Graphing linear relations and functions

Nov 27, 2012Download as pptx, pdf3 likes2,173 views

The document provides an overview of linear relations and functions. It defines relations as sets of ordered pairs and functions as relations where each x-value corresponds to only one y-value. It discusses representing relations as ordered pairs, tables, mappings, and graphs. Key aspects of functions covered include discrete vs continuous functions, the vertical line test, function notation such as f(x), and evaluating functions by finding values such as f(4) given f(x) = x - 2.

2-1: Graphing Linear Relations
and Functions bytarungehlot

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), x y
(-2, 2), (-5, 0), (3, 3)} 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
-6
1
4
0
0
3

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.

Does the graph represent a function?
Name the domain and range.
Yes
x D: all reals
R: all reals
y

Yes
x D: all reals
R: y ≥ -6
y

Does the graph represent a function?
Name the domain and range.
No
x D: x ≥ 1/2
R: all reals
y

No
x D: all reals
R: all reals
y

Does the graph represent a function?
Name the domain and range.
Yes
x D: all reals
R: y ≥ -6
y

No
x D: x = 2
R: all reals
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 - 2
f(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

The document discusses linear relations and functions. It defines relations and functions, and explains how to determine if a relation is a function based on whether the domain contains repeating x-values. It shows how to represent relations as ordered pairs, tables, mappings, and graphs. It introduces the vertical line test to determine if a graph represents a function. It also explains function notation and how to find the value of a function for a given input.

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The document discusses set theory and Venn diagrams. It defines sets, subsets, unions, intersections, and complements. Sets can be described using words, lists, or set-builder notation. Venn diagrams are used to visually represent relationships between sets, such as intersections, unions, and complements. Examples are provided to demonstrate finding intersections, unions, subsets, disjoint sets, and using Venn diagrams to solve problems involving set relationships.

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Graphing linear relations and functions

1. 2-1: Graphing Linear Relations and Functions bytarungehlot 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), x y (-2, 2), (-5, 0), (3, 3)} 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)} 2 -6 1 4 0 0 3

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 x D: all reals R: all reals y Yes x D: all reals R: y ≥ -6 y

13. Does the graph represent a function? Name the domain and range. No x D: x ≥ 1/2 R: all reals y No x D: all reals R: all reals y

14. Does the graph represent a function? Name the domain and range. Yes x D: all reals R: y ≥ -6 y No x D: x = 2 R: all reals y

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

Graphing linear relations and functions

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