Objectives:
1. To evaluate and
graph piecewise
defined functions
2. To internalize the
graphs of various
parent functions
3. To perform
translations on the
graphs of functions
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•
•
•
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Assignment:
P. 49: 35, 37
P. 71-2: 43, 44, 47
P. 73: 71, 72
P. 81: 20, 21, 29, 30,
37, 38, 43, 45, 48
Shifting Worksheet
Read: P. 76-78
Piecewise Functions
Scaling
Reflecting
Translating
You will be able to
graph and evaluate
piecewise functions
Determine whether
the graph shown
represents a
function.
A piecewise defined 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


Evaluate g(x) at the values below.
2 x  1, if x  1
g ( x)  
3x  1, if x  1
1. g(1)
2. g(5)
Graph g(x).
2 x  1, if x  1
g ( x)  
3x  1, if x  1
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.
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.
The absolute value
parent function is
defined by f (x) = |x|.
Write this function as a
piecewise function.
4
2
-5
5
fx = x
-2
-4
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Write a piecewise defined
function for the graph
shown.
This is called a step
function. I wonder
why.
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You will be able to
graph parent
functions and
detail their
important
characteristics
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
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
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
In your groups, graph each parent function by hand
using a few key points. Then list the domain and
range in both interval and set notation. Finally,
indicate each graph’s symmetry.
You will be able
to perform
vertical and
horizontal shifts
on parent
functions
One of the handiest ways to graph a function is
by transformations on parent functions. This
includes scaling, reflecting, and translating;
what we’ll call SRT.
Each of these transformations (scaling,
reflecting, and translating) come in a vertical,
y, and a horizontal, x, variety.
y  a  f (c   x  h)  k
If you are given the graph
of the function y = f (x),
then the graph of
y  f ( x)  k
is translated vertically k
units.
• Upward if k > 0
• Downward if k < 0
By hand, graph each of the following on the
same coordinate plane:
1. y  x
2. y  x  3
3. y  x  3
If you are given the graph
of the function y = f (x),
then the graph of
y  f ( x  h)
is translated horizontally h
units.
• Left if h > 0
• Right if h < 0
It might seem a bit weird that when k is positive,
the graph moves up—in the positive ydirection—while when h is positive, the graph
moves left—in the negative x-direction. This
is not all that weird if you consider that k is
really on the wrong side of the equation:
y  f ( x  h)  k
y  k  f ( x  h)
It might seem a bit weird that when k is positive,
the graph moves up—in the positive ydirection—while when h is positive, the graph
moves left—in the negative x-direction. This
is not all that weird if you consider that k is
really on the wrong side of the equation:
y  x5 3
y 3  x 5
This is still a vertical translation of 3 units up.
By hand, graph each of the following on the
same coordinate plane:
1. y  x
2. y  x  3
3. y  x  3
•
Objectives:
1. To evaluate and
graph piecewise
defined functions
2. To internalize the
graphs of various
parent functions
3. To perform
translations on the
graphs of functions
•
•
•
•
•
•
Assignment:
P. 49: 35, 37
P. 71-2: 43, 44, 47
P. 73: 71, 72
P. 81: 20, 21, 29, 30,
37, 38, 43, 45, 48
Shifting Worksheet
Read: P. 76-78