Chapter 4 - Functions

Algebra I

Table of Contents

• 4.1 - Graphing Relationships• 4.2 - Relations and Functions• 4.3 - Writing Functions• 4.4 - Graphing Functions• 4.5 - Scatter Plots and Trend Lines• 4.6 – Arithmetic Sequences

4.1 - Graphing Relationships

Algebra I

Each day several leaves fall from a tree. One day a gust of wind blows off many leaves.

Eventually, there are no more leaves on the tree. Choose the graph that best represents the situation.

4.1 Example 1 Relating Graphs to Situations

Step 2 List key words in order and decide which graph shows them.

Key Words Segment Description Graphs…

Each day several leaves fall

Wind blows offmany leaves

Eventually no more leaves

Slanting downwardrapidly

Graphs A, B, and C

Never horizontal Graph B

Slanting downward until reaches zero

Graphs A, B, and C

Step 3 Pick the graph that shows all the key phrases in order.

The correct graph is B.

4.1

The air temperature increased steadily for several hours and then remained constant. At the end of the day, the temperature increased slightly before dropping sharply. Choose the graph that best represents this situation.

4.1

Step 2 List key words in order and decide which graph shows them.

Key Words Segment Description Graphs…

Increased steadily

Remained constant

Increased slightly before dropping sharply

Slanting upward Graph C

HorizontalGraphs A, B, and C

Slanting upward and then steeply downward Graphs B and C

Step 3 Pick the graph that shows all the key phrases in order.

The correct graph is graph C.

4.1

Some graphs are connected lines or curves called continuous graphs.

Some graphs are only distinct points. They are called discrete graphs

The graph on theme park attendance is an example of a discrete graph.

It consists of distinct points because each year is distinct and people are counted in whole numbers only.

The values between whole numbers are not included, since they have no meaning for the situation.

4.1

Math Joke

• Q: What do a math teacher and English teacher have in common?

• A: They can both make a “pair-a-graph”

Sketch a graph for the situation. Tell whether the graph is continuous or discrete.

A truck driver enters a street, drives at a constant speed, stops at a light, and then continues.

The graph is continuous.

Spee

d

Time

y

x

As time passes during the trip (moving left to right along the x-axis) the truck's speed (y-axis) does the following:

• initially increases• remains constant• decreases to a stop• increases• remains constant

4.1 Example 2 Sketching Graphs for Situations

Sketch a graph for the situation. Tell whether the graph is continuous or discrete.

A small bookstore sold between 5 and 8 books each day for 7 days.

The graph is discrete.

The number of books sold (y-axis) varies for each day (x-axis).

Since the bookstore accounts for the number of books sold at the end of each day, the graph is 7 distinct points.

4.1

Sketch a graph for the situation. Tell whether the graph is continuous or discrete.

Henry begins to drain a water tank by opening a valve. Then he opens another valve.

Then he closes the first valve. He leaves the second valve open until the tank is empty.

Water tank

Wat

er L

evel

Time

As time passes while draining the tank (moving left to right along the x-axis) the water level (y-axis) does the following:

• initially declines• decline more rapidly• and then the decline slows down.

The graph is continuous.

4.1

Write a possible situation for the given graph.

A car approaching traffic slows down, drives at a constant speed, and then slows down until coming to a complete stop.

Step 1 Identify labels. x-axis: time y-axis: speed

Step 2 Analyze sections. over time, the speed:• initially decreases,• remains constant,• and then decreases to zero.

Possible Situation:

4.1 Example 3 Writing Situations for Graphs

Write a possible situation for the given graph

Possible Situation: When the number of students reaches a certain point, the number of pizzas bought increases.

Step 1 Identify labels x-axis: students y-axis: pizzas

Step 2 Analyze sections. As students increase, the pizzas do the following:• initially remains constant,• and then increases to a new constant.

4.1

HW pg. 237

• 4.1-– 6-16, 21, 22, 29-32, 36, 37

– Follow All HW Guidelines or ½ off

4.2 - Relations and Functions

Algebra I

The domain of a relation is the set of first coordinates (or x-values) of the ordered pairs.

