Transcript 2.7: Use Absolute Value Functions and Transformations

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
Objective 1
You will be able to
evaluate, write, and
graph piecewise
functions
Exercise 1
Determine whether
the graph shown
represents a
function.
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.
 3
if x  2
 2 x  1,

f ( x)   x  1,
if  2  x  1
 3,
if x  1


Exercise 2
Evaluate g(x) at the values below.
2 x  1, if x  1
g ( x)  
3x  1, if x  1
1.
2.
3.
g(1)
g(5)
g(−3)
Exercise 3
Graph g(x).
2 x  1, if x  1
g ( x)  
3x  1, if x  1
Graphing Piecewise Functions
Method 1:
1. Rather than starting at
the 𝑦-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.
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.
Exercise 4
−12𝑥 − 6, 𝑥 ≤ −4
Graph 𝑓 𝑥 =
𝑥 + 5,
𝑥 > −4
Exercise 4
Write a piecewise
function for the
graph shown.
Parent
Function:
The
simplest
member of a
family of
functions
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
𝑦 = −𝑥
𝑦 = 3𝑥– 7
𝑦 = 12𝑥 − 9
𝑦=𝑥
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
𝑦 = −𝑥 2
𝑦 = 3𝑥 2 – 7
𝑦 = 12𝑥 2 − 9
𝑦 = 𝑥2
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
𝑦=−𝑥
𝑦=3 𝑥 −7
𝑦 = 12 𝑥 − 9
𝑦= 𝑥
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
Building the Absolute Value Function
The absolute value
function is defined by
𝑓(𝑥) = |𝑥|.
The graph of the
absolute value function
is similar to the linear
parent function, except
it must always be
positive.
4
2
-5
5
-2
-4
Building the Absolute Value Function
The absolute value
function is defined by
𝑓(𝑥) = |𝑥|.
So we just take the
negative portion of
the graph and reflect
it across the 𝑥-axis
making that part
positive.
4
2
-5
5
-2
-4
Building the Absolute Value Function
The absolute value
function is defined by
𝑓(𝑥) = |𝑥|.
4
2
This is the
absolute value
parent function.
-5
5
fx = x
-2
-4
Parent Function
V-Shape
Symmetric
about the 𝑦axis
The vertex is the
minimum point
Objective 2
You will be able to graph absolute value
functions using SRT transformations on
the parent function.
You will be able
to perform
vertical and
horizontal shifts
on the graph of
a function
Objective 2a
Translation
A translation is a
transformation that
shifts a graph
horizontally or
vertically, but
doesn’t change
the overall shape
or orientation.
Translation
The graph of
𝑦 = 𝒂|𝑥 − 𝒉| + 𝒌
is the graph of 𝑦 =
|𝑥| translated 𝒉
horizontal units and
𝒌 vertical units.
The new vertex is
at 𝒉, 𝒌
You will be able to vertically
scale the graph of
a function
Stretching and Shrinking
The graph of 𝑦 = 𝒂|𝑥| is graph of 𝑦 = |𝑥|
vertically stretched or shrunk depending on
the |𝒂|.
For 𝒂 > 1
•
•
The graph is stretched vertically
The graph of 𝑦 = 𝒂 𝑥 is
narrower than the graph of 𝑦 =
𝑥
For 𝒂 < 1
•
•
The graph is shrunk vertically
The graph of 𝑦 = 𝒂 𝑥 is
wider than the graph of 𝑦 =
𝑥
The value of “𝒂” gets multiplied by each 𝑦-value.
Objective 2c
You will be able to reflect the graph of a function
across the x-axis
Reflection
The graph of 𝑦 = 𝒂|𝑥| is graph of 𝑦 = 𝒂|𝑥|
reflected across the 𝑥-axis when 𝒂 < 0.
4
fx = - x
2
2
-5
5
fx = x
4
-5
5
-2
-2
-4
-4
Minuses and Pluses
You might be wondering why the ℎ’s get a
minus sign while the 𝒌’s get a plus sign.
That’s just because the 𝒌 is on the wrong
side of the equation.
𝑦 = 𝑎|𝑥 − ℎ| + 𝒌
𝑦 − 𝒌 = 𝑎|𝑥 − ℎ|
Just think of the Point-Slope Form:
𝑦 − 𝑦1 = 𝑚 𝑥 − 𝑥1
Minuses and Pluses
You might be wondering why the ℎ’s get a
minus sign while the 𝒌’s get a plus sign.
That’s just because the 𝒌 is on the wrong
side of the equation.
𝑦 = 3|𝑥 − 2| + 𝟓
𝑦 − 𝟓 = 3|𝑥 − 2|
Just think of the Point-Slope Form:
𝑦−7=6 𝑥−4
Just remember that your 𝑥’s always lie!
Objective 2
You will be able to graph absolute value
functions using SRT transformations on
the parent function.
Multiple Transformations
In general, the graph of an absolute value
function of the form 𝑦 = 𝒂|𝑥 − 𝒉| + 𝒌 can involve
translations, reflections, stretches or shrinks.
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.
Graphing Absolute Value Functions
Method 1: Vertex Method
Graphing 𝑦 = 𝒂|𝑥 − 𝒉| + 𝒌 is easy:
Plot
vertex
Step 1
𝒉, 𝒌
Use 𝒂
value
as
“slope”
Step
to plot
another
point
2
Use
symmetry
toStep
find 3
reflected
point
Connect
the
dots4in
Step
a V-shape
Exercise 6
Without a graphing calculator, graph the
following functions. How do they compare
to the parent function?
1. 𝑦 = |𝑥 − 2| + 5
2.
𝑓 𝑥 =
2
−
3
𝑥+4 −2
Exercise 6
1. 𝑦 = |𝑥 − 2| + 5
2. 𝑓 𝑥 = −
2
3
𝑥+4 −2
Objective 3
You will be able to use SRT
transformations to graph any function
Multiple Transformations: SRT
Method 2: SRT Transformations
To graph 𝑦 = 𝒂 ∙ 𝑓 𝑥 − 𝒉 + 𝒌, start with points
on the parent function 𝑓(𝑥):
Scaling:
Multiply
𝑦-values
by 𝒂
Reflecting:
If 𝒂 is
negative,
flip over 𝑥axis
Translating:
Move
left/right for
𝒉, up/down
for 𝒌
Exercise 7
The graph of 𝑦 = 𝑓(𝑥)
is shown. Sketch
the graph of the
given function.
1.
2.
3.
𝑦 = 2𝑓(𝑥)
𝑦 = −𝑓(𝑥 + 2) + 1
𝑦=
1
𝑓
3
𝑥−2 −5
Exercise 7
1. 𝑦 = 2𝑓(𝑥)
Exercise 7
2. 𝑦 = −𝑓(𝑥 + 2) + 1
Exercise 7
3. 𝑦 =
1
𝑓
3
𝑥−2 −5
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