Transcript File

Warm Up 1/4/15
•Turn to a neighbor and tell
them about a fun
experience you had over
break. Be prepared to
share with the class!
Objective
The students will be able to:
graph ordered pairs on a coordinate
plane.
ordered pair – a pair of numbers used to locate a point on a
coordinate plane.
coordinate plane – a plane formed by the intersection of a
horizontal number line called the x-axis and a vertical line
called the y-axis.
x-axis – the horizontal axis on a coordinate plane.
y-axis – the vertical axis on a coordinate plane.
x-coordinate – the first number in an ordered pair; it tells
the units to move right or left from the origin.
y-coordinate – the second number in an ordered pair; it
tells the units to move up or down from the origin.
origin – the point where the x-axis and the y-axis intersect
on the coordinate plane.
The Cartesian Coordinate Plane
- The Cartesian plane was the
brainchild of French mathematician
Rene Descartes trying to combine
algebra and geometry together.
- Descartes took a second number
line and standing it on end, crossed
the lines at zero to form a grid like
pattern.
- The number lines when drawn like
this, are called “axes”. The
horizontal line is called the “x-axis”;
the vertical line is called the “yaxis”.
The x-axis and y-axis separate the
coordinate plane into four regions,
called quadrants.
II
(-, +)
III
(-, -)
I
(+, +)
IV
(+, -)
Ordered pairs are used to locate points in a
coordinate plane.
y-axis (vertical axis)
5
5
-5
x-axis (horizontal
axis)
-5
origin (0,0)
The Coordinate Plane
Graphing an ordered pair, (point): (x, y)
Graph point A at (4, 3)
y
The first number, 4, is called the
x-coordinate
___________.
5
4
(4, 3)
3
It tells the number of units the point lies to
left or right of the origin.
the __________
2
1
x
-5
The second number, 3, is called the
y-coordinate
___________.
It tells the number of units the point lies
_____________
above or below the origin.
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
2
3
4
5
The Coordinate Plane
Graphing an ordered pair, (point): (x, y)
Graph point B at (2, –3)
y
The first number, 2, is called the
x-coordinate
___________.
5
4
3
It tells the number of units the point lies to
left or right of the origin.
the __________
2
1
x
-5
The second number, –3, is called the
y-coordinate
___________.
It tells the number of units the point lies
_____________
above or below the origin.
-4
-3
-2
-1
1
2
3
4
-1
-2
-3
-4
-5
(2, –3)
5
What is the ordered pair for A?
1.
2.
3.
4.
(3, 1)
(1, 3)
(-3, 1)
(3, -1)
5
•A
5
-5
-5
What is the ordered pair for B?
1.
2.
3.
4.
(3, 2)
(-2, 3)
(-3, -2)
(3, -2)
5
5
-5
•B
-5
What is the ordered pair for C?
1.
2.
3.
4.
(0, -4)
(-4, 0)
(0, 4)
(4, 0)
5
5
-5
•
C
-5
What is the ordered pair for D?
1.
2.
3.
4.
(-1, -6)
(-6, -1)
(-6, 1)
(6, -1)
5
5
-5
•D
-5
Write the ordered pairs that name
points A, B, C, and D.
A = (1, 3)
B = (3, -2)
C = (0, -4)
D = (-6, -1)
5
•A
5
-5
•D
•B
•
C
-5
Name the quadrant in which each point
is located
(-5, 4)
1.
2.
3.
4.
5.
6.
I
II
III
IV
None – x-axis
None – y-axis
Name the quadrant in which each point
is located
(-2, -7)
1.
2.
3.
4.
5.
6.
I
II
III
IV
None – x-axis
None – y-axis
Name the quadrant in which each point
is located
(0, 3)
1.
2.
3.
4.
5.
6.
I
II
III
IV
None – x-axis
None – y-axis
Warm Up 1/5/15
•Solve the following rebus
puzzles
Graphing an Equation
Using ordered pairs
Generate ordered pairs for the equation
y = x + 3 for x = –2, –1, 0, 1, and 2.
Graph the ordered pairs.