The range of a relation is the set of second coordinates (or y-values) of the ordered pairs.

4.2 Algebra 1 (bell work)

Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a graph, and as a mapping diagram.

2

4

6

3

7

8

x yTable Graph

2

6

4

3

8

7

Mapping Diagram

x y

4.2 Example 1 Showing Multiple Representations of Relations

Give the domain and range of the relation.

Domain: 1 ≤ x ≤ 5 “and” statement

Range: 3 ≤ y ≤ 4 “and” statement

The domain value is all x-values from 1 through 5, inclusive.

The range value is all y-values from 3 through 4, inclusive.

4.2 Example 2 Finding the Domain and Range of a Relation

Give the domain and range of the relation.

–4

–1

01

2

6

5

Domain: {6, 5, 2, 1} Roster Notation

Range: {–4, –1, 0} Roster Notation

The domain values are all x-values 1, 2, 5 and 6.

The range values are y-values 0, –1 and –4.

4.2

Give the domain and range of the relation.

x y

1 1

4 4

8 1

Domain: {1, 4, 8}

Range: {1, 4}

The domain values are all x-values 1, 4, and 8.

The range values are y-values 1 and 4.

4.2 Optional

Math Joke

• Q: Why did the y-variable leave the city?

• A: He was more at home on the range

4.2

Give the domain and range of the relation. Tell whether the relation is a function. Explain.

{(3, –2), (5, –1), (4, 0), (3, 1)}

R: {–2, –1, 0, 1}

D: {3, 5, 4}

The relation is not a function. Each domain value does not have exactly one range value. The domain value 3 is paired with the range values –2 and 1.

4.2 A function is a special type of relation that pairs each domain value with exactly one range value.

Example 3 Identifying Functions

–4

–8

4

2

1

5

D: {–4, –8, 4, 5} R: {2, 1}

This relation is a function. Each domain value is paired with exactly one range value.

Give the domain and range of the relation. Tell whether the relation is a function. Explain.

4.2

Give the domain and range of the relation. Tell whether the relation is a function. Explain.

D: –5 ≤ x ≤ 3 R: –2 ≤ y ≤ 1

Range

Domain

The relation is not a function. Nearly all domain values have more than one range value.

4.2

HW pg. 243

• 4.2-– 3, 5, 9-14, 17, 19, 20, 25, 27, 28, 39– Ch: 29, 31, 36-38

– Follow All HW Guidelines or ½ off

4.3 - Writing Functions

Algebra I

Determine a relationship between the x- and y-values. Write an equation.

x

y

5 10 15 20

1 2 3 4

Step 1 List possible relationships between the first x and y-values.

5 – 4 = 1 or

4.3 Example 1 Using a Table to Write an Equation

Step 2 Determine which relationship works for the other x- and y- values.

10 – 4 2 and

15 – 4 3 and

20 – 4 4 and

The value of y is one-fifth, , of x.

Step 3 Write an equation.

or

4.3

Determine a relationship between the x- and y-values. Write an equation.

{(1, 3), (2, 6), (3, 9), (4, 12)}

x

y

1 2 3 4

3 6 9 12

Step 1 List possible relationships between the first x- and y-values.

1 3 = 3 or 1 + 2 = 3

4.3

y = 3x

Step 2 Determine which relationship works for the other x- and y- values.

2 • 3 = 63 • 3 = 94 • 3 = 12

2 + 2 6 3 + 2 9 4 + 2 12

The value of y is 3 times x.

Step 3 Write an equation.

4.3

The input of a function is the independent variable.

The output of a function is the dependent variable. The value of the dependent variable depends on, or is a function of, the value of the independent variable.

Identify the independent and dependent variables in the situation.

A painter must measure a room before deciding how much paint to buy.

The amount of paint depends on the measurement of a room.

Dependent: amount of paintIndependent: measurement of the room

4.3

Example 2 Identifying Independent and Dependent Variables

Identify the independent and dependent variables in the situation.

The height of a candle decrease d centimeters for every hour it burns.

Dependent: height of candle Independent: time

The height of a candle depends on the number of hours it burns.

4.3

A veterinarian must weigh an animal before determining the amount of medication.

