
−3
3. y = 6 𝑥 − 2 + 5
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
2𝑥 + 4, 𝑥 < −3
Example: 𝑓 𝑥 =
𝑥 − 5, 𝑥 ≥ −3
PIECEWISE FUNCTIONS
9𝑥 − 4, 𝑥 < 1
Example: 𝑓 𝑥 =
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) = 4 + 2 = 6
PIECEWISE FUNCTIONS
Now you try
5𝑥 + 3,
Example: 𝑓 𝑥 =
2𝑥 − 7,
Evaluate
1. f(-4)
2. f(6)
3. f(3)
𝑥<3
𝑥≥3
ALL DONE