Chapter 2Chapter 2Sections 1- 3Sections 1- 3

Functions and GraphsFunctions and Graphs

Definition of a Relation

A Relation is a mapping, or pairing, of input values with output . A set of ordered pairs is a relation.

The values that make up the set of input values are the domain or also independent variables.

The values that make up the set of output values are the range or also dependent variables

A relation is a function provided there is exactly one output for each input

Input Output

-3

1

3

4

3

1

-2 Do

mai

nR

ang

eGiven: (-3, 3), (1, 1), (3, 1), (4, -2)

It is not a function if at least one input has more than one output.

Input Output

-3

1

4

3

1

-2 Do

mai

nR

ang

eGiven: (-3, 3), (1, 1), (1, -2), (4, 4)

4

To determine if a graph is a function, we perform the vertical line test.

-- Yes, it is a function.

-- No, it is not a function.

Vertical Line Test for Functions

• A relation is a function if and only if no vertical line intersects the graph of the relation at more than one point.

Vertical Line Test:

1.Draw a vertical line through the graph.

2. See how many times the vertical line intersects the graph.

3. Only Once – Pass (function)

More than Once – Fail (not function)

Is this graph a function?

Yes, this is a function because it passes the vertical line test.

Only crosses at one point.

Is this graph a function?

No, this is not a function because it does not pass the vertical line test.

Crosses at more than one point.

The functions in the last two examples are linear functions because it is of the form

y = mx + b Linear Function

where m and b are constants

The graph of a linear function is a line.

By naming a function “f ” you can write the function using function notation.

f(x) = mx + b Function Notation

Function NotationThe Symbolic Form

• A truly excellent notation. It is concise and useful.

y f x

y f x • Output Value• Member of the Range• Dependent Variable

These are all equivalent names for the y.

• Input Value• Member of the Domain• Independent Variable

These are all equivalent names for the x.

Name of the function

Example of Function Notation

• The f notation

f x x 1

f 2 2 1

Decide whether the function is linear. Evaluate the function

when x = -21. f(x) = -x2 – 3x + 5 2. g(x) = 2x + 6

f(-2) = g(-2) =

Slope can be expressed different ways:

2 1

2 1

( ) vertical change

( ) horizontal change

y y risem

x x run

iablestindependeninchangeiablesdependentinchange

varvar

12

12 )()(

xx

xfxf

Slope is sometimesreferred to as the“rate of change”

between 2 points.

Types of Slope

PositiveNegative

Zero

Undefinedor

No Slope

What is the slope of a horizontal line?

The line doesn’t rise!

All horizontal lines have a slope of 0.

f(x) = 3

What is the slope of a vertical line?

The line doesn’t run!

All vertical lines have an undefined slope.

x = -2

SlopeParallel lines

Their slopes will be EQUAL.Perpendicular lines

Their slopes will be the negative reciprocal of each other.

Are the two lines parallel?L1: through (-2, 1) and (4, 5) and

L2: through (3, 0) and (0, -2)

3

2

3

2

21

21

LL

mm

This symbol means Parallel

Write parallel, perpendicular or neither for the pair of lines that passes through (5, -9) and (3, 7) and

the line through (0, 2) and (8, 3).

8-

8

1

21

21 1-

LL

mm

This symbol means Perpendicular

In the Mojave Desert in California, temperatures can drop quickly from day to night. Suppose the

temperature drops from 100ºF at 2 P.M. to 68ºF at 5 A.M. Find the average rate of change and use it to

determine the temperature at 10 P.M.

• Average rate of change = timeinchange

etemperaturinchange

hourperFhourspmam

2

15

32

25

10068

At 10 P.M. the temperature willbe 84ºF

The formula for Slope-Intercept Form is:

• ‘b’ is the y-intercept.

• ‘m’ is the slope.

Graph using the y-intercept and slope.

f(x) = mx + b

f(x) = 2x + 1

Sometimes we must solve the equation for y before we can graph it.

2x y 3

2x y ( 2x) ( 2x) 3

y 2x 3

The constant, b = 3 is the y-intercept.The coefficient, m = -2 is the slope.

f(x) = -2x + 3

The standard form of a linear equation Ax + By = C where A and B are not both 0

To find the y intercept, let x = 0 and solve for y.

Ax + By = C

To find the x intercept. let y = 0 and solve for x.

Ax + By = C

Graphing Equations with Intercepts

1. Write the equation in standard form.2. Find the x-intercept by letting y = 0

and solving for x. Use this x-intercept to plot the point where the line crosses the x-axis.

3. Find the y-intercept by letting x = 0 and solving for y. Use the y-intercept to plot the point where the line crosses the y-axis.

4. Draw a line through the two points.

YOU TRYGraph: 3x - 2y = 6

The equation of a vertical line cannot be written in slope-intercept form because the slope of a vertical line is undefined

Every linear equation, however, can be written in standard form-even the equation of a vertical line.

Horizontal and Vertical LinesHorizontal Lines: The graph of f(x) = c is a

horizontal line through (0, c)

Vertical Lines: The graph of x = c is a vertical line through (c, 0)

f(x) = 5 x = -3

Example

Graph: x = 2

Graph: f(x) = -3