INTRODUCTION TO FUNCTIONS

The coordinate plane is formed by the intersection of two perpendicular number lines called axes . The point of intersection, called the origin , is at 0 on each number line. The horizontal number line is called the x-axis , and the vertical number line is called the y-axis .

Points on the coordinate plane are described using ordered pairs. An ordered pair consists of an x-coordinate and a y-coordinate and is written (x, y). Points are often named by a capital letter.

Graphing Points in the Coordinate Plane 

Example 1 :

Graph each point :

M(3, 4), N(-2, 0)

Solution :

Locating Points in the Coordinate Plane

Example 2 :

Name the quadrant in which each point lies.

Solution :

P -----> Quadrant III

Q -----> Quadrant II

R -----> No Quadrant (x-axis)

S -----> Quadrant IV

T -----> Quadrant I

U -----> Quadrant I

V -----> Quadrant III

W -----> Quadrant II

Function

An equation that contains two variables can be used as a rule to generate ordered pairs. When you substitute a value for x, you generate a value for y. The value substituted for x is called the input , and the value generated for y is called the output .

In a function, the value of y (the output) is determined by the value of x (the input). All of the equations in this lesson represent functions.

Solving Application Problem

Example 3 :

A caricature artist charges his clients a $5 setup fee plus $10 for every person in a picture. Write a rule for the artist’s fee. Write ordered pairs for the artist’s fee when there are 1, 2, 3, and 4 people in the picture.

Solution :

Let y represent the artist’s fee and x represent the number of people in a picture.

Artist’s fee is $5 plus $10 for each person.

y  =  5 + 10 · x

y  =  5 + 10x

Note :

When ordered pairs generated by a function are graphed, they may create a pattern.

Generating and Graphing Ordered Pairs

Generate ordered pairs for each function using the given values for x. Graph the ordered pairs and describe the pattern.

Example 4 :

y  =  4x - 3 ; x = -1, 0, 1, 2

Solution : 

The points form a straight line.

Example 5 :

y  =  2x² + 1 ; x = -2, -1, 0, 1, 2

Solution : 

The points form a U shape.

Example 6 :

y  =  |x + 3| ; x = -5, -4, -3, -2, -1

Solution : 

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