Download Class Notes 2 - Graphing and Writing Absolute Value Functions

Graph and Write Absolute Value Equations
f ( x)  x
What are the key attributes of a Function and define?
Key attributes:
Domain:
Range:
Intercepts:
Symmetry:
Minimum:
Graph the absolute value parent function (2A.2A)
First we will us a table to graph the absolute value linear parent function. Consider all real numbers.
f ( x)  x
Let’s Discuss the Table
Example:
Input x
Process:
Output f(x)
-8
8
8
What would happen if the input was not an
integer?
1
2
What decisions can you make from the pattern
you discovered?
What would this look like on a graph?
3
1
2
3
1
2
3
0
0
0
-122.96
 122.96
122.96
Set Notation and Interval Notation in relation to Domain and Range:
Inequality/ Previous notation
All real numbers
x0
x
-2
-1
0
1
2
f(x)
2
1
0
1
2
Set notation
Interval notation
Read “All numbers x such that x
belongs to the set of real numbers”
“from negative infinity to positive
infinity”
Read “all numbers x such that x is
greater than or equal to 0”
“from 0 to positive infinity, including
0”
x x  
x x  0
Parent Graph: f x   x
 , 
0, 
Key attributes:
Domain:
Range:
Intercepts:
Symmetry:
Minimum:
How is this like something you have done before?
How is the absolute value graph similar and different
to the quadratic?
1
How is the absolute value graph similar and different
to the linear?
Table
x
y
1
4
2
2
3
0
Graph
6
2
f x   3 x  2  1
7
4
What is the minimum value of
this absolute value function?
What is the maximum value on
the interval, 1,5 ?
Equation
What is the axis of symmetry?
What is the domain of this
absolute value function?
What are the x-intercepts?
What is the range?
What is the minimum value on
the interval  9,5 ?
You Try1 Using the graph on the side answer the questions below:
What is the minimum value? ________________
What is the axis of symmetry? _______________
What are the x-intercepts? __________________
What is the domain? _______________________
What is the range? _________________________
What is the y-intercept? _____________________
You Try 2 Using the graph to answer the questions below:
What is the minimum value? ________________
What is the axis of symmetry? _______________
What are the x-intercepts? __________________
What is the domain? _______________________
What is the range? _________________________
What is the y-intercept? _____________________
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What is the y-intercept?
Class Activity
 Open the TI-Nspire document Exploring AbsoluteValue Transformations

Press /
¢ to move to page 1.2 and begin the lesson
1. Write the vertex form of an absolute value function. _______________________________.
Note in this activity b is always assumed to be 1
2. Observe the characteristics of the absolute value
parent function on page 1.2.
List the characteristics observed:
________________________________________
Exploring “a.”
3. Increase and decrease the value of “a.” Describe what is happening to the function.
__________________________________________________________________________________________________
__________________________________________________________________________________________________
__________________________________________________________________________________________________
4. Complete the statements below.
When “a” positive, the function____________________________________________________
_____________________________________________________________________________.
Therefore, when “a” is positive, the function has a ____________________________________.
(Maximum or Minimum)
When “a” negative, the function ___________________________________________________
_____________________________________________________________________________.
Therefore, when “a” is negative, the function has a ____________________________________.
(Maximum or Minimum)
5. What happens when a = 0 and -1 < a < 1? _________________________________________
_____________________________________________________________________________.
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Exploring “h.”
6. Increase and decrease the value of “h.” Describe what is happening to the function.
The function moves _____________________________________________________________.
7. Complete the statements below.
When “h” is positive, the function____________________________________________________.
When “h” is negative, the function ___________________________________________________.
Exploring “k.”
8. Increase and decrease the value of “k.” Describe what is happening to the function.
The function moves_____________________________________________________________.
9. Complete the statements below.
When “k” positive, the function ___________________________________________________.
When “k” negative, the function ___________________________________________________.
10. Use your TI-Nspire to discover how to find the Vertex?
Fill in the chart:
Parameters: a = 1
h=0
k=0
This is called the parent functions.
Vertex form: y  1 x  0  0
Simplify
Parameters: a = .5
h = -3
k=0
Parameters: a = 2
h=1
k = 2.5
1
3
h = -2.3
k = -1.5
Parameters: a = -
y x
Identify the coordinates of the minimum. ( , )
How did the function move?
Vertex form:
Identify the coordinates of the minimum. ( , )
How did the function move?
Vertex form:
Identify the coordinates of the minimum. ( , )
How did the function move?
Vertex form:
Identify the coordinates of the minimum. ( , )
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Vertical Dilation: f x   a x Explore the difference between a vertical stretch and a vertical compression.
