Chapter 2 Section 1: Relations and Graphs
Introduction
A relation is a set of ordered pairs. Curly brackets,
order does not matter). Parentheses,
{ } , are used to enclose the elements of a set (where
( ) , are used to enclose an ordered pair. The pair is called “ordered”
because there is a first element and a second element, and the pair would not be the same if the elements
were listed in the other order. For example, (1, 2 ) is a different ordered pair than ( 2, 1) .
A mathematical relation doesn’t necessarily involve numbers. Marriage can also be considered a relation.
Just let the first element of each ordered pair be a wife, and let the second element be her spouse.
However, in this course, the ordered pairs will be pairs of numbers. Thus, each ordered pair can be
thought of as a point in the coordinate plane. The first element of the ordered pair is the x -coordinate of
the point and the second element is the corresponding y -coordinate of the point.
The Coordinate Plane
In your previous algebra course, you learned about Cartesian coordinates (also referred to as
" x y -coordinates" or as "rectangular coordinates"). Each point in the coordinate plane has a unique
"address," which has a position relative to two perpendicular number lines (the x -axis and the y -axis).
The address of each point is an ordered pair of real numbers. The x -coordinate is the first element of the
ordered pair; it tells how far away the point is from the y -axis. The y -coordinate is the second element
of the ordered pair; it tells how far the point is from the x -axis. The intersection of the two axes is called
the origin and has coordinates ( 0, 0 ) , since it’s at a distance of zero from both axes.
The two axes break the plane into four quadrants, which are specified by Roman numerals. The image
below illustrates the axes and quadrants, and shows a few points and their coordinates.
Note: This image represents an animation that can only be seen in the online course.
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Chapter 2 Section 1: Relations and Graphs
In the graph above, note the following:
In Quadrant I:
Both coordinates are positive.
In Quadrant II:
The x -coordinate is negative and the y -coordinate is positive.
In Quadrant III: Both coordinates are negative.
In Quadrant IV: The x -coordinate is positive while the y -coordinate is negative.
Any set of points in the plane can be considered a relation. The plot of all the points in a relation is called
the graph of the relation.
The set of all the first elements of the ordered pairs in a relation is called the domain of the relation.
The set of all the second elements of the ordered pairs in a relation is called the range of the relation.
Example A
Graph this relation: R =
{(1, 0 ) , (1,1) , ( 0, 2 ) , ( − 2, 2 ) , ( − 3, 0 ) , ( − 3, − 3) , ( 0, − 4 ) , ( 3, − 4 )}
The points in relation R are plotted below:
(0,2)
(-2,2)
(1,1)
(1,0)
(-3,0)
(-3,-3)
(0,-4)
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(3,-4)
Chapter 2 Section 1: Relations and Graphs
Example B
What are the domain and range of relation R below?
R = {(1, 0 ) , (1,1) , ( 0, 2 ) , ( − 2, 2 ) , ( − 3, 0 ) , ( − 3, − 3) , ( 0, − 4 ) , ( 3, − 4 )}
.
The domain of R is the set of all the first coordinates of the points in the relation:
“Domain of R ” = {3, 1, 0, − 2, − 3} = {− 3, − 2, 0, 1, 3} .
With sets, order doesn’t matter. The two sets above are the same since they contain the same
elements. However, it’s a good idea to write the elements of a set in numerical order (or any other
appropriate order), so it’s easier to read.
The range of R is the set of all the second coordinates of the points in the relation:
“Range of R " = {0, 1, 2, − 3, − 4} = {− 4, − 3, 0, 1, 2} .
Defining Relations
Relations are often defined by equations involving the variables x and y . For example, consider this
relation, relation T :
⎧
T = ⎨( x , y )
⎩
y=
x⎫
⎬=
2⎭
“The set of all points ( x, y ) such
that y equals x divided by 2 .”
Relation T is the set of all points ( x, y ) with a y -coordinate that is half of its x -coordinate. For
example, the points ( 2,1) and ( 64, 32 ) are both elements in relation T (in both cases the y -coordinate
is half of the x -coordinate), but the point ( 3, 4 ) is not.
