RECTANGULAR COORDINATE SYSTEM
Quadrant II
(x<0, y>0)
5
Quadrant I
(x > 0, y>0)
4
ORDERED PAIR:
The first number in the
ordered pair is the xcoordinate (aka abscissa)
and the second number
in the ordered pair is the
y-coordinate (aka
ordinate).
3
( -5 , 3)
Origin
( 0 ,0 )
2
(
1
-5
-4
-3
-2
-1
,
x-axis
-1
1
2
3
4
5
-2
-3
Quadrant III
(x<0, y<0)
-4
Quadrant IV
(x>0, y<0)
-5
(
(
,
)
,
)
y-axis
In what Quadrant is the point (-2,-5)?
What is the x-value of any point on the y-axis?
What is the y-value of any point on the x-axis?
A point with a negative x-coordinate is on the ____ side of the y-axis.
A point with a negative y-coordinate is on the _____ side of the x-axis.
How many units away from the x-axis is the point (0,-4)? _________
)
Average Rate of Change of y respect to x is
Change in y / Change in x
= y-value of second point – y-value of first point
x-value of second point – x-value of first point
p. 259 #34
The following table shows the mileage on and
corresponding value of a used 2004 MINI Cooper,
2-door hatchback one year after it was purchased.
Mileage
10,000
20.000
30,000
40,000
50,000
60,000
17,850
16,625
15,125 13,975 13,150
Value, in 18,525
dollars
a) What was the per-mile change in the value of the car between
10,000 mi and 60,000 mi?
When you see “per mile”, think “divided by miles”. Since miles in the
divisor, miles must be the x-coordinate. Therefore Value of the Car
is the y-coordinate.
The first “point” is when the mileage is 10,000:
(10,000, 18,525)
The second “point” is when the mileage is 60,000:
(60,000, 13,150)
The per-mile change = (13,150 – 18,525) dollars / (60,000-10,000) miles
= (-5375) dollars / (50,000) miles
= -.1075 dollars/mile
≈ - $0.11 per mile
b) Explain the meaning of the answer to part a).
The MINI Cooper LOSES 11 cents in value for each mile driven.
Example of “Real World” graphs
The following graph has two sets of data overlaying each
other.
One is a scattered diagram with connected dots. The
other is a bar graph. They both share the same x-values,
which in this case are levels of education. The y-values
for the bar graph (Median Weekly Earnings) are on the
left, since earnings is represented by dollars. The yvalues for the scattered diagram (Unemployment Rates)
are on the right, sinc eunemployment rates are respented
as percentages.
A sample data point would as follows:
People with an Associate Degree have a median weekly
earnings of about $700 and about a 3% unemployment
rate.
Sometimes a set of points in a scatter diagram represents a straight line.
In this case, the points represent a linear equation.
A linear equation can be written in the form
y = mx + b
y and x are the two variables in the equation, and m and b are constants
that represent the slope and y-intercept.
The y-intercept is the point (0,b) where the line crosses the y-axis.
Example y = ⅔ x – 1
The slope, m, is ⅔
b = -1, so the y-intercept is (0,-1).
All points (x,y) that lie on the line y=⅔ x -1 are values of x and y that make
the equation true.
Is the point (-2,-2) a solution of y=⅔ x – 1?
Let x = -2 and y=-2
Does -2 = ⅔(-2) – 1 ?
Example 2
Find the ordered –pair solution of y = ⅔ x -1 that corresponds
to x = 3.
The ordered-pair solution is the point (x,y) where x = 3, and y is
the value of the equation y = ⅔ x -1 when x=3.
y = ⅔ (3) – 1
y =2- 1
y=1
Answer: The ordered –pair solution of y = ⅔ x -1 that
corresponds to x = 3 is
(3,1)
You try this one:
Find the ordered-pair solution of y = -¼ x + 1 that corresponds
to x=4.
Graph -2x + 3y = 6
Get y by itself first.
Add 2x to both sides…
3y = 2x + 6
Now divide both sides by 3
y=⅔x+2
This tells me that the y-intercept is (0,2) and the slope is ⅔
Start at (0,2) and then go UP 2 units then RIGHT 3 units.
right 3
up 2
y-intercept
(0,2)
Graph 3x + y = 1
Get y by itself first.
Subtract 3x from both sides…
y = -3x + 1
This tells me that the y-intercept is (0,1) and the slope is -3
A slope -3 could be written as -3/1 or 3/-1
Which means could draw our line two different ways but get the same
slant.
Using slope=-3/1 Start at (0,1) and then go DOWN 3 units then RIGHT
1 unit.
Alternatively, we could use slope = 3/-1:
Start at (0,1) and then go UP 3 units and then LEFT 1 unit.
left
1
up 3
y-intercept
(0,1)
down 3
right 1
The x-intercept of a linear equation is the point on the line that crosses
the x-axis. At this point y=0.
The x-intercept will be an ordered pair (__, 0)
The y-intercept of a linear equation is the point on the line that crosses
the y-axis. At this point x=0.
The y-intercept will be an ordered pair (0,__)
For example, find the x-intercept and y-intercept of the equation
3x + 4y = 12
The x-intercept is found by setting y=0 and solving for x.
3x + 4(0) = 12
3x = 12
x=4
The x-intercept is (4,0)
The y-intercept if found by setting x=0 and solving for y.
3(0) + 4y = 12
4y = 12
y=3
The y-intercept is (0,3)
This linear equation can easily be
graphed because we now have
two points on the line.