Discover:
a > 1 results in a ________ stretch
(graph gets ________________ )
0  a  1results in a _____________
compression (graph gets ___________)
What characteristic of the graph does “a” also seem to describe?
It seems that “a” defines the ______________ of the right leg of the graph
Horizontal Dilation: f x   bx Explore the difference between a horizontal stretch and a horizontal compression.
Discover:
b  1results in a _____________
compression (graph gets ___________)
0  b  1 results in a ________stretch
(graph gets ___________)
Is there another way you could write the equation of this graph if the value of “b” was set equal to 1?
It seems that value of __________ can be moved outside the |
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| and set the the value of _____________
Reflections: f x    x versus f x    x Explore Horizontal and Vertical reflections.
What would happen if a was negative?
What would happen if b was negative?
Discover:
a  0 results in a reflection over the
___-axis (graph opens down)
b  0 results in a reflection over the
___-axis (graph looks the same due to
symmetry)
Think-Pair-Share Activity
How is the result of f x    x similar to something you have done before?
Why does there appear to be no change with f x    x when compared to f x   x ?
This must mean that if the value “b” is brought outside the |
setting it equal to “a”
| we must first take its _______ value before
Horizontal Translation: f x   x  h Explore the horizontal shifts using positive and negative values for h.
Discover:
h  0 results in a ______ to the ____
(the sign remains negative)
h  0 results in a ____ to the ____
(the sign becomes positive)
What do you notice about the horizontal translation and the sign?
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Vertical Translation: f x  k Explore the vertical shifts using positive and negative values for k.
Discover:
k  0 results in a _______shift ___
(the sign is ___________)
k  0 results in a _______ shift ___
(the sign is ___________)
I Do Example:
Inequality
f x   2 x  1  5
Domain
Describe the transformation:
a: ____________________________
b: ____________________________
h: ____________________________
k: ____________________________
Range
Identify the Key Attributes:
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Set notation
Interval
notation
We Do: x  3  7
Describe the transformation:
a: ____________________________
b: ____________________________
h: ____________________________
k: ____________________________
You do 1: y  2 x  6  5 Describe the transformation and Idenitfy the key attributes of the function.
Transforms 1) _______________________ 2) _______________________ 3) __________________________
Identify the Key Attributes:
_____________________________
_____________________________
_____________________________
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You do 2: f ( x)   x  2  4 Describe the transformation and Idenitfy the key attributes of the function.
Transforms 1) _______________________ 2) _______________________ 3) __________________________
Identify the Key Attributes:
_____________________________
_____________________________
_____________________________
You do 3: f ( x)   x  2  1 Describe the transformation and Idenitfy the key attributes of the function.
Transforms 1) _______________________ 2) _______________________ 3) __________________________
Identify the Key Attributes:
_____________________________
_____________________________
_____________________________
Key attributes based on the Vertex:
Which transformations affect domain?
Which transformations affect range?
Which transformations affect the axis of symmetry? 9
Which transformations affect whether the graph has a maximum or a minimum?
How can the maximum or minimum be determined from the equation?
Solve for x-intercepts and y-intercepts
How can we verify the y-int.
algebraically?
What does a y-int. mean?
Notice from the graph of
f x   2 x  1  5 above, it has a y-intercept at __________?
f 0  2 0  1  5
Let
 2 1  5
x  0 and evaluate
 21  5
7
Notice that the graph of f x   2 x  1  5 does not cross the ____-axis so when solving algebraically we should get __.
2 x 1  5  0
Let y  0 and solve
2 x  1  5
x 1  
Notice the absolute
value is negative so
there is no solution.
5
2
What do we notice about the graph and
x-intercepts?
Explain the relationship between the
algebraic and graphic method for
finding intercepts.
You Try: Determine the min/max, x&y intercepts:
𝟏
𝟐
1. Y = -|x+4| + 4
2. Y = |x-2| + 4
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Graph absolute value with transformations (2A.6C): f x   a b( x  h)  k
WHAT HAPPENS WHEN B DOES NOT EQUAL 1 ?
Are the b and h variables easily identifiable in the equation below?
What are the transformations?
f  x   3x  6
What is the vertex?
What is the axis of symmetry?
How can we make b and c easily identifiable?
How is it related to the quadratic with
the same transformations?
