21: Graphing Linear Relations and Functions

1
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.

2
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.

3
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

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.

5
Table

• (3, 4), (7, 2), (0, -1),
• (-2, 2), (-5, 0), (3, 3)

6
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.

7
Mapping

• (2, -6), (1, 4), (2, 4), (0, 0), (1, -6), (3, 0)

8
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.

9
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.

10
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.

11
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.

12
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

13
Does the graph represent a function? Name the
domain and range.

• No
• D x 1/2
• R all reals
• No
• D all reals
• R all reals

14
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

15
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.

16
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

17
Value of a Function

• If g(s) 2s 3, find g(-2).
• g(-2) 2(-2) 3
• -4 3
• -1
• g(-2) -1

18
Value of a Function

• If h(x) x2 - x 7, find h(2c).
• h(2c) (2c)2 (2c) 7
• 4c2 - 2c 7

19
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