Functions
Unit Overview
In this unit, students will be able
to:
·
identify functions
and non-functions from tables, ordered pair sets, and graphs
·
identify linear
and non-linear functions
Key Concepts
·
relations and
functions
·
vertical line
test
·
linear and
non-linear functions
Ohio’s
Learning Standards
·
8.F.1
Understand that a function is a rule that assigns to each input exactly one
output. The graph of a function is the set of ordered pairs consisting of an
input and the corresponding output. Function notation is not required in Grade
8.
·
8.F.5 Describe
qualitatively the functional relationship between two quantities by analyzing a
graph, e.g., where the function is increasing or decreasing, linear or
nonlinear. Sketch a graph that exhibits the qualitative features of a function
that has been described verbally.
Calculators
· Here is a link
to the Desmos scientific calculator that is provided for the Ohio State
Test for 8th Grade Mathematics in the spring.
·
You are
strongly encouraged to use the Texas Instruments TI-30XIIS handheld scientific
calculator. It is extremely user
friendly.
FUNCTIONS
A relation is a set of ordered pairs.
Example A: Let’s look at an example of a relation written
as a list, an x-y table, in mapping form, and as a graph:
·
List: (0, -3), (1, 4), (3,-2), (4,
2), (7, 6)
·
x-y table:
 |
·
mapping form (seldom used):
numbers are put in numerical order, then the input (x-value) is connected with
its output (y-value)
 |
·
graph on coordinate plane
 |
A function
is a relation where each input is paired with exactly one output.
The example given above in various forms is a
function, because each input has exactly one output.
A relation is not a function if one input is paired with different
outputs.
Example B: Let’s look at an example of a relation that
is not a function. In each form, you
will see that there is an input (3) that is
paired with two different outputs (-2 and 6)
·
List: (0, -3), (1, 4), (3,-2), (4, 2), (3, 6)
·
x-y table:
 |
·
mapping
form:
 |
·
graph on coordinate plane
 |
While A function cannot have one input paired
with two different outputs, a function can have two different inputs paired
with the same output.
Example C: Here is a basic example of each situation:
FUNCTION: (2, 3), (-2, 3)
NOT A
FUNCTION: (5, 2), (5, -2)
·
this is not a function because one input (5) is paired with two
different outputs (2 and -2)
Vertical Line Test
When
determining if a graph is a function, you can use the vertical line test. If you can draw a vertical line anywhere on
the graph, and that line touches more than one point on the graph, that
relation is not a function.
Here are
several examples of functions and non-functions proven by the vertical line
test:
·
Example D:
 |
Example D is not a
function because the vertical line goes through both (3, 6) and (3, -2). An input has two different outputs.
·
Example E:
 |
Example E is not a function because a vertical line goes through both (4, 4)
and (4, -4). An input has two
different outputs.
·
Example F:
 |
Example F
is a function, because no vertical line touches the graphed line more than once.
Click on
the words IDENTIFYING FUNCTIONS for a
further explanation and practice:
Let’s
practice. Determine whether relation is
a function.
For the graph,
determine if the graphed line is a function:
For the graph,
determine if the graphed line is a function:
LINEAR VS. NON-LINEAR FUNCTIONS
Linear
function à a function that creates a straight line when
graphed
·
Example G:
 |
All of the graphed lines above are linear
functions (except for the blue line, because it does not pass the vertical line
test) because they are straight lines.
Non-Linear
function à a function that creates a non-straight line when
graphed
·
Example H:
 |
All of the graphed lines above are non-linear
functions (except for the black line, because it does not pass the vertical
line test) because they are non-straight, curved lines.
Let’s
practice.
| Which lines are not functions, linear, and non-linear? |
 |