Graphing Linear Relations
and Functions
Objectives:
• Understand, draw, and determine if a relation is a
function.
• Graph & write linear equations, determine domain and
range.
• Understand and calculate slope.
Domain & Range
• A relation is a set of ordered pairs.
• Domain: first components in the relation
(independent)
• Range: second components in the relation
(dependent, the value depends on what the
domain value is)
• Functions are SPECIAL relations: A domain
element corresponds to exactly ONE range
element.
EXAMPLE
• Consider the function: eye color
• (assume all people have only one color, and it is
not changeable)
• It IS a function because when asked the eye color
of each person, there is only one answer.
• i.e. {(Joe, brown), (Mo, blue), (Mary, green), (Ava,
brown), (Natalie, blue)}
• NOTE: the range values are not necessarily
unique.
Relations & Functions
Relation: a set of ordered pairs
Domain: the set of x-coordinates
Range: the set of y-coordinates
When writing the domain and range, do not repeat values!
Relations and Functions
Given the relation:
{(2, -6), (1, 4), (2, 4), (0,0), (1, -6), (3, 0)}
State the domain:
D: {0,1, 2, 3}
State the range:
R: {-6, 0, 4}
Relations and Functions
•Relations can be written in several
ways:
• ordered pairs
• table, graph
• mapping.
•We have already seen relations
represented as ordered pairs.
Table
{(3, 4), (7, 2), (0, -1),
(-2, 2), (-5, 0), (3, 3)}
x
3
7
0
-2
-5
3
y
4
2
-1
2
0
3
Mapping
•Create two ovals with the domain on
the left and the range on the right.
•Elements are not repeated.
•Connect elements of the domain with
the corresponding elements in the
range by drawing an arrow.
Mapping
{(2, -6), (1, 4), (2, 4), (0, 0), (1, -6), (3, 0)}
2
1
0
3
-6
4
0
Functions
•A function is a relation in which the
members of the domain (x-values)
DO NOT repeat.
•So, for every x-value there is only
one y-value that corresponds to it.
•y-values can be repeated.
Functions
•Discrete functions consist of
points that are not connected.
•Continuous functions can be
graphed with a line or smooth
curve and contain an infinite
number of points.
Do the ordered pairs represent a
function?
{(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)}
No, 3 is repeated in the domain.
{(4, 1), (5, 2), (8, 2), (9, 8)}
Yes, no x-coordinate is repeated.
Graphing a function
• Horizontal axis: x values
• Vertical axis: y values
• Plot points individually or use a graphing utility
(calculator or computer algebra system)
• Example:
2
y  x 1
Graphs of functions
Graphs of a Function
Vertical Line Test:
If a vertical line is passed
over the graph and it intersects
the graph in exactly one point,
the graph represents a function.
What is the domain & range of the
function with this graph?
1) Domain : (, ), Range : (, )
2) Domain : (3, ), Range : (, )
3) Domain : (3, ), Range :(3, )
4) Domain : (, ), Range : (3, )
Does the graph represent a function?
Name the domain and range.
x
Yes
D: all reals
R: all reals
y
x
y
Yes
D: all reals
R: y ≥ -6
Does the graph represent a function?
Name the domain and range.
No
D: x ≥ 1/2
R: all reals
x
y
No
D: all reals
R: all reals
x
y
Does the graph represent a function?
Name the domain and range.
Yes
D: all reals
R: y ≥ -6
No
D: x = 2
R: all reals
x
y
x
y
Can you identify domain & range
from the graph?
• Look horizontally.
• What all x-values are contained in the graph?
• That’s your domain!
• Look vertically.
• What all y-values are contained in the graph?
• That’s your range!
Function Notation
•When we know that a relation is a
function, the “y” in the equation can be
replaced with f(x).
•f(x) is simply a notation to designate a
function. It is pronounced ‘f’ of ‘x’.
•The ‘f’ names the function, the ‘x’ tells
the variable that is being used.
Value of a Function
Since the equation y = x - 2
represents a function, we can also
write it as f(x) = x - 2.
Find f(4):
f(4) = 4 - 2
f(4) = 2
Value of a Function
If g(s) = 2s + 3, find g(-2).
g(-2) = 2(-2) + 3
=-4 + 3
= -1
g(-2) = -1
Value of a Function
If h(x) = x2 - x + 7, find h(2c).
h(2c) = (2c)2 – (2c) + 7
= 4c2 - 2c + 7
HOMEWORK:
• Complete worksheet packet
• Worksheet packet due November 6th
• Sit quietly AND work until the bell rings!!!!!