1. We will determine1 the graphs of Absolute Value2 Functions and their Transformations.
LEARNING OBJECTIVE
Declare the Objective
A: Read the Objective to B.
B: Define Absolute Value to A
Definition
figure out
the distance a number is from zero (in either direction)
CFU
What are we going to learn today? What is “Absolute Value” mean?
CASS: F-IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
Also F-BF.3 MP.8 Patterns

2. ACTIVATE PRIOR KNOWLEDGE
Remember the Concept
The graph of the absolute value function f(x) =|x|is called the parent graph and is in the shape of a “V”.
Plot and connect these points on a graph. 1 2
ACTIVATE PRIOR KNOWLEDGE
Make the Connection
Students, you already know how to graph the absolute value functions. Today, we will learn how to use the vertex form to transform the graph on the Cartesian coordinate plane.

3. f(x) = a|x – h| + k Absolute Value Function
Vertex from is a way to write the function in a “standard form". The vertex form can help us quickly identify and visualize the effects when graphing a function.
Check for Understanding
B Explain to A: What are the effects of changes in the values of a, h, and k on the graph of f(x) = a|x – h| + k?
Absolute Value Function
h-opposite
sign
On your white board, identify the transformations of f(x)=|x|+3? How did you identify it?
3 units left
3 units right
3 units up
3 units down
The __ represents ____ and translate the function____
CFU
On your white board, identify the transformations of f(x)=|x-2|+3? How did you identify it?
2 units right and 3 units up
2 units up 3 units right
2 units left 3 units up
2 units down 3 units down
CFU
f(x) = a|x – h| + k
k-same
sign
h is the
x-coordinate
of the vertex point
(remember the formula
has a negative sign in it)
k is the
y-coordinate
of the vertex point
CONCEPT DEVELOPMENT
a is the
“slope” of the line
Stretches/Compresses
Makes the graph
Narrow or Wide
Translates
up or down
Translates
Left or Right
Move in opposite
direction
*The vertex (edge) is located at (h, k)
*The axis of symmetry is the line x = h.

4. How did I/you identify the vertex?
How did I/you identify the slope?
How did I/you Graph the two lines?
CFU 1 2 3
Identify & Plot the Vertex of the graph (h, k). Hint: h has the opposite sign.
If “a” is positive, V opens Upward. If “a’ is negative, V opens downward.
Use the slope(a) to find point(s) to the left and right of the vertex. Hint: negative slope
Connect the dots, forming a "V“ for the left.
Steps for graphing Absolute Value Functions f(x) = |x| 1 2 3 4
Example Graph: 1 _ 2 1 3
On your white board, identify the transformations of f(x)=2|x+3|-4? How did you identify it?
What are the coordinates of the vertex of f(x)?
What is the slope of the Right Line?
What is the slope of the Left Line?
Extra: Graph f(x) and describe the effects from parent function |x|?
CFU
Vertex (h, k): (- , )
Parent Function f(x)= |x|
CONCEPT DEVELOPMENT
Slope(a): 1 _ 2
Right:
Vertex
Left(-): -
Definition
Domain: How far it extends on the x-axis.
Range: How far it extends on the y-axis.

5. SKILL DEVELOPMENT / GUIDED PRACTICE
How did I/you identify the vertex?
How did I/you identify the slope?
How did I/you Graph the two lines?
CFU 1 2 3
Identify & Plot the Vertex of the graph (h, k). Hint: h has the opposite sign.
If “a” is positive, V opens Upward. If “a’ is negative, V opens downward.
Use the slope(a) to find point(s) to the left and right of the vertex. Hint: negative slope
Connect the dots, forming a "V“ for the left.
Steps for graphing Absolute Value Functions f(x) = |x| 1 2 3 4 1 2
Remember the Concept
Vertex From
Vertex (h, k): (- , ) 1 3
Vertex (h, k): ( , ) 1 _ 2 1 _ 2
Slope(a):
Slope(a):
Right:
Left(-): -
Right:
Left(-): -
SKILL DEVELOPMENT / GUIDED PRACTICE
Definition
Domain: How far it extends on the x-axis.
Range: How far it extends on the y-axis.
Vertex
a = 1/2
a = -1/2
Remember the Concept
An axis of symmetry of the graph of a function is a vertical line that divides the graph into mirror images.

