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Topic:
Introduction
to graphing:
Plotting
Points
Name:________________________
Per:________________________
Date:________________________
Class:________________________
Notes: (Day 1)
Label
Quadrant _____
Quadrants
Quadrant _____
( ____ , ____ )
( ____ , ____ )
here:
( ____ , ____ )
___________ or ( ___ , ___ )
( ____ , ____ )
( ____ , ____ )
Quadrant _____
Quadrant _____
( ____ , ____ )
Points
Every point has ____ numbers in it
The first number represents the __ value, the second the ___ value.
You always start from the ___________ when plotting points.
The x-value is how much you move ________ or ________
The y-value is how much you move _____ or _________
( ____________ , ____________ )
( __________ or _________ )
Practice
( __________ or _________ )
What quadrant is each point in?
A (2,5)
B ( -5 , 3 )
C ( -2 , -1 )
D ( -1 , 0 )
1
Practice
Plot the following points on the graph below. Label each point with its designated letter.
plotting
A (1,4)
Practice
Give the coordinates of each point.
identifying
points
B(6,3)
C ( -4 , 2 )
D ( -1 , -6 )
E ( 7 , -8 )
A ( ____ , ____ )
B ( ____ , ____ )
C ( ____ , ____ )
D ( ____ , ____ )
E ( ____ , ____ )
F ( ____ , ____ )
G ( ____ , ____ )
H ( ____ , ____ )
2
Domain
The ___ values.
Range
The ___ values.
What is the domain of the function shown below?
STAR released
question #79
Domain:
{ ___ , ___ , ___ , ___ }
Which line goes through the point (-2,-4)?
D
C
B
A
Summary: The points (–4 , 3 ) and ( 3 , –4 ) appear to be similar, but are two unique
points. Discuss the differences of the two points.
3
Topic: Graphing
Lines: Slope-intercept
form (Day 2)
Name:________________________
Per:________________________
Date:________________________
Class:________________________
Notes:
Slope-intercept form of a
y = _______ + _____
line
y-intercept
The y-intercept is represented by the letter _____ .
The y-intercept is where the line ________ the __-axis.
It is also where you __________ each graph.
slope
The slope is represented by the letter ____.
The slope is
.
You go ____ or ________, then to the __________.
Positive slopes always go _____ and to the _________.
They look like this.
Positive slopes
Negative slopes always go _______ and to the _________.
They look like this.
Negative slopes
4
Graphing a line
1) Always start at the ____________________.
2) From there graph the _____________.
Practice
A) Graph y 
2
x4
3
1
B) Graph y   x  5
2
C) Graph y 
2
x6
1
5
D) Graph y  3 x  1
E) Graph y  x  2
F) Graph y  2 x
6
G) Graph y 
7
x4
8
What is the y-intercept of 4 x  2 y  12 ?
STAR released question
#23
How can we get the equation into slope-intercept form? _____________
Teacher guided question
4 x  2 y  12
If
Then, 2 y = _______________
And, y = _______________
and our y-intercept is ______
Summary: Given the graph below, write the equation of the line.
y
5
4
3
2
1
–5 –4 –3 –2 –1
–1
1
2
3
4
5
x
Equation: ____________________________
–2
–3
–4
–5
Hint: What is the slope?
Hint: What is the y-intercept?
7
Topic: Identifying
equations of lines and
slope
(Day 3)
Name:________________________
Per:________________________
Date:________________________
Class:________________________
Notes:
Review
Positive slopes go ___ and to the _________.
Negative slopes go _______ and to the _________.
Positive y-intercepts are above the ___________.
Negative y-intercepts are below the ___________.
Similar to STAR released
What linear equation is shown on the graph below?
question #24
Concept review
1
x 1
2
A
y
B
1
y   x 1
2
C
y
D
1
y   x 1
2
1
x 1
2
When graphing a line, START with the __________________,
and then use the _____________ after that.
To find the slope, find two ____________ where the line goes
through. Count how far ______________ and to the _________
the line goes between those two points. ____________ too.
8
Slope concept
The slope is represented by the letter ____
The slope is
You go ____ or ________, then to the __________
Slope formula
To find the slope, you are talking about finding the
As a formula, this looks like
Where do the 1s and 2s
When you are given two ordered pairs (two points), in each
come from?
ordered pair you can label the __ and __. But since you have two
different __s, and two different __s, you label the _________
ordered pair with ones, and the second ordered pair with ____.
 2,1
3,9
Try it
Label:
Now find the slope
Use the formula to find the slope of the line passing through  2,1
and
and  3,9 . Simplify your answer if possible.
Try again
Find the slope passing through the points  4, 2  and 1, 6
Harder examples
A)
1, 4
and  2, 10
9
B) 14, 25 and  37,60 
C)
STAR released question
 8, 12
and 14, 20
Which best represents the graph of y  2 x  2 ?
#25
A
C
B
D
10
STAR released question
#27
Which equation best represents the graph above?
A yx
Think it through
B y  2x
C y  x2
D y  2x  2
Every line in the form of y = mx+b has ____ y-intercept.
Every line in the form of y = mx+b has ____ x-intercept.
Can a line pass through the x-axis twice? Why?
Sometimes/always/never
The graph of a linear function in the form y  mx  b has one xintercept. _______________________
11
STAR released question
Which equation represents the line shown in the graph below?
