Relations, Functions and Graphs, Exams of Nursing

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RELATIONS, FUNCTIONS AND

GRAPHS

Aufmann^ (COLLEGE ALGEBRA AND TRIGONOMETRY,, Barker and Nation 7th ed., page 175 - 178, 184-189, 252-259)^1

WEEK 1 and Graphs - Relations, Functions

OBJECTIVES

At the end of the lesson, the student is expected to be able to:

• familiarize with the use of Cartesian Coordinate System
• determine the distance between two points
• determine the coordinates of a point of division of a line segment.
• distinguish functions and relations
• identify domain and range of a function/relation
• evaluate functions/relations
• perform operation on functions/relations
• graph functions/relations

• The coordinates of a point are determined by the point’s position relative to a horizontal coordinate axis called the x-axis and a vertical axis called the y-axis.
• The axes intersect at the point ( 0 , 0 ), called the origin.
• The four regions formed by the axes are called quadrants and are numbered counter clockwise.
• The two-dimensional coordinate system is referred to as a Cartesian Coordinate System.

Aufmann, Barker and Nation 7th ed., page^ (COLLEGE ALGEBRA AND TRIGONOMETRY, 175 - 178, 184-189, 252-259)^4

WEEK 1 and Graphs - Relations, Functions

Graph of an Equation by Point-Plotting

Definition of the Graph of an Equation The graph of an equation in the two variables and is the set of all points ( x , y ) whose coordinates satisfy the equation. Example 1: Consider the equation 𝑦 = 2𝑥 + 1 , graph by point- plotting.

Aufmann, Barker and Nation 7th ed., page^ (COLLEGE ALGEBRA AND TRIGONOMETRY, 175 - 178, 184-189, 252-259)^5

𝒙 𝒚 = 𝟐𝒙 + 𝟏 𝒚 (𝒙, 𝒚) − 2 2 − 2 + 1 − 3 (− 2 , − 3 ) − 1 2 − 1 + 1 − 1 (− 1 , − 1 ) 0 2 0 + 1 1 ( 0 , 1 ) 1 2 1 + 1 3 ( 1 , 3 ) 2 2 2 + 1 5 ( 2 , 5 )

WEEK 1 and Graphs - Relations, Functions

Example 2. Graph: 𝑦 = 𝑥 + 3

Example 3. Graph: 𝑦 = 2 − 𝑥

Example 4. Graph: 𝑥^2 + 𝑦 = 5

Example 5. Graph: 𝑦^2 = 4𝑥

Aufmann, Barker and Nation 7th ed., page^ (COLLEGE ALGEBRA AND TRIGONOMETRY, 175 - 178, 184-189, 252-259)^7

WEEK 1 and Graphs - Relations, Functions

DISTANCE BETWEEN TWO POINTS

The length of a horizontal line segment is

the abscissa (x-coordinate) of the point on

the right minus the abscissa (x-coordinate)

of the point on the left.

1. Horizontal

2. Vertical

The length of a vertical line segment

is the ordinate (y-coordinate) of the

upper point minus the ordinate (y-

coordinate) of the lower point.

SAMPLE PROBLEMS

1. Determine the distance between a. (- 2 , 3 ) and ( 5 , 1 ) b. ( 6 , - 1 ) and (- 4 , - 3 )

2. Show that points A ( 3 , 8 ), B (- 11 , 3 ) and C (- 8 , - 2 ) are vertices of an isosceles triangle.

3. Show that the triangle A ( 1 , 4 ), B ( 10 , 6 ) and C ( 2 , 2 ) is a right triangle.

4. Find the point on the y-axis which is equidistant from A(- 5 , - 2 ) and B( 3 , 2 ).

DIVISION OF A LINE

SEGMENT

1 2

1

P P

r  PP

P 2  x 2 ,y 2 

P  x,y 

P 1  x 1 ,y 1  M  x,y 1  N  x 2 ,y 1 

Internal Point of Division

 

1 2

1 2 2 1 1 2

1 2 2 1 2

1 2

1

2 1

1 2

2 1

1 1

1 1 2

1 1 2

1 1 2

1

If P ,then and

Formula :

andforexternalpoint 1. 2

and. Forinternalpoint 1 ,formidpoint

wherein. But ; ;

iscomputed with the useof ratio and proportion,

thefigure isasimilartriangle,the coordinatesof point

r r

y r y r y r r

x r x r x r

r PP

P

Alternative

r r

r y y

y y P N

PM

x x

x x P N

r P M P P

P P

P N

PM

P N

P M

P P

P P

P x, y

Since

^ 

 ^ 

^ 

   ^ 

Examples :

1. The line segment joining (- 5 , - 3 ) and ( 3 , 4 ) is to be

divided into five equal parts. Find the point of

division closest to (- 5 , - 3 ).

2. Find the midpoint of the segment joining ( 7 , - 2 ) and

3. The line segment from ( 1 , 4 ) to ( 2 , 1 ) is extended a

distance equal to twice its length. Find the terminal

point.

4. On the line joining ( 4 , - 5 ) to (- 4 , - 2 ), find the point

which is three-seventh the distance from the first to

the second point.

5. Find the trisection points of the line joining (- 6 , 2 )

and ( 3 , 8 ).