The amount of medication depends on the weight of an animal.

Dependent: amount of medicationIndependent: weight of animal

An algebraic expression that defines a function is a function rule.

If x is the independent variable and y is the dependent variable, then,

Function notation for y is f(x), read “f of x,” where f names the function.

When an equation in two variables describes a function, you can use function notation to write it.

4.3 Algebra 1 (bell work) Day 2

Helpful HintThere are several different ways to describe the variables of a function.

Independent

Variable

Dependent

Variable

x-values y-values

Domain Range

Input Output

x f(x) or y

4.3

The dependent variable is a function of the independent variable.

y is a function of x.

y = f (x)

y = f(x)

4.3

Identify the independent and dependent variables. Write a rule in function notation for the situation.

A math tutor charges $35 per hour.

The function for the amount a math tutor charges is f(h) = 35h.

The amount a math tutor charges depends on number of hours.

Dependent: chargesIndependent: hours

Let h represent the number of hours of tutoring.

4.3 Example 3 Writing Functions

A fitness center charges a $100 initiation fee plus $40 per month.

The function for the amount the fitness center charges is f(m) = 40m + 100.

Identify the independent and dependent variables. Write a rule in function notation for the situation.

The total cost depends on the number of months, plus $100.

Dependent: total costIndependent: number of months

Let m represent the number of months

4.3

You can think of a function as an input-output machine.

input

10

x

functionf(x)=5x

output

5x

6

30

2

4.3

Evaluate the function for the given input values.

For f(x) = 3x + 2, find f(x) when x = 7 and when x = –4.

= 21 + 2

f(7) = 3(7) + 2

f(x) = 3(x) + 2

= 23

f(x) = 3(x) + 2

f(–4) = 3(–4) + 2

= –12 + 2

= –10

4.3 Example 4 Evaluating Functions

Evaluate the function for the given input values.

For , find h(r) when r = 600 and when r = –12.

= 202= –2

4.3

Math Joke

• Q: Why did the x-variable move back home?

• A: She was more comfortable in her own domain.

4.3

Write a function to describe the situation. Find a reasonable domain and range of the function.

Joe has enough money to buy 1, 2, or 3 DVDs at $15.00 each.

Money spent is $15.00 for each DVD.

f(x) = $15.00 • x

If Joe buys x DVDs, he will spend f(x) = 15x dollars.

Joe only has enough money to purchase 1, 2, or 3 DVDs. A reasonable domain is {1, 2, 3}.

x 1 2 3

f(x) 15(1) = 15 15(2) = 30 15(3) = 45

A reasonable range for this situation is {$15, $30, $45}.

4.3 Example 5 Finding the Reasonable Domain and Range of a Function

The settings on a space heater are the whole numbers from 0 to 3. The total of watts used for each setting is 500 times the setting number.

Write a function rule to describe the number of watts used for each setting. Find a reasonable domain and range for the function.

Number of

watts usedis 500 times the setting #.

watts

f(x) = 500 • x

For each setting, the number of watts is f(x) = 500x watts.

4.3

There are 4 possible settings 0, 1, 2, and 3, so a reasonable domain would be {0, 1, 2, 3}.

A reasonable range for this situation is {0, 500, 1,000, 1,500} watts.

HW pg. 253

• 4.3-– Day 1: 3-6, 15, 16, 38-44– Day 2: 7-12, 17-19, 28, 31– Ch: 24, 29, 30, 32

– Follow All HW Guidelines or ½ off

4.4 - Graphing Functions

Algebra I

Graph the function for the given domain.

x – 3y = –6; D: {–3, 0, 3, 6}

Step 1 Solve for y since you are given values of the domain, or x.

–x –x

–3y = –x – 6

x – 3y = –6

4.4 Example 1 Graphing Solutions Given a Domain

Step 2 Substitute the given value of the domain for x and find values of y.

x (x, y)

–3 (–3, 1)

0 (0, 2)

3 (3, 3)

6 (6, 4)

Graph the function for the given domain.4.4

••

••

y

x

Step 3 Graph the ordered pairs.