#31
A
y
2
x4
3
B
y
2
x6
3
C
y
3
x4
2
D
y
3
x6
2
STAR based skill
What is the slope of the line y 
STAR based skill
What is the y-intercept of the equation y  2 x  3 ?
A
 0, 2 
B
 0, 2
C
 0,3
D
 0, 3
4
x7 ?
5
______________
12
Review
The equation y  3 x  3 crosses the __ -axis at ____ and then
goes _____ and to the _______.
Summary: Explain how you determine which of the four numbers goes in each
position of the formula when finding the slope.
13
Topic: Graphing
linear equations:
Standard form
(Day 4)
Name:________________________
Per:________________________
Date:________________________
Class:________________________
Notes:
Review
Slope-intercept form is ___ = _____ + _____.
To be in slope-intercept form, _____ always has to be by itself!
Standard form
If an equation is given in standard form, the ___s and ____s are on the
______ side of the equation. It looks like:
OR
What to do
OR
Usually the best thing to do when an equation is given in standard form
is to ____________ it to ________________ form.
Practice
Convert 2 x  3 y  12 to slope-intercept form.
Are there any like terms? ____________________________
How many terms should this have in slope-intercept form? ________
What is the first thing to do? ________________________________
Show your work here:
Try it on your own
Convert 3x  4 y  24 to slope-intercept form.
14
Thinking it through
Look at the problem at the bottom of this page. What equation is
given? _____________________________
What will you need to do to determine the correct graph? ___________
_________________________________________________________
_________________________________________________________
Do that here:
What do graphs A & C have in common? _______________________
What do graphs A & B have in common? _______________________
STAR based skill
Which graph represents 5 x  6 y  12 ?
y
–5
–5
–4
–4
–3
–3
–2
–2
y
5
5
4
4
3
3
2
2
1
1
–1
–1
1
2
3
4
5
x
–5
–4
–3
–2
–1
–1
–2
–2
–3
–3
–4
–4
–5
–5
A
C
y
y
5
5
4
4
3
3
2
2
1
1
–1
–1
1
2
3
4
5
x
–5
–4
–3
–2
–1
–1
–2
–2
–3
–3
–4
–4
–5
–5
B
1
2
3
4
5
x
1
2
3
4
5
x
D
15
Practice converting
1) 4 x  y  2
2) 4 x  y  2
3) 2 x  3 y  1
4) 2 x  3 y  1
5)  x  y  4
6)  x  y  4
to slope-intercept
form
Summary: Sally says that 2 x  y  4 in slope-intercept form is  y  2 x  4 .
Explain why this is not correct.
16
Topic: Graphing
Linear Equations:
Special Cases
(Day 5)
Name:________________________
Per:________________________
Date:________________________
Class:________________________
Notes:
2 unique cases
Not every line has a y-intercept and a slope.
A ___________________ line goes from ______ to ________
A ______________________ lines goes ____ and _________
Vertical lines
Vertical lines pass through the ___ - axis, thus their equation
looks like x = a, where a is a number. For example, x  3 is a
vertical line.
Example
Look at the line passing through the points (4,5) and (4,0)
How much does the line rise between thos two points?
______________ run? ________
So its slope is
, which is ___________________.
Thus we call the slope of a vertical line _________________.
Please don’t call this ______ __________!
Graph
1) x  2
What does x always
If we have the line x  7 , the graph goes _____ and ________,
equal in a vertical line?
and every x value is equal to _______. So, no matter what y is,
x will always be _____.
17
Horizontal lines
Horizontal lines pass through the ____ - axis, thus their equation
looks like y = b. For example, y  1 is a horizontal line.
Example
Look at the line passing through the points (0,5) and (7,5)
How much does the line rise? ______________ run? ________
So its slope is
, which is _____.
Thus a horizontal line has a slope of ________________.
Please don’t call this ____ ____________!
Graph
1) y  4
What does y always
If we have the line y  6 , the graph goes _____ and ________,
equal in a horizontal line?
and every y value is equal to _______. So, no matter what x is,
y will always be _____.
Summary: How can you tell whether a line will be horizontal or vertical?
18
Topic: Another way
of graphing: Pointslope form
(Day 6)
Name:________________________
Per:________________________
Date:________________________
Class:________________________
Notes:
Another form of a line
y  mx  b is not the only way to graph a line.
Instead of always having to start with the ___- intercept, there is
another way to write an equation of a line that can start from any
point on the graph. It is called __________ - __________ form.
Point-slope form
For any point  x, y  , and slope m, an equation of the line passing
through the point with the slope can be written:
How to use point-slope
Instead of always having to start at the ___-_____________ and
form
then graphing the slope, you could start graphing at any
________ that the graph goes through, and then graph the
________ from there!
When plugging the point  2,3 into the equation, the signs
always _____________.
For example, the line that passes through  2,3 with a slope of 4
looks like ____________________________
Try this
1)  3,5 , m = -2
Graph
Graph the line that passes
2)
 2, 4 , m = 1
through  2,3 with a slope of
1
 .
2
19
STAR released question
Which point lies on the line defined by 3x  6 y  2 ?
28
#28
A
 0, 2 
B
 0, 6 
1