A relation like relation T given above is usually not written so formally (in set-builder notation).
x
. This equation is a linear equation
2
("linear" because its graph is a straight line, as we'll see in Example C). Points like ( 2,1) and ( 64, 32 )
Ordinarily, it would simply be referred to by the equation y =
are said to satisfy the equation. Point ( 3, 4 ) does not. For example, we can plug the x and y values of
point ( 2,1) into the given equation:
y=
x
2
We get a true result; ( 2,1) satisfies the equation.
→ 1=
2
2
But plugging in ( 3, 4 ) does not give a true result: y =
x
3
→ 4≠ .
2
2
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Chapter 2 Section 1: Relations and Graphs
Example C
Graph the equation y =
x
2
. In other words, graph the relation T that was defined on the previous
screen: T = ⎧ x, y y = x ⎫ .
)
⎨(
⎬
⎩
2⎭
You graph a relation by plotting points. The graph of an equation is simply the set of all points in
the plane whose x - and y -coordinates satisfy the equation. First make an x | y table. Put a
number in the x column. In this case, put half that number in the y column to satisfy the given
equation. Repeat several times.
y
x
0
2
4
6
8
10
0
1
2
3
4
5
Plotting and connecting these points, we get:
Notice how these
points seem to line
up. Adding more
points would
confirm this.
Example D
1
?
x
⎛ 1⎞
⎛ 1⎞
⎛2 3⎞
⎛2 ⎞
A = (1, 1) , B = ⎜ 1, ⎟ , C = ⎜ 2, ⎟ , D = ⎜ , ⎟ , E = ⎜ , 3 ⎟
⎝ 2⎠
⎝ 2⎠
⎝3 2⎠
⎝3 ⎠
Which of the following points satisfy the equation y =
To see if a point satisfies an equation, substitute the first coordinate (the
x -coordinate) into the equation, and see if the resulting y is in fact the
y -coordinate of the point.
The x -coordinate of point A is 1 . Substituting this into the equation we get:
y=
1
= 1.
1
Since the y -coordinate of point A is 1 , this point — point (1, 1) — does satisfy the equation, so
it is on the equation’s graph.
continued…
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Chapter 2 Section 1: Relations and Graphs
Example D, continued…
y=
1
⎛ 1⎞
⎛ 1⎞
⎛2 3⎞
⎛2 ⎞
, A = (1, 1) , B = ⎜ 1, ⎟ , C = ⎜ 2, ⎟ , D = ⎜ , ⎟ , E = ⎜ , 3 ⎟
x
⎝ 2⎠
⎝ 2⎠
⎝3 2⎠
⎝3 ⎠
The x -coordinate of point B is 1 . Substituting this into the equation we get:
y=
1
= 1.
1
(
Since the y -coordinate of point B is 1 (rather than 1 ), the point 1, 1
2
2
equation, so it is not on the equation’s graph.
) does not satisfy the
The x -coordinate of point C is 2 . Substituting this into the equation we get:
y=
1
.
2
Since the y -coordinate of point C is 1 , this point does satisfy the equation.
2
2 . Substituting this into the equation we get:
3
1
3
3
= 1⋅ = .
y=
2
2
⎛2⎞
⎜ ⎟
⎝3⎠
3 , point D does satisfy the equation.
Since the y -coordinate of point D is
2
The x -coordinate of point D is
2 . As above, substituting this into the equation, we get:
3
1
3
3
y=
= 1⋅ = .
2
2
2
3
3 ), this point does not satisfy the equation
Since the y -coordinate of point E is 3 (rather than
2
(and so it is not on the equation’s graph).
The x -coordinate of point E is also
( )
Extended Example 1a
1
, over the interval x = 0 to x = 5 .
x
Hint: Make an x | y table and fill in some x -values. Use the values of the given interval:
0, 1, 2, 3, 4, 5 .
Graph the equation y =
Step 1:
x
y
0
1
2
3
4
5
continued…
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Chapter 2 Section 1: Relations and Graphs
Extended Example 1a, continued…
Hint: Fill in the corresponding y -values. (Use decimal approximations, since they are easier to plot.)
