Transcript 5 • A

Bell Ringer 10/8/14
Bell Ringer 10/9/14
Name the locations of the four quadrants on a graph.
Objective 1
The student will be able to:
graph ordered pairs on a coordinate
plane.
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)
In an ordered pair, the first number is
the x-coordinate. The second number
is the y-coordinate.
Graph. (-3, 2)
5
•
5
-5
-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
The x-axis and y-axis separate the
coordinate plane into four
regions, called quadrants.
II
(-, +)
III
(-, -)
I
(+, +)
IV
(+, -)
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
HW
Objective 2 and 3
Linear Equations in Two Variables
Students will complete a table for a
linear equation and graph ordered
pairs.
List some pairs of numbers that will
satisfy the equation x + y = 4.
x = 1 and y = 3
x = 2 and y = 2
x = 4 and y = 0
What about negative numbers?
If x = -1 then y = ?
y=5
x+y=4
What about decimals?
If x = 2.6 then y = ?
y = 1.4
Now, let’s graph the pairs of numbers
we have listed.
(1, 3) (2, 2) (4, 0) (-1, 5) (2.6, 1.4)
•
••
•
•
Connect the points on your graph.
What does the graph look like?
It is a straight line! It is a linear relation.
•
What does the line
represent?
••
•
•
All solutions for the
equation x+y=4!
Is (3, -1) a solution to this equation?
NO! You can check by graphing it or
plugging into the equation!
1) Which is a solution to
2x – y = 5?
1.
2.
3.
4.
(2, 1)
(3, 2)
(4, 3)
(5, 4)
Answer Now
2) Which ordered pair is
not a solution to the graph
shown?
1.
2.
3.
4.
(0, -1)
(3, 5)
(-2, -5)
(-3, -1)
Answer Now
Bell Work 10/10/14
1. Write the following equation in standard
form and check if the ordered pair (4 , 4) is
a solution the linear equation:
3(y - 5) + 2(x + 2) = 10
Bell Ringer 10/13/14
1. For the linear equation -2x + 3y = 8,
determine whether the ordered pair is a
solution.
• A. (-4 , 0)
• B. (2 , -4)
2. Solve for y
-4x + 5y = -1
Objectives 3 and 4
The student will be able to:
1. graph linear functions.
2. write equations in standard form.
Graphing Steps
1) Isolate the variable (solve for y).
2) Make a t-table. If the domain is
not given, pick your own values.
3) Plot the points on a graph.
4) Connect the points.
1) Review: Solve for y
1. Draw “the river”
2. Subtract 2x from
both sides
2) Solve for y:
1.
2.
3.
4.
2x + y = 4
- 2x
- 2x
y = -2x + 4
4x + 2y = -6
- 4x
- 4x
Subtract 4x
2y = -4x - 6
Simplify
Divide both sides by 2
2
2
Simplify
y = -2x - 3
3) Solve for y:
x - 3y = 6
x
x
Subtract x
Simplify
-3y = -x + 6
Divide both sides by -3
-3
-3
1.
2.
3.
4. Simplify
x  6
y
3
or
x
y  2
3
4) Review: Make a t-table
If f(x) = 2x + 4, complete a table
using the domain {-2, -1, 0, 1, 2}.
x
-2
-1
0
1
2
f(x)
ordered pair
2(-2) + 4 = 0
(-2, 0)
2(-1) + 4 = 2
(-1, 2)
2(0) + 4 = 4
(0, 4)
2(1) + 4 = 6
(1, 6)
2(2) + 4 = 8
(2, 8)
5) Given the domain {-2, -1, 0, 1, 2},
graph 3x + y = 6
1. Solve for y:
Subtract 3x
2. Make a table
x -3x + 6
-2 -3(-2) + 6 = 12
-1 -3(-1) + 6 = 9
0
-3(0) + 6 = 6
1
-3(1) + 6 = 3
2
-3(2) + 6 = 0
3x + y = 6
- 3x
- 3x
y = -3x + 6
ordered pair
(-2, 12)
(-1, 9)
(0, 6)
(1, 3)
(2, 0)
5) Given the domain {-2, -1, 0, 1, 2},
graph 3x + y = 6
3. Plot the points
(-2,12), (-1,9), (0,6), (1,3),
(2,0)
4. Connect the points.
Bonus questions!
What is the x-intercept?
(2, 0)
What is the y-intercept?
(0, 6)
Does the line increase or decrease?
Decrease
Which is the graph of y = x – 4?
1.
2.
3.
4.
.
.
.
.
Standard Form
Ax + By = C
A, B, and C have to be integers
An equation is LINEAR (the graph
is a straight line) if it can be written
in standard form.
This form is useful for graphing
(later on…).
Determine whether each equation is
a linear equation.
1) 4x = 7 + 2y
Can you write this in the form
Ax + By = C?
4x - 2y = 7
A = 4, B = -2, C = 7
This is linear!
here
Determine whether each equation is
a linear equation.
2) 2x2 - y = 7
Can you write it in standard form?
NO - it has an exponent!
Not linear
3) x = 12
x + 0y = 12
A = 1, B = 0, C = 12
Linear
Here’s the cheat sheet! An equation that is
linear does NOT contain the following:
1. Variables in the denominator
3
y  2
x
2. Variables with exponents
y  x 3
2
3. Variables multiplied with other
variables.
xy = 12
Is this equation linear?
x  4y  3
1. Yes
2. No
Standard Form
x – 4y = 3
Is this equation linear?
2
9  4y  x
1. Yes
2. No
Exponents are
not allowed!
Is this equation linear?
y = -3
1. Yes
2. No
Standard Form
0x + y = -3
Bell Ringer 10/14/14
Solve for y.
Evaluate the following expression
Domain: -4, -2, 0, 2, 4
Objective 4 and 5
The student will be able to:
find the x- and y-intercepts of linear
equations.
What does it mean to INTERCEPT
a pass in football?
The path of the defender crosses the path
of the thrown football.
In algebra, what are x- and y-intercepts?
What are the x- and y-intercepts?
The x-intercept is where
the graph crosses the
x-axis.
The y-coordinate is
always 0.
The y-intercept is where
the graph crosses the
y-axis.
The x-coordinate is
always 0.
(2, 0)
(0, 6)
Find the x- and y-intercepts.
1. x - 2y = 12
x-intercept: Plug in 0 for y.
x - 2(0) = 12
x = 12; (12, 0)
y-intercept: Plug in 0 for x.
0 - 2y = 12
y = -6; (0, -6)
Find the x- and y-intercepts.
2. -3x + 5y = 9
x-intercept: Plug in 0 for y.
-3x - 5(0) = 9
-3x = 9
x = -3; (-3, 0)
y-intercept: Plug in 0 for x.
-3(0) + 5y = 9
5y = 9
9
9
y = ; (0, )
5
5
Find the x- and y-intercepts.
3. y = 7
***Special case***
x-intercept: Plug in 0 for y.
Does 0 = 7?
No! There is no x-intercept. None
What type of lines have no x-intercept?
Horizontal!
Horizontal lines…y = 7…y-int = (0, 7)
1.
2.
3.
4.
What is the x-intercept of
3x – 4y = 24?
(3, 0)
(8, 0)
(0, -4)
(0, -6)
1.
2.
3.
4.
What is the y-intercept of
-x + 2y = 8?
(-1, 0)
(-8, 0)
(0, 2)
(0, 4)
1.
2.
3.
4.
What is the y-intercept of
x = 3?
(3, 0)
(-3, 0)
(0, 3)
None
Objective
The student will be able to:
find the slope of a line given 2 points and
a graph.
What is the meaning of this sign?
1.
2.
3.
4.
Icy Road Ahead
Steep Road Ahead
Curvy Road Ahead
Trucks Entering
Highway Ahead
What does the 7% mean?
7% is the slope of the road.
It means the road drops 7 feet
vertically for every 100 feet
horizontally.
7%
7 feet
100 feet
So, what is slope???
Slope is the steepness of a line.
Slope can be expressed different ways:
( y2  y1 ) rise
vertical change
m