C  1,  
6

1

D  1,  
3

List two ways that you could determine if a point is on a line.
1) _______________________________________________________
2) _______________________________________________________
Attempt #1: ______________ the ___________ into the ____________.
A
 0, 2 
B
 0, 6 
1

C  1,  
6

1

D  1,  
3

Attempt #2: ________________ the line and ____ which ________ the
Hint: Begin by getting the
line _____________ through.
equation into _______________________ form.
3x  6 y  2
20
Try this
Write the equation of a line that has a slope of -3 and passes
through the point  2, 1 .
STAR released question
#29
STAR based question
What is the equation for the line passing through the points graphed
below?
y
6
5
4
3
2
1
–6
–5
–4
–3
–2
–1
–1
1
2
3
4
5
6
x
–2
–3
–4
–5
–6
STAR based question
What is an equation of the line that passes through the points
?
21
Summary: Pat is given the points  2, 3 and 1,6  . Explain how Pat finds the
equation of the line is y  3x  3 .
22
Topic: Another way
of graphing: Coverup Method
(Optional)
Name:________________________
Per:________________________
Date:________________________
Class:________________________
Notes:
Standard form
When equations are given in standard form, sometimes you can
graph the equation by finding the __ and __ intercepts.
Note: This does not help you graph _________________, and it
doesn’t always help you find the solution to a ______________.
Intercepts
When a point is on the x-axis (an x-intercept), the coordinates
always look like ( __ , __ ).
When a point is on the y-axis (an y-intercept), the coordinates
always look like ( __ , __ ).
This means
That when you solve for an x-intercept, you plug in ___ for y,
which means it _____________, and you can _________ it up.
So when you solve for the y-intercept, you _________ up the x.
Try this
Graph 4 x  2 y  12
Find the intercepts only
1) 2 x  y  4
2)
1
x  3y  6
2
23