UNIT 1.8 INTRODUCTION TO UNIT 1.8 INTRODUCTION TO

EQUATIONS/FUNCTIONSEQUATIONS/FUNCTIONS

Warm UpAdd.

1. Draw and label a number line. Then plot the points –2, 0, and 4.

Evaluate each expression for the given value of x.

210123456 6543- - - - - -• • •

2. 2x + 1 for x = 314

3. – x + 3 for x = 8

4. |x + 6| for x = –10

7

1

4

Graph ordered pairs in the coordinate plane.Graph functions from ordered pairs.

Objectives

coordinate planeaxesoriginx-axisy-axisordered pairx-coordinate

Vocabulary

y-coordinate

quadrant

input

output

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.

The x-coordinate tells how many units to move left or right from the origin. The y-coordinate tells how many units to move up or down.

Reading Math

Example 1: Graphing Points in the Coordinate Plane

Graph each point.

A. T(–4, 4)

Start at the origin.Move 4 units left and 4 units up.

B. U(0, –5)Start at the origin.Move 5 units down.

•T(–4, 4)

•U(0, –5)

C. V (–2, –3)Start at the origin.Move 2 units left and 3 units down.

•V(–2, −3)

Check It Out! Example 1Graph each point.

A. R(2, –3)

B. S(0, 2)

Start at the origin.Move 2 units right and 3 units down.

Start at the origin.Move 2 units up.

C. T(–2, 6)

Start at the origin.Move 2 units left and6 units up.

•R(2, –3)

S(0,2)

T(–2,6)

Look at the graph at the top of this lesson. The axes divide the coordinate plane into four quadrants. Points that lie on an axis are not in any quadrant.

Example 2: Locating Points in the Coordinate Plane

Name the quadrant in which each point lies.

A. EQuadrant ll

B. Fno quadrant (y-axis)

C. GQuadrant l

D. HQuadrant lll

•E

•F

•H

•Gx

y

Name the quadrant in which each point lies.

A. Tno quadrant (y-axis)

B. UQuadrant l

C. VQuadrant lll

D. WQuadrant ll

Check It Out! Example 2

•T

•W

•V

•U

x

y

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.

y = 10x + 5

Output Input

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.

Example 3: Art Application

An engraver charges a setup fee of $10 plus $2 for every word engraved. Write a rule for the engraver’s fee. Write ordered pairs for the engraver’s fee when there are 5, 10, 15, and 20 words engraved.

Let y represent the engraver’s fee and x represent the number of words engraved.

Engraver’s fee is $10 plus $2 for each word

y = 10 + 2 · x

y = 10 + 2x

The engraver’s fee is determined by the number of words in the engraving. So the number of words is the input and the engraver’s fee is the output.

Writing Math

Example 3 Continued

Number ofWords

EngravedRule Charges

Ordered Pair

x (input) y = 10 + 2x y (output) (x, y)

y = 10 + 2(5)5 20 (5, 20)

y = 10 + 2(10)10 30 (10, 30)

y = 10 + 2(15)15 40 (15, 40)

y = 10 + 2(20)20 50 (20, 50)

Check It Out! Example 4

What if…? The caricature artist increased his fees. He now charges a $10 set up fee plus $20 for each person in the picture. Write a rule for the artist’s new fee. Find the artist’s fee when there are 1, 2, 3 and 4 people in the picture.

y = 10 + 20x

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

Artist’s fee is $10 plus $20 for each person

y = 10 + 20 · x

Number of People in Picture

Rule ChargesOrdered

Pair

x (input) y = 10 + 20x y (output) (x, y)

y = 10 + 20(1)1 30 (1, 30)

y = 10 + 20(2)2 50 (2, 50)

y = 10 + 20(3)3 70 (3, 70)

y = 10 + 20(4)4 90 (4, 90)

Check It Out! Example 4 Continued

When you graph ordered pairs generated by a function, they may create a pattern.

Example 4A: Generating and Graphing Ordered Pairs Generate ordered pairs for the function using the given values for x. Graph the ordered pairs and describe the pattern.

y = 2x + 1; x = –2, –1, 0, 1, 2

–2

–101

2

2(–2) + 1 = –3 (–2, –3)(–1, –1)(0, 1)

(1, 3)(2, 5)

2(–1) + 1 = –12(0) + 1 = 1

2(1) + 1 = 32(2) + 1 = 5

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Input OutputOrdered

Pairx y (x, y)

The points form a line.

Example 4B: Generating and Graphing Ordered Pairs

y = x2 – 3; x = –2, –1, 0, 1, 2

–2

–101

2

(–2)2 – 3 = 1 (–2, 1)

(–1, –2)(0, –3)

(1, –2)(2, 1)

(–1)2 – 3 = –2(0)2 – 3 = –3

(1)2 – 3 = –2(2)2 – 3 = 1

Input OutputOrdered

Pairx y (x, y)

The points form a U shape.

Example 4C: Generating and Graphing Ordered Pairs

y = |x – 2|; x = 0, 1, 2, 3, 4

0

123

4

|0 – 2| = 2 (0, 2)

(1, 1)(2, 0)

(3, 1)(4, 2)

|1 – 2| = 1|2 – 2| = 0

|3 – 2| = 1

|4 – 2| = 2

Input OutputOrdered

Pairx y (x, y)

The points form a V shape.

–4

–202

4

–2 – 4 = –6 (–4, –6)

(–2, –5)(0, –4)

(2, –3)

(4, –2)

–1 – 4 = –50 – 4 = –4

1 – 4 = –3

2 – 4 = –2

Input OutputOrdered

Pairx y (x, y)

The points form a line.

Check It Out! Example 4a

y = x – 4; x = –4, –2, 0, 2, 412

–3

–101

3

3(–3)2 + 3 = 30 (–3, 30)

(–1, 6)(0, 3)

(1, 6)(3, 30)

3(–1)2 + 3 = 63(0)2 + 3 = 3

3(1)2 + 3 = 63(3)2 + 3 = 30

Input OutputOrdered

Pairx y (x, y)

The points form a U shape.

Check It Out! Example 4b

y = 3x2 + 3; x = –3, –1, 0, 1, 3

0

123

4

|0 – 2| = 2 (0, 2)

(1, 1)(2, 0)

(3, 1)(4, 2)

|1 – 2| = 1|2 – 2| = 0

|3 – 2| = 1

|4 – 2| = 2

Input OutputOrdered

Pairx y (x, y)

The points form a V shape.

Check It Out! Example 4c

y = |x – 2|; x = 0, 1, 2, 3, 4

Lesson Quiz: Part 1

Graph each point. Name the quadrant in which each point lies.

1. (2, 0)

2. (–3, –4)

3. (1, –1)

4. (–5, 4)

None

lll

lV

ll

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Lesson Quiz: Part 2

5. A cable company charges $50 to set up a movie channel and $3.00 per movie watched. Write a rule for the company’s fee. Write ordered pairs for the fee when a person watches 1, 2, 3, or 4 movies.

y = 50 + 3x; (1, 53), (2, 56), (3, 59), (4, 62)

Lesson Quiz: Part 3

6. Generate ordered pairs for y = x² –5 using x = –2, –1, 0, 1, and 2. Graph the ordered pairs, and describe the pattern.

(–2, –1)

(–1, –4)

(0, –5)

(1, –4)

(2, –1)

The pattern is U-shaped.

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