( x2  x1 ) run horizontal change
A line has a positive slope if it is
going uphill from left to right.
A line has a negative slope if it is
going downhill from left to right.
1) Determine the slope of the line.
When given the graph, it is easier to apply
“rise over run”.
Determine the slope of the line.
Start with the lower point and count how
much you rise and run to get to the other
point!
6
3
rise
3
1
=
=
run
6
2
Notice the slope is positive
AND the line increases!
2) Find the slope of the line that passes
through the points (-2, -2) and (4, 1).
When given points, it is easier to use the formula!
( y2  y1 )
m
( x2  x1 )
y2 is the y coordinate of the 2nd ordered pair (y2 = 1)
y1 is the y coordinate of the 1st ordered pair (y1 = -2)
(1  (2)) (1  2) 3 1
m

 
(4  (2)) (4  2) 6 2
Did you notice that Example #1 and
Example #2 were the same problem
written differently?
6
3
(-2, -2) and (4, 1)
1
slope 
2
You can do the problems either way!
Which one do you think is easiest?
Find the slope of the line that passes
through (3, 5) and (-1, 4).
1.
2.
3.
4.
4
-4
¼
-¼
3) Find the slope of the line that goes
through the points (-5, 3) and (2, 1).
y2  y1
m
x2  x1
1 3
m
25
1 3
m
2  (5)
2
m
7
Determine the slope of the line shown.
1.
2.
3.
4.
-2
-½
½
2
Determine the slope of the line.
-1
2
Find points on the graph.
Use two of them and
apply rise over run.
rise 2

 2
run 1
The line is decreasing (slope is negative).
What is the slope of a horizontal line?
The line doesn’t rise!
0
m
0
number
All horizontal lines have a slope of 0.
What is the slope of a vertical line?
The line doesn’t run!
number
m
 undefined
0
All vertical lines have an undefined slope.
Draw a line through the point (2,0)
that has a slope of 3.
1
3
1. Graph the ordered pair (2, 0).
2. From (2, 0), apply rise over
run (write 3 as a fraction).
3. Plot a point at this location.
4. Draw a straight line through
the points.