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Functions & Relations
Warm Up
State whether each word or phrase represents an amount
that is increasing, decreasing, or constant.
1. stays the same constant
2. rises
increasing
3. drops decreasing
4. slows down
decreasing
Warm Up
Generate ordered pairs for the function
y = x + 3 for x = –2, –1, 0, 1, and 2. Graph the ordered
pairs.
(–2, 1)
(–1, 2)
(0, 3)
(1, 4)
(2, 5)
Objectives
Match simple graphs with situations.
Graph a relationship.
Identify functions.
Find the domain and range of relations and
functions.
Vocabulary
continuous graph
discrete graph
relation
domain
range
function
Graphs can be used to illustrate many different
situations. For example, trends shown on a
cardiograph can help a doctor see how a patient’s heart
is functioning.
To relate a graph to a given situation, use key words in
the description.
Example 1
The air temperature increased steadily for several hours and
then remained constant. At the end of the day, the
temperature increased slightly before dropping sharply.
Choose the graph that best represents this situation.
Step 1 Read the graphs from left to right to show time passing
.
Example 1 Continued
Step 2 List key words in order and decide which graph
shows them.
Key Words
Increased
steadily
Remained
constant
Increased
slightly before
dropping
sharply
Segment Description
Graphs…
Slanting upward
Graph C
Horizontal
Graphs A, B,
and C
Slanting upward and then
steeply downward
Graphs B and C
Step 3 Pick the graph that shows all the key phrases in
The correct graph is graph C.
order.
As seen in Example 1, some graphs are connected lines
or curves called continuous graphs. Some graphs are
only distinct points. They are called discrete graphs
The graph on theme park
attendance is an example of a
discrete graph. It consists of distinct
points because each year is distinct
and people are counted in whole
numbers only. The values between
whole numbers are not included,
since they have no meaning for the
situation.
Sketching Graphs for Situations
Sketch a graph for the situation. Tell whether the graph is
continuous or discrete.
A truck driver enters a street, drives at a constant speed,
stops at a light, and then continues.
• initially increases
• remains constant
• decreases to a stop
• increases
• remains constant
y
Speed
As time passes during the trip (moving
left to right along the x-axis) the
truck's speed (y-axis) does the
following:
Time
The graph is continuous.
x
Helpful Hint
When sketching or interpreting a graph, pay close
attention to the labels on each axis.
Sketching Graphs for Situations
Sketch a graph for the situation. Tell whether the graph is
continuous or discrete.
A small bookstore sold between 5 and 8 books each day
for 7 days.
The number of books sold
(y-axis) varies for each day
(x-axis).
Since the bookstore accounts
for the number of books sold
at the end of each day, the
graph is 7 distinct points.
The graph is discrete.
Try This!
Sketch a graph for the situation. Tell whether the graph is
continuous or discrete.
Jamie is taking an 8-week keyboarding class. At the end of
each week, she takes a test to find the number of words she
can type per minute. She improves each week.
Each week (x-axis) her typing
speed is measured. She gets
a separate score (y-axis) for
each test.
Since each score is separate, the
graph consists of distinct units.
The graph is discrete.
Both graphs show a relationship about a child going
down a slide. Graph A represents the child’s distance
from the ground related to time. Graph B represents the
child’s Speed related to time.
You have seen relationships represented by graphs.
Relationships can also be represented by a set of
ordered pairs called a relation.
In the scoring systems of some track meets, for first place
you get 5 points, for second place you get 3 points, for
third place you get 2 points, and for fourth place you get 1
point. This scoring system is a relation, so it can be
shown by ordered pairs. {(1, 5), (2, 3), (3, 2) (4, 1)}. You
can also show relations in other ways, such as tables,
graphs, or mapping diagrams.
Showing Multiple Representations of Relations
Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a
graph, and as a mapping diagram.
Table
x
y
2
3
4
7
6
8
Write all x-values under “x” and all
y-values under “y”.
Continued
Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a
graph, and as a mapping diagram.
Graph
Use the x- and y-values to
plot the ordered pairs.
Continued
Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a
graph, and as a mapping diagram.