Graph the function for the given domain.4.4

Graph the function for the given domain.

f(x) = x2 – 3; D: {–2, –1, 0, 1, 2}

Step 1 Use the given values of the domain to find values of f(x).

f(x) = x2 – 3 (x, f(x))x

–2

–1

0

1

2

f(x) = (–2)2 – 3 = 1

f(x) = (–1)2 – 3 = –2

f(x) = 02 – 3 = –3

f(x) = 12 – 3 = –2

f(x) = 22 – 3 = 1

(–2, 1)

(–1, –2)

(0, –3)

(1, –2)

(2, 1)

4.4 Optional

•

•

•

•

•

y

x

Step 2 Graph the ordered pairs.

Graph the function for the given domain.

f(x) = x2 – 3; D: {–2, –1, 0, 1, 2}

4.4

Math Joke

• Psychology Teacher: Can anyone use “dysfunction” in a sentence?

• Math Student: I can! “Dysfunction” is really hard to graph

x –3x + 2 = y (x, y)

Graph the function –3x + 2 = y.

–3(1) + 2 = –11 (1, –1)

0 –3(0) + 2 = 2 (0, 2)

Step 1 Choose several values of x and generate ordered pairs.

–1 (–1, 5)–3(–1) + 2 = 5

–3(2) + 2 = –42 (2, –4)

3 –3(3) + 2 = –7 (3, –7)

–3(–2) + 2 = 8–2 (–2, 8)

4.4

Graph the function –3x + 2 = y.4.4

Graph the function y = |x – 1|.

y = |–2 – 1| = 3–2 (–2, 3)

y = |1 – 1| = 01 (1, 0)

0 y = |0 – 1| = 1 (0, 1)

Step 1 Choose several values of x and generate ordered pairs.

y = |–1 – 1| = 2–1 (–1, 2)

y = |2 – 1| = 12 (2, 1)

x y = |x – 1| (x, y)

4.4 Optional

Step 2 Plot enough points to see a pattern.

Graph the function y = |x – 1|.4.4

HW pg. 260

• 4.4-– 1, 2, 9, 12, 23, 27, 44, 64-71– Ch: 54, 55– (Graph at least 5 points)

– Follow All HW Guidelines or ½ off

4.5 - Scatter Plots and Trend Lines

Algebra I

The table shows the number of cookies in a jar from the time since they were baked. Graph a scatter plot using the given data.

4.5 Example 1 Graphing a Scatter Plot from Given Data

A correlation describes a relationship between two data sets.

A graph may show the correlation between data.

The correlation can help you analyze trends and make predictions. There are three types of correlations between data.

4.5

Describe the correlation illustrated by the scatter plot.

There is a positive correlation between the two data sets.

4.5 Example 2 Describing Correlations from Scatter Plots

The average temperature in a city and the number of speeding tickets given in the city

You would expect to see no correlation. The number of speeding tickets has nothing to do with the temperature.

Identify the correlation you would expect to see between the pair of data sets. Explain.

4.5 Example 3 Identifying Correlations

the number of people in an audience and ticket sales

You would expect to see a positive correlation. As ticket sales increase, the number of people in the audience increases.

a runner’s time and the distance to the finish line

You would expect to see a negative correlation. As a runner’s time increases, the distance to the finish line decreases.

a runner’s time and the distance to the finish line

You would expect to see a negative correlation. As a runner’s time increases, the distance to the finish line decreases.

Identify the correlation you would expect to see between the pair of data sets. Explain.

4.5

Math Joke

• Q: What should you title a graph showing the relative diameters and weights of a batch of pancakes?

• A: The batter Plot!

Graph A Graph B Graph C

Graph A shows negative values, so it is incorrect.

Graph C shows negative correlation, so it is incorrect.

Graph B is the correct scatter plot.

Choose the scatter plot that best represents the relationship between the age of a car and the amount of money spent each year on repairs. Explain.

4.5 Example 4 Matching Scatter Plots to Situations

Choose the scatter plot that best represents the relationship between the number of minutes since a pie has been taken out of the oven and the temperature of the pie. Explain.

Graph A Graph B Graph CGraph B shows the pie cooling while it is in the oven, so it is incorrect.

