Graphs of Functions

The graph of a function gives you a visual representation

of its rule. A set of points generated like we did in the previous

section.

You can determine if a given graph is a true function by the

VERTICAL LINE TEST. Given the graph of a relation, the

vertical line test is used to check if that relation is a function.

The vertical line CAN NOT touch the graph in more than 1 spot.

Graphs of Functions

VERTICAL LINE TEST – a vertical line drawn anywhere on the graph

CAN NOT intersect the graph more than once.

Here is the graph of a relation.

Is it a true function ?

Graphs of Functions

VERTICAL LINE TEST – a vertical line drawn anywhere on the graph

CAN NOT intersect the graph more than once.

Here is the graph of a relation.

Is it a true function ?

YES…I can not draw a vertical line that passes thru the graph more than once

Graphs of Functions

VERTICAL LINE TEST – a vertical line drawn anywhere on the graph

CAN NOT intersect the graph more than once.

Here is the graph of a relation.

Is it a true function ?

Graphs of Functions

VERTICAL LINE TEST – a vertical line drawn anywhere on the graph

CAN NOT intersect the graph more than once.

Here is the graph of a relation.

Is it a true function ?

NO…I only need to find one spot where a vertical line crosses more than one time.

Graphs of Functions

One-to-One Functions – a function where each range value has one and ONLY one domain value.

Horizontal line test checks for one-to-one…

Here is the graph of a function.

Is it one-to-one ?

Graphs of Functions

One-to-One Functions – a function where each range value has one and ONLY one domain value.

Horizontal line test checks for one-to-one…

Here is the graph of a function.

Is it one-to-one ?

NO…I only need to find one spot where a horizontal line crosses more than one time.

Graphs of Functions

One-to-One Functions – a function where each range value has one and ONLY one domain value.

Horizontal line test checks for one-to-one…

Here is the graph of a function.

Is it one-to-one ?

Graphs of Functions

One-to-One Functions – a function where each range value has one and ONLY one domain value.

Horizontal line test checks for one-to-one…

Here is the graph of a function.

Is it one-to-one ?

YES…I can not draw a vertical line that crosses more than one time.

Graphs of Functions

One-to-One Functions – a function where each range value has one and ONLY one domain value.

Horizontal line test checks for one-to-one…

Functions that are not one-to-one have NO INVERSE !!!

Graphs of Functions

Graphing functions falls into two categories…

1. Continuous or constant functions ( no breaks )

2. Piecewise functions…breaks in the graph occur

We will first look at the Continuous Functions…

Steps : 1. Create a table of values for f(x)

2. Plot the points and sketch your graph

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x2 – 5

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x2 – 5

x f(x)

- 2 -1

f(-2) = (-2)2-5

f(-2) = 4 – 5

f(-2) = -1

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x2 – 5

x f(x)

- 2 -1

- 1 - 4

f(-1) = (-1)2-5

f(-1) = 1 – 5

f(-1) = - 4

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x2 – 5

x f(x)

- 2 -1

- 1 - 4

0 - 5

f(0) = (0)2-5

f(0) = 0 – 5

f(0) = - 5

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x2 – 5

x f(x)

- 2 -1

- 1 - 4

0 - 5

1 - 4

f(1) = (1)2-5

f(1) = 1 – 5

f(1) = - 4

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x2 – 5

x f(x)

- 2 -1

- 1 - 4

0 - 5

1 - 4

2 - 1

f(2) = (2)2-5

f(2) = 4 – 5

f(2) = -1

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x2 – 5

x f(x)

- 2 -1

- 1 - 4

0 - 5

1 - 4

2 - 1

Plot the points…

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x2 – 5

x f(x)

- 2 -1

- 1 - 4

0 - 5

1 - 4

2 - 1

Sketch your graph…

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x3 + 2

x f (x)

-3 - 25

f(-3) = (-3)3 + 2

f(-3) = - 27 + 2

f(-3) = -25

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x3 + 2

x f (x)

-3 - 25

-2 -6

f(-2) = (-2)3 + 2

f(-2) = -8 + 2

f(-2) = -6

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x3 + 2

x f (x)

-3 - 25

-2 -6

-1 1

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

f(-1) = -1+ 2

f(-1) = 1

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x3 + 2

x f (x)

-3 - 25

-2 -6

-1 1

0 2

f(0) = (0)3 + 2

f(0) = 0+ 2

f(0) = 2

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x3 + 2

x f (x)

-3 - 25

-2 -6

-1 1

0 2

1 3

f(1) = (1)3 + 2

f(1) = 1 + 2

f(1) = 3

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x3 + 2

x f (x)

-3 - 25

-2 -6

-1 1

0 2

1 3

2 10

f(2) = (2)3 + 2

f(2) = 8 + 2

f(2) = 10

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x3 + 2

x f (x)

-3 - 25

-2 -6

-1 1

0 2

1 3

2 10

3 29

f(3) = (3)3 + 2

f(3) = 27 + 2

f(3) = 29

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x3 + 2

x f (x)

-3 - 25

-2 -6

-1 1

0 2

1 3

2 10

3 29

Plot your points…

Graphs of Functions

Steps : 1. Create a table of values for ƒ(x)2. Plot the points and sketch your graph

EXAMPLE : Graph ƒ(x) = x3 + 2

x f (x)

-3 - 25

-2 -6

-1 1

0 2

1 3

2 10

3 29

Sketch the graph…