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Honors Algebra 2 Unit 1
Mod1.1: Domain, Range, and End Behavior
Learning Target: I can relate the domain, range, and end behavior of a function to its graph.
Mod1.2 – Characteristics of Functions and Graphs
Learning Target: I can relate the characteristics of real world phenomena to characteristics of its function graph.
Ex: Which of the following represent identical graphs? How the third is different?
{"|" ≥ 5 } (−∞, 5 ] 5 ≥ "
Characteristics
Increasing Moving from left to right on the graph, the y values are increasing
Decreasing Moving from left to right on the graph, the y values are decreasing
Turning point Point where the function changes from increasing to decreasing or vice versa
Maximum Highest point on a graph (could be a turning point or an endpoint)
Minimum Lowest point on a graph (could be a turning point or an endpoint)
Zero The x – value where the y – value is zero.
Average rate of
change
Ratio of movement restricted to a specific timeframe
Domain The set of all x – values (inputs) for which the function is defined
Range The set of all y – values (also called f(x) or outputs) for which the function is defined
End
Behavior
Describes what happens to the values of the outputs ( or f(x) /y – coordinates) as the x
values increase and decrease.
Notation:
→ _______
→ _______
Interval An unbroken portion of a number line
- Finite: two endpoints which may/may not be included in the interval
- Infinite: unbounded on at least one end
Representations: inequality 2 < " ≤ 5
Set notation {"| 2 < " ≤ 5 }
Read as: the set of all real numbers x such that x is greater
than 2 but less than or equal to 5.
Interval
notation
( 2 , 5 ]
- Parenthesis – end point is NOT included;
o Use for infinity (−∞ or ∞)
- Brackets [ ] – end point IS included
Other symbols: ∈ “is an element of”; ℝ denotes the set of all real numbers
down
vertical
up
Left
DIRECTIONS
Right used
for
Always
1st
Always
Last
Domain
(D)
A
Range(R)
tall #sincluded (-
,
c)
A
↑
b
b
A
A
Honors Algebra 2 Unit 1
Examples: Use interval notation to write the domain and range of the function shown in the graph.
Describe the end behavior of the graph.
Restricted Domain
!
O A
↑
y
A
.
&
A
D
:
1
,
R
: (-
,
1]
xvalues yvalves
D
: +
,
R
:
( ,
End
EB
:
As XTEY
,
+(x)
Behavior
ASx
,
f(x)
EB
:
As
X-
,
f(x)
right
As
x
0
,
+(x)
0
in
DiF
,
b)
R
:
[
, 4)
EB
:
As
x
,
f(x)
Asx
,
f(x)
Honors Algebra 2 Unit 1
4. Describe how to transform the graph of 0 (") shown to obtain the graph of
M(") = − 0 (" + 2 ) − 4
and show the graph. Include each parameter on the graph of the
image. Also include the effects of each parameter on the
coordinates of the reference points shown on the graph of 0 (").
Mod 1. 4 – Transformations of Absolute Value and Quadratic Functions
Learning Target: I can use parameters to identify changes in the key characteristics of a function.
Parent
Functions
The most basic form of each type of function.
Q = "
Q = "
!
Q = "
&
Q =
Q = |"|
Absolute
value
function
Reminder: absolute value of a number is its distance from 0 on the number line.
Parent function: 0
= S
" H0 " ≥ 0
−" H0 " < 0
- Graph is V – shaped. (check it out on the online graphing calculator)
- This is an example of a piecewise – defined function (the definition is broken into
pieces)
Quadratic
function
The graph of a quadratic function is a curve called a parabola.
!
!
+ J" + T
Vertex The point at which the graph of an absolute value or quadratic function changes direction.
- What is another name for a point where a function changes direction?
You will use the transformation information from 1.3 to interpret, describe, and graph from parent
functions.
①
⑤
g(x)
=
af((x
n))
O
4
Describe
:
translation
left
2 units
horizontal
(vertical)
reflection
across
the
X-axis ,
vertical
translation
down
units
on
f
(
f(x)
( ,
M
O
O
(
,
4
(
,
quadratic
absolute
value
focus &
, 0)
parabola
I
Vshape
a
Honors Algebra 2 Unit 1
Examples:
1. The graphs of f and g are shown below. Write the function M
as a
transformation of 0
- Describe the transformations on 0 (") or ℎ(") that create M(") given 0 (") = "
!
and ℎ(") = |"|.
a. M
$
%
①
in
function
f(2)
=
(2) 2
<
f(l)
=
=
g(
= H)
4
!
=
9
(1)
=
a
K
g(x)
=
4 x
A
n(-g(x)
=
((x
horizontal
translation
left
3 units
,
vertical compression
by
a
factor
of
14
,
(vertical)
reflection
across
the
X-axis
vertical
translation
up
2
unit
o
f(x)
=x=
#f(x)
f(x)
, g(x)
X
h(x)
=
(x)
X
3
h(x)(x,
g(x)
3
(-512)
(
41-1)
( ,
3 ,
0
F
(
,
-2)
(
,
9/4)
(
2 ,
-2n
X
Y
T
6
4
2
O 2 4
O ↑
T
· ·
2
· ⑨
2
⑧
⑧ O
O ⑧
· ⑧
·
2
B
D
4
No
Which function !(#) represents the transformed function ℎ(#) after a
reflection across the x-axis, vertical stretch by a factor of 3, and a vertical
translation down 5 units?
The graph of !(#) is transformed by a horizontal translation right 7 units, a
horizontal compression by a factor of
!
"
, and a vertical translation up 1 unit.
Which is the function, ℎ
, of the new graph?
A. ℎ
, + 1 B. ℎ
C. ℎ(#) =! /
!
"
(# − 7 ) 0 + 1 D. ℎ(#) =! /
!
"
D: R:
EB:
d(x)
=
=
3h(y)
=
5
&
A
(X)
to
, v
R
A
AsX
w
,
fWxe0,
As
x
> 0
,
f(x)
Describe each transformation of the parent function 1 (#) = |#| OR 3 (#) =
. Show your work in a table, then graph the parent function and the new
image.
Ib
.
T
= 2
. b
1
= 2b
= b
X
un
horizontal compression by
a
factor
of"
vertical
translation
down
units
&
T
X f(x)
=
(x)t
-X
f(x)
4
3
I
2
(
1
2
⑧ ⑧
3
(
y
,
-3)
↓
A
& &
E
n(t)
F
( ,
OB
⑧
D
3 (
,
-3)
2 ( ,
-2)
·
⑧
Y
3
· ①
4
⑧
S
D
3
2
1 01234
g(x)
= x
horizontal
translation
right
4 units
vertical
stretch
by
a
factor of
3
S
(vertical)
reflection
across
the x-axis
X
g(x)
=
x
x
4
3g(x)
g(x)
2
4
2
12 (
,
-12)
(
,
(
,
01
211
+2)
(
,
