2-1 Relations and FunctionsObjectives

Students will be able to:

1) Analyze and graph relations

2) Find functional values

Note: You cannot spell function without “fun”

Terminology

• 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

Functions

• 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

Mappings

A mapping is a way of showing how each member of the domain is paired with each member of the range.

Mappings

Example 1

State the relation shown in the graph. Then list the domain and range. Is the relation a function?• Relation:

• Domain:

• Range:

• Function???

You try

State the relation shown in the graph. Then list the domain and range. Is the relation a function?

Relation:

Domain:

Range:

Function???

{(-4, -2), (-2, 3), (2, -3), (2, 1)}

{-4, -2, 2}

{-3, -2, 1, 3}

No! The x value of 2 repeats

Vertical Line Test

• 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

Example 2: Vertical line Test

Yes, is a function Not a function

Try these:

Yes, is a function Not a function

Function Notation

Example 3

You Try!

2-2 Linear EquationsObjective

Students will be able to identify and graph linear equations and functions

Linear Equations

• 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?

Graphing w/ Intercpets

• One way to graph a linear equation is by finding its x-intercept 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).

Example 1:

Find the x-intercept and the y-intercept for each equation. Then use the intercepts to graph the equation.

x-intercept:

y-intercept:

You try.

x-intercept:

y-intercept:

Problems w/ Intercepts!

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…

Graph?!

x-intercept:

y-intercept:

Other Graphing Methods

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 y-intercept and slope to graph the line. If it is not in slope-intercept form, get it in slope-intercept form!

2-3 SlopeObjectives

Students will be able to:

1) Find and use the slope of a line

2) Graph linear equations using slope-intercept form

Slope…

• The slope of a line is the ratio of the change in y-coordinates to the corresponding change in x-coordinates

• Slope is also referred to as rate of change.

Four types of slope

Example 1:

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)

You Try

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)

Slope-Intercept Form

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 y-intercept. If we have these pieces of information we can graph any linear equation.

Example 2: Graph each equation.

You Try

Try More…

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