IN ORDER TO DETERMINE h FOR VALUES OF B ≠ 1 YOU MUST PUT THE ABSOULTE VALUE EQUATION INTO VERTEX
FORM:
f(x) = a|b(x-h)| + k
 vertex form for absolute value
Step 1: Put parentheses around the argument: 
f x  (3x  6)  2

f x  3( x  2)  2
Step 2: Factor out b from the parentheses:
or
f(x) = 3|x+2| +2
We Do: Determine the x & y intercepts, the horizontal & vertical shifts and the min or max of the function:
f(x) = -2|4x -12| + 3
y intercept = f(0) = ______________________
x intercept(s) by solving 0 = ____________________
Vertical shift = k = _______________________
Horizontal shift = h  f(x) = -___|___(x-____)| + 3
Or f(x) = _______________________
Min or Max occurs at the vertex = (h,k) = (_____,_____) if a is negative then the vertex is a ________________
Axis of symmetry: x = ___________________________
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You Try: Determine the x & y intercepts, the horizontal & vertical shifts and the min or max of the function:
f(x) = 2|3x + 9| -4
y intercept = f(0) = ______________________
x intercept(s) by solving 0 = ____________________
Vertical shift = k = _______________________
Horizontal shift = h  f(x) = -|___(x-____)| - ______
Min or Max occurs at the vertex = (h,k) = (_____,_____) if a is _________ then the vertex is a ________________
Axis of symmetry: x = ___________________________
Graph the function:
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Practice Problems
I. For each equation: state the transformations, domain, range and intercepts and graph the equation.
1
x 3
2
1) y  x  4  5
2) f ( x ) 
Trans:
Trans:
Trans:
D: _______ R:________
x-int:________________
y-int:________________
Axis of sym:__________
D: _______ R:________
x-int:________________
y-int:________________
Axis of sym:__________
D: _______ R:________
x-int:________________
y-int:________________
Axis of sym:__________
5) y   x  3  6
6) f ( x)  2 x  1  3
Trans:
Trans:
Trans:
D: _______ R:________
x-int:________________
y-int:________________
Axis of sym:__________
D: _______ R:________
x-int:________________
y-int:________________
Axis of sym:__________
D: _______ R:________
x-int:________________
y-int:________________
Axis of sym:__________
4) f ( x )  
1
x3
2
3) y  2 x  10  1
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Writing equations from a graph:
How is the graph different from the parent?
Which transformations would cause each change?
Opens down (___________), Slope of  2 (__________), vertex at ______
(moves ______ 3 and ____ 4)
How can we represent the changes in an equation?
f x   2 x  3  4 or f x    2x  3  4 or f x    2 x  6  4
How would the graph of a quadratic look with the same transformations?
What is the linear equation of each “piece” of the graph? How do the linear
equations relate to the absolute value equation?
Steps in developing equations from Graphs: (This applies to any function we will study):
1. Identify the function type and graph its parent function. In our case f(x) = |x|
2. Compare the function graph with the parent function graph and list the transforms that are apparent:
- Magnitude of the slope of the branches? This will be the “a” value
- Reflection across the x axis: Yes, we negate “a”
- How has the x coordinate of the vertex changed from the parent (size & direction) this will be “h”
- -How has the y coordinate of the vertex changed from the parent (size & direction) this will be “k”
3. Plug in the values of a, h, and k into the vertex form of the function
f(x) = a|b(x-h)| + k
Note: b will always be 1 with this process
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Example We Do: Determine the equation that defines the graph below:
Step1: The graph above is a transform of the parent function f(x) = ________________
Step 2. Magnitude of the slope of the branches = _______________ so “a” seems to be ______________
Step 3: Relection on x? _____________ so the sign of “a” _________ change so “a” = ____________
Apparent horizontal shift = _____________ to the ____________ so h = _____________
Apparent verical shift = _____________ ____________ so k = _____________
Step 4: The f(x) = ____(x - ________) + _________
You Do: Determine the equation that defines the graph below:
Step1: The graph above is a transform of the parent function f(x) = ________________
Step 2. Magnitude of the slope of the branches = _______________ so “a” seems to be ______________
Step 3: Reflection on x? _____________ so the sign of “a” _________ and “a” = ________________
Apparent horizontal shift = _____________ to the ____________ so h = _____________
Apparent vertical shift = _____________ ____________ so k = _____________
Step 4: The f(x) = ____|x - ________| + _________
You Do: Determine the equation that defines the graph below:
Step1: The graph above is a transform of the parent function f(x) = ________________
Step 2. Magnitude of the slope of the branches = _______________ so “a” seems to be ______________
Step 3: Relection on x? _____________ so the sign of “a” _________ and “a” = ________________
Apparent horizontal shift = _____________ to the ____________ so h = _____________
Apparent verical shift = _____________ ____________ so k = _____________
Step 4: The f(x) = ____|x - ________| + _________
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Practice Problems:
Determine the equation for the following graphs when b = 1 and an equivalent equation when b≠1: Verify you answer by
graphing on the Nspire.
A
B
C
D
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