6. SKILL DEVELOPMENT / GUIDED PRACTICE
How did I/you identify the vertex?
How did I/you identify the slope?
How did I/you write the function f(x)?
CFU 1 2 3
Identify the Vertex of the graph (h, k). Hint: h has the opposite sign.
If “a” is positive, V opens Upward. If “a’ is negative, V opens downward.
Find the slope(a) of the right side line.
Plug the values into
Write an absolute value function from a graph of the function. 1 2 3 4
f(x) = a|x – h| + k 1 2
f(x) = |x – 2| + 3
Remember the Concept
Vertex From
Vertex (h, k):
Vertex (h, k):
Slope of Right Line (a): 2
Slope(a):
Right: 1
SKILL DEVELOPMENT / GUIDED PRACTICE
(4, 5) 2
(2, 3) 2
(2, 0) 4
(1,
-2) 2

7. SKILL DEVELOPMENT / GUIDED PRACTICE
How did I/you identify the vertex?
How did I/you identify the slope?
How did I/you Graph the two lines?
CFU 1 2 3
Identify & Plot the Vertex of the graph (h, k). Hint: h has the opposite sign.
If “a” is positive, V opens Upward. If “a’ is negative, V opens downward.
Use the slope(a) to find point(s) to the left and right of the vertex. Hint: negative slope
Connect the dots, forming a "V“ for the left.
Steps for graphing Absolute Value Functions f(x) = |x| 1 2 3 4 1 2
Remember the Concept
Vertex From
Vertex (h, k):
Definition
Domain: How far it extends on the x-axis.
Range: How far it extends on the y-axis.
Slope(a): m
SKILL DEVELOPMENT / GUIDED PRACTICE

8. d(x) = -5|x – 2| + 10, 4 hours Turlock Vertex (2, 10):
Relevance Reason #1: Absolute values are often used in problems involving distance:
Suppose you plan to ride your bicycle from Delhi to Turlock, and back to Delhi. The distance between Turlock and Delhi is 10 miles. You plan to ride 5 miles an hour. Write an absolute value function d(x), where x is the number of hours into the ride, that describes your distance from Turlock and use your function to determine the number of hours it will take to complete your ride.
Miles(y)
a = -5
Delhi
Delhi
Hours(x)
RELEVANCE
Relevance Reason #1: Know how to graph absolute value functions will help you do well on tests (PSAT, SAT, ACT, GRE, GMAT, LSAT, etc..).
d(x) = -5|x – 2| + 10, 4 hours
Sample Item
Check for Understanding
Does anyone else have another reason why it is relevant to use verb tense correctly?
Which reason is most relevant to you? Why?
Which statement is true about the graph of the function f(x)= -|x + 2| + 3 ?
Its vertex is at (2,3)
Its vertex is at (-2,-3)
It opens down
It is wider than the graph of y=|x|.

9. SKILL/CONCEPT CLOSURE

10. What are the effects of parameter a, h, & k changes on the graph of f(x) = a|x – h| + k ?
Word Bank
horizontal shift
vertical shift
stretch
shrink
Use: Changes to h result in a ________ in the graph; changes to k result in a ________ in the graph; and changes to a result in a vertical ______, or ______, of the graph.
SUMMARY CLOSURE
Use: Changes to h result in a horizontal shift in the graph; changes to k result in a vertical shift in the graph; and changes to a result in a vertical stretch, or shrink, of the graph.

11. What did you learn today about graph of absolute value functions and their transformations?
Word Bank
Absolute-value function
Axis of symmetry
Vertex Form
Vertex
Parent Function
Domain & Range
SUMMARY CLOSURE

12. INDEPENDENT PRACTICE

13. INDEPENDENT PRACTICE