Step 2:
For example, the first y -value in the following table is found by substituting 0 into the equation in place
of x : y =
1
1
=
= undefined .
x
0
y
x
0
1
2
3
4
5
Undefined
1
0.5
0.333
0.25
0.2
Hint: To see what happens between 0 and 1 , extend your table to include x = 0.5 and
x = 0.25 .
Step 3:
x
y
0
1
2
3
4
5
0.5
0.25
Undefined
1
0.5
0.333
0.25
0.2
2
4
Hint: Plot the points you just found, and smoothly connect the dots.
Answer:
Note: This time, the points of the graph are not along a
straight line. This is the graph of a nonlinear equation.
y = 1x
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Chapter 2 Section 1: Relations and Graphs
Extended Example 1b
Graph the equation y =
1
x2 + 1
over the interval x = − 3 to x = 3 .
Hint: Make an x | y table and fill in some x -values. Use the values of the given interval:
− 3, − 2, − 1, 0, 1, 2, 3 .
Step 1:
y
x
–3
–2
–1
0
1
2
3
Hint: Fill in the corresponding y -values (in decimal form, since it’s easier to plot).
Step 2:
For example, the first y -value in the following table is found by substituting ( − 3) into
y=
the equation in place of x :
x
y
–3
–2
–1
0
1
2
3
0.1
0.2
0.5
1
0.5
0.2
0.1
1
x2 + 1
=
1
( − 3) 2 + 1
=
1
1
=
= 0.1 .
9 + 1 10
Hint: Plot the points you just found, and smoothly connect the dots.
Answer:
y=
1
x 2 +1
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Chapter 2 Section 1: Relations and Graphs
Extended Example 1c
4
over the interval x = − 5 to x = 5 .
x2 + 2
Hint: Make an x | y table and fill in some x -values. Use − 5, − 3, − 2, − 1, 0, 1, 2, 3, 5 .
Graph the equation y = −
Step 1:
y
x
–5
–3
–2
–1
0
1
2
3
5
Hint: Fill in the corresponding y -values (in decimal form, rounded to the nearest hundredth,
since it’s easier to plot that way).
Step 2:
For example, the first y -value in the following table is found by substituting ( − 5) into the equation in
place of x :
y=−
4
x2 + 2
=−
x
y
–5
–3
–2
–1
0
1
2
3
5
−.15
−.36
−.67
−1.33
−2
−1.33
−.67
−.36
−.15
4
( − 5) 2 + 2
=−
4
4
=−
= − 0. 148 ≅ − 0.15 .
27
25 + 2
Hint: Plot the points you just found, and smoothly connect the dots.
Answer:
y=−
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4
x +2
2
Chapter 2 Section 1: Relations and Graphs
Example E
Graph the equation y = x − 2 over the interval x = − 1 to x = 5 .
Note: This image represents an animation that can only be seen in the online course.
Extended Example 2a
Graph the equation y = x + 3 over the interval x = − 6 to x = 0 .
Hint: Make an x | y table and fill in some x -values. Use the values of the given interval:
− 6, − 5, − 4, − 3, − 2, − 1, 0 .
Step 1:
x
y
–6
–5
–4
–3
–2
1
0
Hint: Fill in the corresponding y -values.
Step 2:
For example, the first y -value in the following table is found by substituting ( − 6 ) into the equation in
place of x : y = x + 3 = ( − 6 ) + 3 = − 3 = 3 .
x
y
–6
–5
–4
–3
–2
1
0
3
2
1
0
1
2
3
continued…
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Chapter 2 Section 1: Relations and Graphs
Extended Example 2a, continued…
Hint: Plot the points you just found; remember that the graph is V-shaped.
Answer:
y = x+3
Extended Example 2b
Graph the equation y = x − 4 over the interval x = 0 to x = 8 .
Hint: Make an x | y table and fill in some x -values. This time, let's use 0, 2, 4, 6, 8 (we'll
get the idea of the graph even without using all x -values in the interval).
Step 1:
x
y
0
2
4
6
8
Hint: Fill in the corresponding y -values.
Step 2:
For example, the first y -value in the following table is found by substituting 0 into the equation in place
of x : y = x − 4 = 0 − 4 = − 4 = 4 .
x
y
0
4
2
2
4
0
6
2
8
4
Hint: Plot the points you just found; remember that the graph is V-shaped.