Mapping Diagram
x
y
2
3
4
7
6
8
Write all x-values under “x” and all yvalues under “y”. Draw an arrow
from each x-value to its
corresponding y-value.
Try This!
Express the relation {(1, 3), (2, 4), (3, 5)} as a table, as a
graph, and as a mapping diagram.
Table
x
y
1
3
2
4
3
5
Write all x-values under “x” and all
y-values under “y”.
Try This! Continued
Express the relation {(1, 3), (2, 4), (3, 5)} as a table, as a
graph, and as a mapping diagram.
Graph
Use the x- and y-values to
plot the ordered pairs.
Try This! Continued
Express the relation {(1, 3), (2, 4), (3, 5)} as a table, as a
graph, and as a mapping diagram.
Mapping Diagram
y
x
1
3
2
4
3
5
Write all x-values under “x” and all yvalues under “y”. Draw an arrow
from each x-value to its
corresponding y-value.
The domain of a relation is the set of first
coordinates (or x-values) of the ordered pairs. The
range of a relation is the set of second coordinates
(or y-values) of the ordered pairs. The domain of
the track meet scoring system is {1, 2, 3, 4}. The
range is {5, 3, 2, 1}.
Finding the Domain and Range of a Relation
Give the domain and range of the relation.
The domain value is all x-values from 1
through 5, inclusive.
The range value is all y-values from 3
through 4, inclusive.
Domain: 1 ≤ x ≤ 5
Range: 3 ≤ y ≤ 4
Try This!
Give the domain and range of the relation.
6
5
2
1
–4
–1
0
Domain: {6, 5, 2, 1}
Range: {–4, –1, 0}
The domain values are all xvalues 1, 2, 5 and 6.
The range values are y-values
0, –1 and –4.
Try This!
Give the domain and range of the relation.
x
y
1
1
4
4
8
1
Domain: {1, 4, 8}
Range: {1, 4}
The domain values are all xvalues 1, 4, and 8.
The range values are yvalues 1 and 4.
A function is a special type of relation
that pairs each domain value with
EXACTLY ONE range value.
Identifying Functions
Give the domain and range of the relation. Tell whether the
relation is a function. Explain.
{(3, –2), (5, –1), (4, 0), (3, 1)}
D: {3, 5, 4}
R: {–2, –1, 0, 1}
Even though 3 is in the domain twice,
it is written only once when you are
giving the domain.
The relation is not a function. Each domain value does not
have exactly one range value. The domain value 3 is paired
with the range values –2 and 1.
Identifying Functions
Give the domain and range of the relation. Tell whether the
relation is a function. Explain.
–4
–8
4
5
2
1
D: {–4, –8, 4, 5}
Use the arrows to determine
which domain values correspond
to each range value.
R: {2, 1}
This relation is a function. Each domain value is paired with
exactly one range value.
Try This!
Give the domain and range of each relation. Tell whether the
relation is a function and explain.
a. {(8, 2), (–4, 1), (–6, 2),(1, 9)}
b.
D: {–6, –4, 1, 8}
R: {1, 2, 9}
The relation is a function.
Each domain value is
paired with exactly one
range value.
D: {2, 3, 4}
R: {–5, –4, –3}
The relation is not a function.
The domain value 2 is paired
with both –5 and –4.
Identifying Functions
Give the domain and range of the relation. Tell whether the
relation is a function. Explain.
Range
Draw in lines to
see the domain
and range
values
Domain
D: –5 ≤ x ≤ 3 R: –2 ≤ y ≤ 1
The relation is not a function. Nearly all domain values have
more than one range value.
The "Vertical Line Test"
Looking at this function stuff graphically, what if we had the
relation {(2, 3), (2, –2)}? We already know that this is not a
function, since x = 2 goes to both y = 3 and y = –2.
If we graph this relation, it looks like:
Notice that you can draw a vertical
line through the two points, like
this:
Given the graph of a relation, if you can draw a vertical line that
crosses the graph in more than one place, then the relation is
not a function. Here are a couple examples:
The "Vertical Line Test“ continued
This one is a
function. There is
no vertical line that
will cross this graph
twice.
This one is not a
function. Any
number of vertical
lines will intersect
this oval twice. For
instance, the y-axis
intersects twice.