2-1: 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.

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, or mapping.

• We have already seen relations represented as ordered pairs.

Table

{(3, 4), (7, 2), (0, -1),

(-2, 2), (-5, 0), (3, 3)}

x y 3 4 7 2 0 -1 -2 2 -5 0 3 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.

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.

x

y

x

y

Does the graph represent a function? Name the domain and range.

Yes

D: all reals

R: all reals

Yes

D: all reals

R: y ≥ -6

x

y

x

y

Does the graph represent a function? Name the domain and range.

NoD: x ≥ 1/2R: all reals

NoD: all realsR: all reals

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

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 - 2f(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

Value of a Function

If f(k) = k2 - 3, find f(a - 1)

f(a - 1)=(a - 1)2 - 3

(Remember FOIL?!)

=(a-1)(a-1) - 3

= a2 - a - a + 1 - 3

= a2 - 2a - 2