Answer:
y = x−4
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Chapter 2 Section 1: Relations and Graphs
Extended Example 2c
Graph the equation y = − x − 2 over the interval x = − 2 to x = 6 .
Hint: Make an x | y table and fill in some x -values. Use − 2, 0, 2, 4, 6 .
Step 1:
x
y
–2
0
2
4
6
Hint: Fill in the corresponding y -values.
Step 2:
For example, the first y -value in the following table is found by substituting − 2 into the equation in
place of x : y = − x − 2 = − ( − 2 ) − 2 = − − 4 = − 4 = − 4 .
x
y
–2
0
2
4
6
–4
2
0
2
–4
Hint: Plot the points you just found; remember that the graph is V-shaped, but this V is upsidedown.
Answer:
y =− x−2
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Chapter 2 Section 1: Relations and Graphs
Example F
Graph the equation y = 10 x − x 2 over the interval x = 0 to x = 10 .
Make an x | y table and fill in the x -values 0, 1, 2,..., 10 .
x
y
0
1
2
3
Etc...
Plug the x -values into the equation to find the corresponding y -values:
x
y
0
1
2
3
4
5
6
7
8
9
10
0
9
16
21
24
25
24
21
16
9
0
For example, the y-value for
x = 2 is found this way:
y = 10 ( 2 ) − 2 2 = 20 − 4 = 16
Plot and connect the points you just found:
Note: This is the graph of a parabola. An object thrown in the air follows a path that is shaped like a
parabola.
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Chapter 2 Section 1: Relations and Graphs
Extended Example 3a
Graph the equation y = 4 x − x 2 over the interval x = − 1 to x = 5 .
Hint: Make an x | y table and fill in some x -values. Use − 1, 0, 1, 2, 3, 4, 5 .
Step 1:
x
y
–1
0
1
2
3
4
5
Hint: Fill in the corresponding y -values.
Step 2:
For example, the first y -value in the following table is found by substituting − 1 into the equation in
place of x :
y = 4 x − x 2 = 4 ( − 1) − ( − 1) = − 4 − ( − 1) = − 4 − 1 = − 5 .
y
x
2
–1
0
1
2
3
4
5
2
–5
0
3
4
3
0
–5
Hint: Plot the points you just found. The graph is parabola-shaped.
Answer:
y = 4x − x 2
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Chapter 2 Section 1: Relations and Graphs
Extended Example 3b
Graph the equation y = x 2 − 3x − 5 over the interval x = − 2 to x = 5 .
Hint: Make an x | y table and fill in some x -values. Use − 2, − 1, 0, 1, 2, 3, 4, 5 .
Step 1:
y
x
–2
–1
0
1
2
3
4
5
Hint: Fill in the corresponding y -values.
Step 2:
For example, the first y -value in the following table is found by substituting − 2 into the equation in
2
place of x : y = x − 3x − 5 = ( − 2 ) − 3 ( − 2 ) − 5 = 4 + 6 − 5 = 5 .
2
x
y
–2
–1
0
1
2
3
4
5
5
–1
–5
–7
–7
–5
–1
5
Hint: Plot the points you just found. The graph is parabola-shaped.
Answer:
y = x 2 − 3x − 5
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Chapter 2 Section 1: Relations and Graphs
Extended Example 3c
Graph the equation y = x 2 − 2 x + 2 over the interval x = − 1 to x = 3 .
Hint: Make an x | y table and fill in some x -values. Use − 1, 0, 1, 2, 3 .
Step 1:
x
y
–1
0
1
2
3
Hint: Fill in the corresponding y -values.
Step 2:
For example, the first y -value in the following table is found by substituting − 2 into the equation in
2
place of x : y = x − 2 x + 2 = ( − 1) − 2 ( − 1) + 2 = 1 + 2 + 2 = 5 .
2
x
y
–1
0
1
2
3
5
2
1
2
5
Hint: Plot the points you just found. The graph is parabola-shaped.
Answer:
y = x 2 − 2x + 2
End of Lesson
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