use absolute value functions and transformations objectives: 1.to evaluate, write, and graph...
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Use Absolute Value Functions and Transformations
Objectives:
1. To evaluate, write, and graph piecewise functions
2. To graph an absolute value function by performing SRT transformations on the parent
3. To apply SRT transformations to graphing any function
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Objective 1You will be able to
evaluate, write, and graph piecewise
functions
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Exercise 1
Determine whether the graph shown represents a function.
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Piecewise Functions
A piecewise function is defined by more than one equation. Each equation corresponds to a different part of the domain of the function.
1 if,3
12 if,1
2 if,12
3
)(
x
xx
xx
xf
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Exercise 2
Evaluate g(x) at the values below.
1. g(1)
2. g(5)
3. g(−3)
1 if,13
1 if,12)(
xx
xxxg
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Exercise 3
Graph g(x).
1 if,13
1 if,12)(
xx
xxxg
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Graphing Piecewise Functions
Method 1:
1. Rather than starting at the -intercept, start at the domain’s breaking point. Use the slope to graph the partial line in the correct direction.
2. Repeat for each piece of your function.
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Graphing Piecewise Functions
Method 2:
1. Graph one of the equations in the piecewise function as you normally would.
2. Erase the part of the graph that you don’t need according to the domain of the piece.
3. Repeat for each piece of your function.
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Exercise 4
Graph
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Exercise 4
Write a piecewise function for the graph shown.
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Parent Function:
The simplest member of a family of functions
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Parent Functions
Who is the simplest member of your family? Well, in math, the simplest member of a family of functions is called the parent function.
Family of Linear Functions Linear Parent Function
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Parent Functions
Who is the simplest member of your family? Well, in math, the simplest member of a family of functions is called the parent function.
Family of Quadratic Functions Quadratic Parent Function
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Parent Functions
Who is the simplest member of your family? Well, in math, the simplest member of a family of functions is called the parent function.
Family of Absolute Value Functions Absolute Value Parent Function
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Parent Functions
Who is the simplest member of your family? Well, in math, the simplest member of a family of functions is called the parent function.
Family of Functions Parent Function
A group of functions that share common characteristics
Simplest member of the family
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Building the Absolute Value Function
The absolute value function is defined by .
The graph of the absolute value function is similar to the linear parent function, except it must always be positive.
4
2
-2
-4
-5 5
4
2
-2
-4
-5 5
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4
2
-2
-4
-5 5
Building the Absolute Value Function
The absolute value function is defined by .
So we just take the negative portion of the graph and reflect it across the -axis making that part positive.
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Building the Absolute Value Function
The absolute value function is defined by .
This is the absolute value
parent function.
4
2
-2
-4
-5 5
f x = x
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Parent Function
V-ShapeThe vertex is the minimum pointSymmetric
about the -axis
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Objective 2
You will be able to graph absolute value functions using SRT transformations on the parent function.
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Objective 2a
You will be able to perform vertical and horizontal shifts on the graph of a function
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Translation
A translation is a transformation that
shifts a graph horizontally or vertically, but
doesn’t change the overall shape
or orientation.
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Translation
The graph of
is the graph of translated horizontal units and vertical units.
The new vertex is at
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Yo
u w
ill be
able to vertically
scale the
grap
h o
f
a fu
nction
Objective 2b
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Stretching and Shrinking
The graph of is graph of vertically stretched or shrunk depending on the .
The value of “” gets multiplied by each -value.
For For
• The graph is stretched vertically• The graph of is narrower than
the graph of
• The graph is shrunk vertically• The graph of is wider than
the graph of
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Objective 2c
You will be able to reflect the graph of a function across the x-axis
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Reflection
The graph of is graph of reflected across the -axis when .
4
2
-2
-4
-5 5
f x = x
4
2
-2
-4
-5 5
f x = - x
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Minuses and Pluses
You might be wondering why the ’s get a minus sign while the ’s get a plus sign. That’s just because the is on the wrong side of the equation.
𝑦=𝑎∨𝑥−h∨+𝒌 𝑦−𝒌=𝑎∨𝑥− h∨¿
𝑦− 𝑦1=𝑚 (𝑥−𝑥1 )Just think of the Point-Slope Form:
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Minuses and Pluses
You might be wondering why the ’s get a minus sign while the ’s get a plus sign. That’s just because the is on the wrong side of the equation.
𝑦=3∨𝑥− 2∨+𝟓 𝑦−𝟓=3∨𝑥− 2∨¿
𝑦−7=6 (𝑥− 4 )Just think of the Point-Slope Form:
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Just remember that your ’s always lie!
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Objective 2
You will be able to graph absolute value functions using SRT transformations on the parent function.
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Multiple Transformations
Here are two methods that you could use to graph and absolute value function. The first method will only work for absolute value functions. The second is more general and will work for any function.
In general, the graph of an absolute value function of the form can involve translations,
reflections, stretches or shrinks.
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Step 4Step 3Step 2Step 1
Graphing Absolute Value Functions
Method 1: Vertex Method
Graphing is easy:
Plot vertex
Use value
as “slope” to plot
another point
Use symmetry
to find reflected
point
Connect the dots in a V-shape
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Exercise 6
Without a graphing calculator, graph the following functions. How do they compare to the parent function?
1.
2.
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Exercise 6
1. 2.
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Objective 3 You will be able to use SRT transformations to graph any function
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Multiple Transformations: SRT
Method 2: SRT Transformations
To graph , start with points on the parent function :
S R TScaling: Multiply -values
by
Reflecting: If is
negative, flip over -
axis
Translating: Move
left/right for ,
up/down for
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Exercise 7
The graph of is shown. Sketch the graph of the given function.
1.
2.
3.
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Exercise 7
1.
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Exercise 7
2.
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Exercise 7
3.
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Use Absolute Value Functions and Transformations
Objectives:
1. To evaluate, write, and graph piecewise functions
2. To graph an absolute value function by performing SRT transformations on the parent
3. To apply SRT transformations to graphing any function