Transcript 1.3 Graphing Equations - Warren County Schools

§ 1.3
Graphing Equations
The Rectangular Coordinate System
In the rectangular coordinate system, the horizontal number line is the x-axis.
The vertical number line is the y-axis. The point of intersection of these axes is
their zero points, called the origin. The axes divide the plane into 4 quarters,
called quadrants.
y-axis
2nd quadrant
1st quadrant
x-axis
3rd quadrant
4th quadrant
Blitzer, Algebra for College Students, 6e – Slide #2 Section 1.3
The Rectangular Coordinate System
Each point in the rectangular coordinate system corresponds to an ordered pair
of real numbers (x,y). Note the word “ordered” because order matters. The first
number in each pair, called the x-coordinate, denotes the distance and direction
from the origin along the x-axis. The second number, the y-coordinate, denotes
vertical distance and direction along a line parallel to the y-axis or along the y-axis
Itself.
Point
Movement from origin
(3.5)
3 units right and 5 units up
(4,-3)
4 units right and 3 units down
(-2,-7)
2 units left and 7 units down
(0,0)
0 right or left and 0 up or down
In plotting points,
we move across first
(either left or right), and
then move either up or
down, always starting
from the origin.
Blitzer, Algebra for College Students, 6e – Slide #3 Section 1.3
Plotting Points
EXAMPLE
Plot the points (3,2) and (-2,-4).
SOLUTION
 (3,2)
(-2,-4) 
Blitzer, Algebra for College Students, 6e – Slide #4 Section 1.3
The Graph of an Equation
The graph of an equation in two variables is the set of
points whose coordinates satisfy the equation.
An ordered pair of real numbers (x,y)
is said to satisfy the equation when substitution of the x and y coordinates
into the equation makes it a true statement.
For example, in the equation
y = 2x + 6, the ordered pair (1,8) is a solution. When we substitute this point
the sentence reads 8 = 8, which is true.
The ordered pair (2,3) is not a solution. When we substitute this point, the
sentence reads 3 = 10, which is not true.
Blitzer, Algebra for College Students, 6e – Slide #5 Section 1.3
Graphing an Equation
EXAMPLE
Graph y = 2|x| + 1.
SOLUTION
x
y = 2|x| + 1
Ordered Pair (x,y)
-2
y = 2|-2| + 1 = 2(2) + 1 = 4 + 1 = 5
(-2,5)
-1
y = 2|-1| + 1 = 2(1) + 1 = 2 + 1 = 3
(-1,3)
0
y = 2|0| + 1 = 2(0) + 1 = 0 + 1 = 1
(0,1)
1
y = 2|1| + 1 = 2(1) + 1 = 2 + 1 = 3
(1,3)
2
y = 2|2| + 1 = 2(2) + 1 = 4 + 1 = 5
(2,5)
Blitzer, Algebra for College Students, 6e – Slide #6 Section 1.3
Graphing an Equation
CONTINUED
(-2,5)
 (2,5)
(-1,3)

(1,3)

In graphing an equation, we
try to get enough ordered pairs
to get a good idea of what the
graph looks like. Next,
we plot these points. Finally, we
connect these points with a
smooth curve or line, always
moving from left to right. This
often gives us a picture of all
ordered pairs that satisfy the
equation.
(0,1)
Blitzer, Algebra for College Students, 6e – Slide #7 Section 1.3
Graphing an Equation
EXAMPLE
Graphs in the rectangular coordinate system can also be used to tell a
story. Try to select the graph that best illustrates the story of the
population of the U.S.A.
Years
Population
(c)
Population
(b)
Population
(a)
Years
SOLUTION
Graph (c)
Blitzer, Algebra for College Students, 6e – Slide #8 Section 1.3
Years