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**Introduction to Functions**

1-8 Introduction to Functions Holt Algebra 1 Warm Up Lesson Presentation Lesson Quiz

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Warm Up Add. 1. Draw and label a number line. Then plot the points –2, 0, and 4. 2 1 3 4 5 6 - • • • Evaluate each expression for the given value of x. 2. 2x + 1 for x = 3 7 1 4 3. – x + 3 for x = 8 1 4. |x + 6| for x = –10 4

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**Objectives Graph ordered pairs in the coordinate plane.**

Graph functions from ordered pairs.

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**Vocabulary coordinate plane y-coordinate axes quadrant origin input**

x-axis y-axis ordered pair x-coordinate y-coordinate quadrant input output

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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.

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**Points on the coordinate plane are described using ordered pairs**

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

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**Example 1: Graphing Points in the Coordinate Plane**

Graph each point. A. T(–4, 4) Start at the origin. T(–4, 4) • Move 4 units left and 4 units up. B. U(0, –5) Start at the origin. Move 5 units down. C. V (–2, –3) • V(–2, −3) Start at the origin. • U(0, –5) Move 2 units left and 3 units down.

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**Move 2 units right and 3 units down.**

Check It Out! Example 1 Graph each point. A. R(2, –3) Start at the origin. T(–2,6) Move 2 units right and 3 units down. S(0,2) B. S(0, 2) Start at the origin. Move 2 units up. C. T(–2, 6) • R(2, –3) Start at the origin. Move 2 units left and 6 units up.

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**Look at the graph at the top of this lesson**

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.

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**Example 2: Locating Points in the Coordinate Plane**

Name the quadrant in which each point lies. •E •F •H •G y A. E Quadrant ll B. F no quadrant (y-axis) x C. G Quadrant l D. H Quadrant lll

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

Check It Out! Example 2 Name the quadrant in which each point lies. y A. T no quadrant (y-axis) •W •U •T B. U Quadrant l x C. V Quadrant lll •V D. W Quadrant ll

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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. Output Input y = 10x + 5 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.

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**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 = x

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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

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**Example 3 Continued Number of Words Engraved Rule Charges Ordered Pair**

x (input) y = x y (output) (x, y) 5 y = (5) 20 (5, 20) 10 y = (10) 30 (10, 30) 15 y = (15) 40 (15, 40) 20 y = (20) 50 (20, 50)

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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. 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 y = x

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**Number of People in Picture**

Check It Out! Example 4 Continued Number of People in Picture Rule Charges Ordered Pair x (input) y = x y (output) (x, y) 1 y = (1) 30 (1, 30) 2 y = (2) 50 (2, 50) 3 y = (3) 70 (3, 70) 4 y = (4) 90 (4, 90)

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

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**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 • Input Output Ordered Pair x y (x, y) –2 2(–2) + 1 = –3 (–2, –3) –1 2(–1) + 1 = –1 (–1, –1) 2(0) + 1 = 1 (0, 1) 1 2(1) + 1 = 3 (1, 3) 2(2) + 1 = 5 2 (2, 5) The points form a line.

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**Example 4B: Generating and Graphing Ordered Pairs**

y = x2 – 3; x = –2, –1, 0, 1, 2 Input Output Ordered Pair x y (x, y) –2 (–2)2 – 3 = 1 (–2, 1) –1 (–1)2 – 3 = –2 (–1, –2) (0)2 – 3 = –3 (0, –3) 1 (1)2 – 3 = –2 (1, –2) (2)2 – 3 = 1 2 (2, 1) The points form a U shape.

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**Example 4C: Generating and Graphing Ordered Pairs**

y = |x – 2|; x = 0, 1, 2, 3, 4 Input Output Ordered Pair x y (x, y) |0 – 2| = 2 (0, 2) 1 |1 – 2| = 1 (1, 1) 2 |2 – 2| = 0 (2, 0) 3 |3 – 2| = 1 (3, 1) 4 |4 – 2| = 2 (4, 2) The points form a V shape.

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**Input Output Ordered Pair**

Check It Out! Example 4a 1 2 y = x – 4; x = –4, –2, 0, 2, 4 Input Output Ordered Pair x y (x, y) –4 –2 – 4 = –6 (–4, –6) –2 –1 – 4 = –5 (–2, –5) 0 – 4 = –4 (0, –4) 2 1 – 4 = –3 (2, –3) 4 2 – 4 = –2 (4, –2) The points form a line.

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**Input Output Ordered Pair**

Check It Out! Example 4b y = 3x2 + 3; x = –3, –1, 0, 1, 3 Input Output Ordered Pair x y (x, y) –3 3(–3)2 + 3 = 30 (–3, 30) –1 3(–1)2 + 3 = 6 (–1, 6) 3(0)2 + 3 = 3 (0, 3) 1 3(1)2 + 3 = 6 (1, 6) 3(3)2 + 3 = 30 3 (3, 30) The points form a U shape.

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**Input Output Ordered Pair**

Check It Out! Example 4c y = |x – 2|; x = 0, 1, 2, 3, 4 Input Output Ordered Pair x y (x, y) |0 – 2| = 2 (0, 2) 1 |1 – 2| = 1 (1, 1) 2 |2 – 2| = 0 (2, 0) 3 |3 – 2| = 1 (3, 1) 4 |4 – 2| = 2 (4, 2) The points form a V shape.

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

Lesson Quiz: Part 1 Graph each point. Name the quadrant in which each point lies. 1. (2, 0) None 2. (–3, –4) lll • 3. (1, –1) lV • • 4. (–5, 4) 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 = x; (1, 53), (2, 56), (3, 59), (4, 62)

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**The pattern is U-shaped.**

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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