Download 8.3 powerpoint for notes File

3-1 Lines and Angles
 Paper for notes
 Graph paper
 Pearson 8.3
 Graphing Calc.
Holt Geometry
TOPIC:
3-1 Lines
8.3
Name: Daisy Basset
and
Angles
Period:
Date :
Subject:
Rational
Functions &
Their Graphs
Objective:
Holt Geometry
Notes
Create equations in two
or more variables to
represent relationships
between quantities; graph
equations on coordinate
axes with labels and
scales.
3-1 Lines and Angles
Vocabulary
 Rational Function
 Continuous Graph
 Discontinuous
Graph
Holt Geometry
3-1 Lines and Angles
Key Concepts
 Point of Discontinuity
 Vertical Asymptotes of
Rational Functions
 Horizontal Asymptotes
of Rational Functions
Holt Geometry
3-1 Lines and Angles
1. What are the points
of discontinuity of
each rational
function?
Sketch the graph
showing the points
of discontinuity.
Holt Geometry
A.
x3
y 2
x  4x  3
3-1 Lines and Angles
Factor the numerator and denominator to
check for common factors.

x  3
y
x  1x  3
Holt Geometry


x3
3-1 Lines and Angles
y
x  1x  3
The function is
undefined where
x – 1 = 0 and where
________
x – 3 = 0 .
________
Holt Geometry
3-1 Lines and Angles
There are points
of discontinuity
x = 1 and
at _____
_____.
x = 3
Holt Geometry
x  3
3-1 Lines and Angles
y1 
Holt Geometry
x
2

 4x  3
Notice
the
graph
“breaks” at x = 1 and
Lines
and
Angles
3-1
x = 3, the points of discontinuity.
Holt Geometry
B.
x 5
y 2
x 1
3-1 Lines and Angles
The numerator and denominator
can not be factored.
0  x 1
2
There are no values of x to make
the denominator 0.
Holt Geometry
3-1 Lines and Angles
no
There are __
points of
________
__________.
discontinuity
Holt Geometry
 x  5
3-1 Lines and Angles
y1 
Holt Geometry
x
2

1
Notice
there
no “breaks” in the graph.
Lines
and are
Angles
3-1
Holt Geometry
C.
2
3-1 Lines and Angles
x  3x  4
y
x4
Factor the numerator and denominator to
check for common factors.

x  4x  1
y
x  4
Holt Geometry

y
x  1
x  4
3-1 Lines and Angles
x4
The function is
undefined where
________
x – 4 = 0 .
Holt Geometry

y
x  1
x  4
3-1 Lines and Angles
x4
There is a point
of discontinuity
x = 4
at _____.
Holt Geometry


2
3-1 Lines and Angles
x  3x  4
y1
x  4

The graph “breaks” x = 4, the point
of discontinuity.
Holt Geometry
D.
3-1 Lines and Angles
1
y 2
x  16
Factor the numerator and denominator to
check for common factors.
y
Holt Geometry
1
x  4x  4
3-1 Lines and Angles
y
1
x  4x  4
The function is
undefined where
x + 4 = 0 and
________
x – 4 = 0
where ________.
Holt Geometry
3-1 Lines and Angles
The points of
discontinuity
x = -4
are at ______
x = 4
and _____.
Holt Geometry
3-1 Lines and Angles
1
y1  2
x  16

Holt Geometry

The
“breaks”
Lines and
Anglesat x = -4 and x = 4,
3-1 graph
the points of discontinuity.
Holt Geometry
3-1 Lines and Angles
Summary
D
L
I
Q
Holt Geometry
Summarize/reflect
What did I do?
What did I learn?
What did I find most
interesting?
What questions do I still
have? What do I need
clarified?
3-1 Lines and Angles
Notes 8.3
Graphing Calc.
Holt Geometry
2. Find the
vertical
asymptotes
and holes for
the graph?
3-1 Lines and Angles
Holt Geometry
Vertical asymptotes
are nonremovable
points of discontinuity.
3-1 Lines and Angles
Holes are removable
points of discontinuity.
Holt Geometry
A.


3-1 Lines and Angles x  1
y
x  2x  3
The function is undefined where
________
x - 2 = 0 and where ________.
x – 3 = 0
The zeros for the denominator
are at ________.
x = 2, 3
Neither is a zero for the
no
numerator. So, there are ___
holes.
Holt Geometry
3-1 Lines and Angles
The vertical
asymptotes for
the graph are
x = 2 and ____.
x = 3
____
Holt Geometry
B.

3-1 Lines and Angles x  2
y

x  1x  3
The function is undefined where
x + 3 = 0
________
x - 1 = 0 and where ________.
The zeros for the denominator
are at ________.
x = 1, -3
Neither is a zero for the
no
numerator. So, there are ___
holes.
Holt Geometry
3-1 Lines and Angles
The vertical
asymptotes for
the graph are
= -3
x = 1 and x____.
____
Holt Geometry
C.
3-1 Lines and Angles
x3
y 2
x  7 x  12
Factor the numerator and denominator.

x  3
y
x  3x  4
Holt Geometry

x  3
y
x  3x  4
3-1 Lines and Angles
The function is undefined where
x + 4 = 0
x + 3 = 0 and where ________.
________
The zeros for the denominator
x = -3, -4
are __________.
-3 is a zero for the numerator.
___
Holt Geometry
3-1 Lines and Angles
The vertical
asymptote for the
x
=
-4
graph is ______.
The hole for the
x = -3
graph is ______.
Holt Geometry
3-1 Lines and Angles
Summary Summarize/reflect
D
What did I do?
L
I
Q
Holt Geometry
What did I learn?
What did I find most
interesting?
What questions do I still
have? What do I need
clarified?
3-1 Lines and Angles
Notes 8.3
Graphing Calc.
Holt Geometry
3-1 Lines and Angles
3. Find the
horizontal
asymptotes
for the graph?
Holt Geometry
3-1 Lines and
2 xAngles
A.
y
x3
1
2x
y 1
x 3
Find the degrees of the
numerator & denominator.
degree: 1
degree: 1
The degrees of the numerator &
same so the
denominator are the
_______,
a
horizontal asymptote is at y  .
b
Holt Geometry
and Angles
3-1 Lines a
b
2x
y
1x  3
a 2
y 
b 1
The horizontal
asymptote is at
y = 2
_____.
Holt Geometry
3-1 Lines and Angles
x2
B.
y
x  2x  3
1
x 2
y 2
x  2x  3
2
Find the degrees
of the numerator &
denominator.
degree: 1
degree: 2
The degree of the
is less than
numerator _________
than the degree of the
denominator.
Holt Geometry
3-1 Lines and Angles
The horizontal
asymptote is
x-axis or
the ______,
y0
_____.
Holt Geometry
2
3-1 Lines and Angles
x
C.
y
2x  5
2
x
y 1
2x  5
Find the degrees
of the numerator &
denominator.
degree: 2
degree: 1
The degree of the
is greater than
numerator ___________
than the degree of the
denominator.
Holt Geometry
3-1 Lines and Angles
There is __
no
horizontal
asymptote.
Holt Geometry
Name: Daisy Basset
Lines
and
Angles
3-1
Period:
TOPIC:
Date :
Subject:
8.4
Rational
Expressions
Objective:
Notes
Interpret parts of
an expression, such
as terms, factors,
and coefficients.
Holt Geometry
Vocabulary
3-1 Lines and Angles
 Rational Expression
 Simplest Form of a
Rational Expression
Holt Geometry