Download 2-1 Relations and Functions

2-1 Relations and
Functions
Objectives
Students will be able to:
1) Analyze and graph relations
2) Find functional values
Note: You cannot spell function without “fun”
• Ordered pair: a pair of coordinates, written
in the form (x, y), used to locate any point
on a coordinate plane
• Relation: a set of ordered pairs
• Domain: the set of all x-coordinates of the
ordered pairs of a relation
• Range: the set of all y-coordinates of the
ordered pairs of a relation
Terminology
• A function is a special type of relation in
which each element of the domain is paired
with exactly one element of the range.
• One-to-one function: a function where
each element of the range is paired with
exactly one element of the domain
Functions
A mapping is a way of showing how each member of the
domain is paired with each member of the range.
Mappings
Mappings
State the relation shown in the graph. Then list the domain
and range. Is the relation a function?
• Relation:
• Domain:
• Range:
• Function???
Example 1
State the relation shown in the graph. Then list the domain and range.
Is the relation a function?
Relation:
{(-4, -2), (-2, 3), (2, -3), (2, 1)}
Domain:
{-4, -2, 2}
Range:
{-3, -2, 1, 3}
Function???
No! The x value of 2 repeats
You try
• When given a graph of a relation, one can
perform a vertical line test to determine whether
a relation is a function.
• If a vertical line, does not intersect the graph in
more than one point, then the relation is a
function. If they do intersect the graph in more
than one point, then the relation is not a function.
Vertical Line Test
Vertical Line Test
Yes, is a function
Not a function
Example 2: Vertical line
Test
Yes, is a function
Try these:
Not a function
Function Notation
Example 3
You Try!
2-2 Linear
Equations
Objective
Students will be able to identify and graph linear equations
and functions
• A linear equation is the equation of a straight line.
• The only operations that exist in linear equations are
addition, subtraction, and multiplication of a variable by a
constant.
• Linear equations are often written in slope-intercept form
(y=mx + b).
• Linear functions can be written in the form f(x)=mx + b.
• What linear equations would not be linear functions?
Linear Equations
• One way to graph a linear equation is by finding its xintercept and y-intercept.
• The x-intercept is the point at which the graph crosses the
x-axis. At this point, the y value will be 0. The ordered
pair will be (x, 0).
• The y-intercept is the point at which the graph crosses the
y-axis. At this point, the x value will be 0. The ordered
pair will be (0, y).
Graphing w/ Intercpets
Find the x-intercept and the y-intercept for each equation.
Then use the intercepts to graph the equation.
x-intercept:
y-intercept:
Example 1:
x-intercept:
y-intercept:
You try.
NOTE: When finding intercepts, there are times when you
will not attain two ordered pairs. Remember, to graph a
linear equation, you need at least two ordered pairs. Times
you will not attain two ordered pairs occur when:
1) The equation is vertical
x=constant
2) The equation is horizontal
y=constant
3) Both intercepts occur at (0, 0)
Let’s look at an example…
Problems w/ Intercepts!
x-intercept:
y-intercept:
Graph?!
When you do not attain two ordered pairs via the intercept
method, you have a few options.
1) You can create a table of x and y values. This is a way
of attaining a few ordered pairs to help you graph the
line.
2) If the equation is in slope-intercept form, use the yintercept and slope to graph the line. If it is not in
slope-intercept form, get it in slope-intercept form!
Other Graphing Methods
2-3 Slope
Objectives
Students will be able to:
1)
Find and use the slope of a line
2)
Graph linear equations using slope-intercept form
• The slope of a line is the ratio of the change in ycoordinates to the corresponding change in x-coordinates
• Slope is also referred to as rate of change.
Slope…
Four types of slope
Find the slope of the line that passes through each pair of
points.
a) (-1, 4) and (1, -2)
b) (1, 3) and (-2, -3)
c) (6, 4) and (-3, 4)
Example 1:
Find the slope of the line that passes through each pair of
points.
d) (-6, -3) and (6, 7)
e) (5, 8) and (5, 0)
You Try
y = mx + b
Why is it so useful?
The equation gives us two pieces of information we
need to graph a linear equation: it’s slope, and it’s yintercept. If we have these pieces of information we can
graph any linear equation.
Slope-Intercept Form
Example 2: Graph each
equation.
You Try
Try More…
Oh man! Try some More