Graphing LinesGraphing Lines

How To Find How To Find THE SLOPETHE SLOPE of of a Line Given Two Pointsa Line Given Two Points

Slope of a Line

x

y

x1 x2

y2

y1

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x2 − x1

€

y2 − y1

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m =rise

run=y2 − y1

x2 − x1

€

x1,y1( )

€

x2 ,y2( )

Slope of a Line

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m =rise

run=y2 − y1

x2 − x1

Definition of Slope

• The slope of the line is it’s measure of steepness. It measures the rate of change of the line. In all lines the slope is constant, it doesn’t change no matter where you are at on the line.

Find the Slope of the line containing the points

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a( ) −3,8( ) and 2,−7( )

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b( ) 3.24,0.13( ) and 2.01,−0.07( )

BicyclesIn groups, generate the line that relates the number of wheels to the number of bicycles. Represent it Graphically.

What does the slope of this line represent?

Graph of a Line

x y

-1 -5

0 -3

1 -1

2 1

3 3

Intercepts

• Where the graph crosses the x-axis is the x-intercept. It has coordinates (a,0).

• Where the graph crosses the y-axis is the y-intercept. It has coordinates (0,b).

y

xx-intercept

y-intercept

(a,0)

(0,b)

The x and y intercepts of a line

Example

• Find the x and y intercepts of the line given by:

y =2x−6

Slope Intercept Form

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y = mx + b

x

y

€

x − 0

€

y − b

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x,y( )

Any stinkingpoint on the line

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m =y − b

x − 0

€

m =y − b

x

€

mx = y − b

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0,b( )

y-intercept

Determine the slope and y-intercept of the following

€

1( ) y = −x +1

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(2) 3x − 4y = 8

Point Slope Form

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y − y1 = m x − x1( )

x

y

€

x − x1

€

y − y1

€

x,y( )

Any stinkingpoint on the line

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m =y − y1x − x1

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m x − x1( ) = y − y1

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x1,y1( )

Any stinking fixedpoint on the line

Find the Equation of the Line Through the Points

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1( ) 6,9( ) and 2,−1( )

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2( ) −2,1( ) and 2,−1( )

Line Formulas

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m =y2 − y1

x2 − x1

Slope Formula

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y = mx + b Slope Intercept Form

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y − y1 = m x − x1( ) Point Slope Form

Definition of FunctionDefinition of Function

oror

How to RelateHow to Relate

Definition of a Relation

Relation

(1)32 mpg(2)8 mpg(3)16 mpg

(A)

(C)

(B)

Domain and Range

• The values that make up the set of independent values are the domain

• The values that make up the set of dependent values are the range.

• State the domain and range from the 4 examples of relations given.

DomainDomain RangeRange

Correspondenceor

Relation

Definition of a Relation

• A Relation maps a value from the domain to the range. A Relation is a set of ordered pairs.

• The most common types of relations in algebra map subsets of real numbers to other subsets of real numbers.

Example

Domain Range

3 π

11 - 2

1.618 2.718

Definition of a Function

• If a relation has the additional characteristic that each element of the domain is mapped to one and only one element of the range then we call the relation a Function.

Definition of a Function

• If we think of the domain as the set of boys and the range the set of girls, then a function is a monogamous relationship from the domain to the range. Each boy gets to go out with one and only one girl.

• But… It does not say anything about the girls. They get to live in Utah.

Decide if the Relation is a Function.

• The relation is the year and the cost of a first class stamp.

• The relation is the weight of an animal and the beats per minute of it’s heart.

• The relation is the time of the day and the intensity of the sun light.

• The relation is a number and it’s square.• The relation is time since you left your house

for work and your distance from home.

Examples Please

• Give three examples from the real world of relations. Be sure and state the domain, the range, and the definition of how the variables are related.

• Decide which if any of your examples are functions.

x

DOMAIN

y1

y2

RANGE

R

NOT A FUNCTION

y

RANGE

f

FUNCTION

x1

DOMAIN

x2

Mathematical Examples

• Decide if the following relations are functions.

X Y

1 2

-5 7

-1 2

3 3

X Y

1 1

-5 1

-1 1

3 1

X Y

1 2

1 7

1 2

1 3

X Y

1 π

π 1 -1 5

π 3

Ways to Represent a FunctionWays to Represent a Function• SymbolicSymbolic

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x,y( ) y = 2x{ }or

y = 2x

X Y

1 2

5 10

-1 -2

3 6

• GraphicalGraphical

• NumericNumeric

• VerbalVerbalThe cost is twice the original amount.

Function NotationThe Symbolic Form

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

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y = f x( )

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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

Examples of Function Notation

• The f notation

• Find f(2), g(-1), f(-0.983),

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f x( ) = 3x − x 2

g x( ) = x 2 − 4x +1

Your Turn!Your Turn!

€

f x( ) =x +1

2x − 3Given:

Evaluate the following:

€

a( ) f 1( )

€

b( ) f −2( )

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c( ) f 1.5( )

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d( ) f a( )

Graphical Representation

• Graphical representation of functions have the advantage of conveying lots of information in a compact form. There are many types and styles of graphs but in algebra we concentrate on graphs in the rectangular (Cartesian) coordinate system.

Graphs and Functions

Domain

Range

Determine the Domain and Range for Each Function

From Their Graph

Vertical Line Test for Functions

• If a vertical line intersects a graph once and only once for each element of the domain, then the graph is a function.

How to determine Domain and Range of a function.

• Graph the following on your calculator. Also give the algebra.

€

a( ) f x( ) =x

x − 3

€

c( ) f x( ) =1

2− x

€

b( ) f x( ) = x 3 − 5x

Big Deal!Big Deal!

• A point is in the set of A point is in the set of ordered pairs that make up ordered pairs that make up the function if and only if the function if and only if the point is on the graph of the point is on the graph of the function.the function.

Key Points

• Definition of a function

• Ways to represent a functionSymbolicallyGraphicallyNumericallyVerbally