Graphing

1
Graphing
Chapter Eight

• 8.3 Rectangular Coordinate System Paired Data
• 8.4 Graphing Linear Equations in Two Variables

2
The Rectangular Coordinate System and Paired Data
Section 8.3
3
The Rectangular Coordinate System
y-axis
5
4
quadrant I
quadrant II
3
2
origin
1
(0,0)
x-axis
1
2
3
4
5
-1
-2
-3
- 4
- 5
-2
-3
- 4
quadrant III
quadrant IV
- 5
4
Plotting Points
y
5
4
(4,3)
3
2
1
x
3
1
4
5
-1
-2
-3
- 4
- 5
2
-2
-3
- 4
- 5
5
In general, to plot the ordered pair (x,y), start
at the origin. Next,
(x,y)
move x units left or right and then move y units
up or down.
right if x is positive, left if x is negative
up if y is positive, down if y is negative
Martin-Gay, Prealgebra, 5ed
6
Since the first number, or x-coordinate, of an
ordered pair is associated with the x-axis, it
tells how many units to move left or right.
Similarly, the second number, or y-coordinate,
tells how many units to move up or down.
Martin-Gay, Prealgebra, 5ed
7
Plot (2,1), (- 3,4), (5,0), (0,- 2), (1,- 3), and
(- 4,- 5)
y
5
4
(- 3,4)
3
2
(2,1)
1
(5,0)
x
1
2
3
4
5
-1
-2
-3
- 4
- 5
(0,- 2)
-2
-3
(1,- 3)
- 4
(- 4,- 5)
- 5
8
Remember that each point in the rectangular
coordinate system corresponds to exactly one
ordered pair and that each ordered pair
corresponds to exactly one point.
Martin-Gay, Prealgebra, 5ed
9
If an ordered pair has a y-coordinate of 0, its
graph lies on the x-axis. If an ordered pair has
an x-coordinate of 0, its graph lies on the
y-axis. Order is the key word in ordered pair.
The first value always corresponds to the x-value
and the second value always corresponds to the
y-value.
Martin-Gay, Prealgebra, 5ed
10
Completing Ordered Pair Solutions
An equation in two variables, such as 3x y 9,
has solutions consisting of two values, one for x
and one for y.
For example, x 1 and y 6 is a solution of 3x
y 9, because, if x is replaced with 1 and y
is replaced with 6, we get a true statement.
The solution x 1 and y 6 can be written as
(1,6), an ordered pair of numbers.
3x y 9
9 9
True
Martin-Gay, Prealgebra, 5ed
11
In general, an ordered pair is a solution of an
equation in two variables if replacing the
variables by the values of the ordered pair
results in a true statement.
Martin-Gay, Prealgebra, 5ed
12
If the x-value of an ordered pair is known, then
the y-value can be determined, and vice-versa.
Complete each ordered pair so that it is a
solution to the equation 2x - y 6.
(0, )
( , 4)
Let x 0 and solve for y.
Let y 4 and solve for x.
2x - y 6
2x - y 6
2(0) - y 6
2x - 4 6
0 - y 6
2x 10
y - 6
x 5
The ordered pair is (0,- 6).
The ordered pair is (5,4).
13
Helpful Hint
Have you noticed?
Equations in two variables can have more than one
solution.
Martin-Gay, Prealgebra, 5ed
14
Graphing Linear Equations in Two Variables
Section 8.4
15
Linear Equation in Two Variables

• A linear equation in two variables is an
  equation that can be written in the form
• ax by c where a, b, and c are numbers,
  and a and b are not both 0.
• Examples
• 3x 2y 6 y 8 3x 4

Martin-Gay, Prealgebra, 5ed
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Graphing Linear Equations by Plotting Points

• Every linear equation in two variables has
  infinitely many ordered-pair solutions. Since it
  is impossible to list every solution, we graph
  the solutions instead.
• The pattern described by the solutions of a
  linear equation makes seeing the solutions
  possible by graphing because all the solutions of
  a linear equation in two variables correspond to
  points on a single straight line.
• If we plot a few of these points and draw the
  straight line connecting them, we have a complete
  graph of all the solutions.

Martin-Gay, Prealgebra, 5ed
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To graph the equation x y 5, we plot a few
ordered-pair solutions, say (2,3), (0,5), and
(-1, 6). Then we connect the points.
y
(0,5)
(- 1,6)
5
4
(2,3)
3
2
1
x
1
2
3
4
5
-1
-2
-3
- 4
- 5
Martin-Gay, Prealgebra, 5ed
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To Graph a Linear Equation in Two Variables

• Find three ordered-pair solutions.
• Graph the solutions.
• Draw a line through the plotted points.

Martin-Gay, Prealgebra, 5ed
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To Find an Ordered-Pair Solution of an Equation

• Choose either an x-value or y-value of the
  ordered pair.
• Complete the ordered pair by replacing the
  variable with the chosen value and solving for
  the unknown variable.

Martin-Gay, Prealgebra, 5ed
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All three points should fall on the same straight
line. If not, check your ordered-pair solutions
for a mistake, since every linear equation is a
line.
Martin-Gay, Prealgebra, 5ed
21
Horizontal Lines/Vertical Lines
y b
x a
Martin-Gay, Prealgebra, 5ed