X
-2
-1
0
1
2
y
Complete the table of values to determine the ordered pairs.
Graph the equation on a coordinate plane.
x
-2
-1
0
1
2
y = 2x + 3
y
(x, y)
X
Y
-2
-1
-1
1
0
3
1
5
2
7
x
-2
-1
0
1
2
y = x^2
y
(x, y)
X
Y
-2
4
-1
1
0
0
1
1
2
4
When graphing, you always want to
draw neatly.
Cartography, the science of map making, is an
application of graphing on a coordinate plane.
Cartographers map a region of the surface of the
earth onto part of a plane.
Warm Up 1/6/15
•Moving or not?
Definition: A relation is any set of
ordered pairs.
A={(1,3),(2,4),(3,5)}
A relation can also be represented by: a table of
values, a graph, a mapping diagram, and an
equation (equations come later).
Domain: In a set of ordered
pairs, (x, y), the domain is
the set of all x-coordinates.
Range: In a set of ordered
pairs, (x, y), the range is the
set of all y-coordinates.
Given the following set of ordered
pairs, find the domain and range.
Ex:{(2,3),(-1,0),(2,-5),(0,-3)}
Domain: {-1,0,2}
Range: {-5,-3,0,3}
If a number occurs
more than once,
you do not need to
list it more than
one time.
Practice: Find the domain and
range of the following set of
ordered pairs.
1. {(3,7),(-3,7),(7,-2),(-8,-5),(0,-1)}
Domain:{-8,-3,0,3,7}
Range:{-5,-2,-1,7}
State the domain and range of the following
Relation (represented by a mapping diagram)
x
y
1
5
2
6
3
8
11
State the domain and range of the following relation.
x
y
-3
8
-2
7
-1
6
0
5
1
4
domain :  3,  2,1,0,1
range : 4,5,6,7,8
Give the domain and range for the following
relation.
The domain value is all
x-values from 1
through 5, inclusive.
The range value is all
y-values from 3
through 4, inclusive.
Domain: 1 ≤ x ≤ 5
Range: 3 ≤ y ≤ 4
^^This is known as interval notation^^
Give the domain and range for the following
relation.
D: –5 ≤ x ≤ 3
R: –2 ≤ y ≤ 1
Give the domain and range for the following
relation.
D:
R:
Give the domain and range for the following
relation.
D:
R:
Warm Up 1/7/15
•https://www.youtube.com
/watch?v=vJG698U2Mvo
Functions
•A relation is a function provided there is
exactly one output for each input.
•A relation is NOT a function if at least
one input has more than one output
In order for a relationship to be a function…
EVERY INPUT MUST HAVE AN OUTPUT
TWO DIFFERENT INPUTS CAN HAVE THE
SAME OUTPUT
ONE INPUT CAN HAVE ONLY ONE
OUTPUT
INPUT
(DOMAIN)
Functions
FUNCTION
MACHINE
OUTPUT (RANGE)
Which of the following relations are
functions?
R= {(9,10, (-5, -2), (2, -1), (3, -9)}
S= {(6, a), (8, f), (6, b), (-2, p)}
T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)}
No two ordered pairs can have the same
first coordinate
(and different second coordinates).
Identify the Domain and Range. Then tell if the
relation is a function.
Input
Output
-3
3
1
1
3
-2
4
Domain = {-3, 1,3,4}
Range = {3,1,-2}
Function?
Yes: each input is mapped
onto exactly one output
Identify the Domain and Range. Then
tell if the relation is a function.
Input
Output
-3
3
1
-2
4
1
4
Domain = {-3, 1,4}
Range = {3,-2,1,4}
Notice the set notation!!!
Function?