Graph C shows the temperature of the pie increasing, so it is incorrect.

Graph A is the correct answer.

4.5 Optional

The scatter plot shows a relationship between the total amount of money collected at the concession stand and the total number of tickets sold at a movie theater.

Based on this relationship, predict how much money will be collected at the concession stand when 150 tickets have been sold.

Draw a trend line and use it to make a prediction.

Based on the data, $750 is a reasonable prediction of how much money will be collected when 150 tickets have been sold.

4.5

HW pg. 270

• 4.5-– 1-13, 23, 27, 34-38– Do not need to copy original tables or graphs

– Follow All HW Guidelines or ½ off

4.6 - Arithmetic Sequences

Algebra I

Distance (mi)

1 542 6 7 83

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Time (s)

+0.2 +0.2 +0.2 +0.2 +0.2 +0.2 +0.2

Notice that in the distance sequence, you can find the next term by adding 0.2 to the previous term.

When the terms of a sequence differ by the same nonzero number d, the sequence is an arithmetic sequence and d is the common difference.

So the distances in the table form an arithmetic sequence with the common difference of 0.2.

Time (s)

Distance (mi)

A sequence is a list of numbers that often forms a pattern. Each number in a sequence is a term.

4.6

Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms.

9, 13, 17, 21,…

Step 1 Find the difference between successive terms.

9, 13, 17, 21,…

+4 +4 +4 +4 +4 +4

25, 29, 33,…

The sequence appears to be an arithmetic sequence with a common difference of 4. The next three terms are 25, 29, 33.

4.6 Example 1 Identifying Arithmetic Sequences

Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms.

10, 8, 5, 1,…

Find the difference between successive terms.

This sequence is not an arithmetic sequence.

10, 8, 5, 1,…

–2 –3 –4

4.6

Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms.

Step 1 Find the difference between successive terms.

Step 2 Use the common difference to find the next 3 terms.

4.6

4, 1, –2, –5,…

Step 1 Find the difference between successive terms.

4, 1, –2, –5,…

–3 –3 –3

Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms.

The sequence appears to be an arithmetic sequence with a common difference of –3. The next three terms are –8, –11, –14.

–8, –11, –14,…

4.6 Optional

The variable a is often used to represent terms in a sequence. The variable a9, read “a sub 9,” is the ninth term, in a sequence. To designate any term, or the nth term in a sequence, you write an, where n can be any number.

1 2 3 4… n Position

The sequence above starts with 3.

The common difference d is 2. You can use the first term and the common difference to write a rule for finding an.

3, 5, 7, 9… Term

a1 a2 a3 a4 an

4.6

Math Joke

• Teacher: Is something wrong with your printer? Your history paper is covered in dots.

• Student: Those ellipses. After the first paragraph, imagine the rest!

The pattern in the table shows that to find the nth term,

add the first term to the product of (n – 1) and the common difference.

4.6

4.6

Find the indicated term of the arithmetic sequence.

The 25th term: a1 = –5; d = –2

The 25th term is –53.

an = a1 + (n – 1)d

a25 = –5 + (25 – 1)(–2)

= –5 + (24)(–2)

= –5 + (–8) 4

= –53

4.6

16th term: 4, 8, 12, 16, …

Step 1 Find the common difference.4, 8, 12, 16,…

+4 +4 +4

The 16th term is 64.

an = a1 + (n – 1)d

a16 = 4 + (16 – 1)(4)

= 4 + (15)(4)

= 4 + 60

= 64

Example 2 Finding the nth Term of a Arithmetic Sequence

Find the indicated term of the arithmetic sequence.

60th term: 11, 5, –1, –7, …

Step 1 Find the common difference.

11, 5, –1, –7,…

–6 –6 –6

Step 2 Write a rule to find the 60th term.

The 60th term is –343.

an = a1 + (n – 1)d

a60 = 11 + (60 – 1)(–6)

= 11 + (59)(–6)

= 11 + (–354)

= –343

4.6 Optional

HW pg. 279

• 4.6-– 3-15 (Odd), 48-51– Ch: 38-41

– Follow All HW Guidelines or ½ off