No: input 1 is mapped onto
Both -2 & 1
State whether or not the following
relations are functions or not.
x
y
x
y
x
y
-3
2
1
3
-2
3
-3
8
2
5
-1
7
0
1
3
6
0
8
1
9
4
4
1
-5
5
6
5
3
2
7
Relation (not a
function)
Function
Function
State whether or not the following
relations are functions or not.
x
y
x
y
x
y
-2
-7
-4
16
1
1
0
-3
-2
4
1
-1
3
3
0
0
4
2
4
5
2
4
4
-2
5
7
3
9
9
3
function
function
Relation (not
a function)
1. {(2,5) , (3,8) , (4,6) , (7, 20)}
2. {(1,4) , (1,5) , (2,3) , (9, 28)}
3. {(1,0) , (4,0) , (9,0) , (21, 0)}
The Vertical Line Test
If it is possible for a vertical line to
intersect a graph at more than one
point, then the graph is NOT the
graph of a function.
Use the vertical line test to visually check if the
relation is a function.
(-3,3)
(4,4)
(1,1)
(1,-2)
Function?
No, Two points are on
The same vertical line.
Use the vertical line test to visually check if the
relation is a function.
(-3,3)
(1,1)
(3,1)
(4,-2)
Function?
Yes, no two points are
on the same vertical line
Examples
• I’m going to show you a series of graphs.
• Determine whether or not these graphs
are functions.
• You do not need to draw the graphs in
your notes.
#1
Function?
#2 Function?
#3 Function?
#4
Function?
#5
Function?
#6 Function?
#7 Function?
#8
Function?
Warm Up 1/8/15
•Why don’t you ever see
hippopotamus hiding in
trees?
•Because they’re good at it.
What time is it
when you go to the
dentist?
Tooth-hurty.
What do you call a
big pile of kittens?
A meowntain.
Graphing Relationships
Graphs can be used to illustrate many different
situations. For example, trends shown on a
cardiograph can help a doctor see how a
patient’s heart is functioning.
To relate a graph to a given situation, use key
words in the description.
Example 1: Relating Graphs to Situations
Each day several leaves fall from a tree. One
day a gust of wind blows off many leaves.
Eventually, there are no more leaves on the
tree. Choose the graph that best represents
the situation.
Step 1 Read the graphs from left to right to show
time passing.
Check It Out! Example 1
The air temperature increased steadily for
several hours and then remained constant. At
the end of the day, the temperature increased
slightly before dropping sharply. Choose the
graph that best represents this situation.
Step 1 Read the graphs from left to right to show
time passing .
As seen in Example 1, some graphs are
connected lines or curves called continuous
graphs. Some graphs are only distinct points.
They are called discrete graphs
The graph on theme park
attendance is an example of a
discrete graph. It consists of
distinct points because each
year is distinct and people are
counted in whole numbers only.
The values between whole
numbers are not included, since
they have no meaning for the
situation.
Example 2B: Sketching Graphs for Situations
Sketch a graph for the situation. Tell whether
the graph is continuous or discrete.
A small bookstore sold between 5 and 8
books each day for 7 days.
The number of books sold
(y-axis) varies for each day
(x-axis).
Since the bookstore can only
sell whole numbers of books,
the graph is 7 distinct points.
The graph is discrete.
Check It Out! Example 2a
Sketch a graph for the situation. Tell whether
the graph is continuous or discrete.
Jamie is taking an 8-week keyboarding class.
At the end of each week, she takes a test to
find the number of words she can type per
minute. She improves each week.
Each week (x-axis) her typing
speed is measured. She gets
a separate score (y-axis) for
each test.
Since each test score is a whole
number, the graph consists of 8
distinct points.
The graph is discrete.
Check It Out! Example 2b
Sketch a graph for the situation. Tell whether
the graph is continuous or discrete.
Henry begins to drain a water tank by opening
a valve. Then he opens another valve. Then he
closes the first valve. He leaves the second
valve open until the tank is empty.
As time passes while draining the
tank (moving left to right along the
x-axis) the water level (y-axis) does
the following:
• initially declines
• decline more rapidly
• and then the decline slows down.
The graph is continuous.
Lesson Quiz: Part I
1. Write a possible situation for the given graph.
Possible Situation: The level of water in a bucket
stays constant. A steady rain raises the level. The
rain slows down. Someone dumps the bucket.
Lesson Quiz: Part II
2. A pet store is selling puppies for $50 each. It has
8 puppies to sell. Sketch a graph for this situation.