holt mcdougal algebra 1 4-1 graphing relationships match simple graphs with situations. graph a...

Holt McDougal Algebra 1

4-1 Graphing Relationships

Match simple graphs with situations.Graph a relationship.

Objectives

continuous graphdiscrete graph

Vocabulary

Graphs can be used to illustrate many different situations. For example, trends shown on a cardiograph can help a doctor see how a patient’s heart is functioning.

To relate a graph to a given situation, use key words in the description.

Example 1: Relating Graphs to SituationsEach 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.

Step 1 Read the graphs from left to right to show time passing.

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 downward

rapidly

Graphs A, B, and C

Never horizontal Graph B

Slanting downward until reaches zero

Graphs A, B, and C

Example 1 Continued

Never horizontal

Slanting downward rapidly

Slanting downward until it reaches zero

Check It Out! Example 1The 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.

Step 1 Read the graphs from left to right to show time passing .

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.

Check It Out! Example 1 Continued

As seen in Example 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.

Example 2A: Sketching Graphs for SituationsSketch 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.

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

When sketching or interpreting a graph, pay close attention to the labels on each axis.

Helpful Hint

Example 2B: Sketching Graphs for SituationsSketch 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 can only sell whole numbers of books, the graph is 7 distinct points.

Check It Out! Example 2aSketch a graph for the situation. Tell whether the graph is continuous or discrete.Jamie is taking an 8-week keyboarding class. At the end of each week, she takes a test to find the number of words she can type per minute. She improves each week.

The graph is discrete.

Each week (x-axis) her typing speed is measured. She gets a separate score (y-axis) for each test.

Since each test score is a whole number, the graph consists of 8 distinct points.

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

Check It Out! Example 2b

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.

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.

Both graphs show a relationship about a child going down a slide. Graph A represents the child’s distance from the ground related to time. Graph B represents the child’s Speed related to time.

Example 3: Writing Situations for Graphs

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:

Check It Out! Example 3

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.

4-2 Relations and Functions

Identify functions.

Find the domain and range of relations and functions.

Objectives

relationdomainrange function

Vocabulary

In Lesson 4-1 you saw relationships represented by graphs. Relationships can also be represented by a set of ordered pairs called a relation.

In the scoring systems of some track meets, for first place you get 5 points, for second place you get 3 points, for third place you get 2 points, and for fourth place you get 1 point. This scoring system is a relation, so it can be shown by ordered pairs. {(1, 5), (2, 3), (3, 2) (4, 1)}. You can also show relations in other ways, such as tables, graphs, or mapping diagrams.

Example 1: Showing Multiple Representations of Relations

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

Write all x-values under “x” and all y-values under “y”. 2

x yTable

Example 1 Continued

Use the x- and y-values to plot the ordered pairs.

Mapping Diagram

x y Write all x-values under “x” and all y-values under “y”. Draw an arrow from each x-value to its corresponding y-value.

Example 1 Continued

Write all x-values under “x” and all y-values under “y”.

Use the x- and y-values to plot the ordered pairs.

Mapping Diagramx y

Write all x-values under “x” and all y-values under “y”. Draw an arrow from each x-value to its corresponding y-value.

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. The domain of the track meet scoring system is {1, 2, 3, 4}. The range is {5, 3, 2, 1}.

Example 2: Finding the Domain and Range of a

RelationGive the domain and range of the relation.

Domain: 1 ≤ x ≤ 5

Range: 3 ≤ y ≤ 4

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

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

Check It Out! Example 2aGive the domain and range of the relation.

Domain: {6, 5, 2, 1}

Range: {–4, –1, 0}

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

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

Give the domain and range of the relation.

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.

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

Example 3A: Identifying Functions

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} Even though 3 is in the domain twice, it is written only once when you are giving the domain.

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.

D: {–4, –8, 4, 5} R: {2, 1}

Use the arrows to determine which domain values correspond to each range value.

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

Example 3B: Identifying Functions

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

Example 3C: Identifying FunctionsGive the domain and range of the relation. Tell whether the relation is a function. Explain.

Draw lines to see the domain and range values.

D: –5 ≤ x ≤ 3 R: –2 ≤ y ≤ 1

Domain

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

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

a. {(8, 2), (–4, 1),

(–6, 2),(1, 9)}

The relation is not a function. The domain value 2 is paired with both –5 and –4.

D: {–6, –4, 1, 8}

R: {1, 2, 9}

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

D: {2, 3, 4}

R: {–5, –4, –3}

4-3 Writing Functions

Identify independent and dependent variables.

Write an equation in function notation and evaluate a function for given input values.

Objectives

independent variabledependent variablefunction rule function notation

Vocabulary

Example 1: Using a Table to Write an Equation

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

5 10 15 20

1 2 3 4

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

5 – 4 = 1 or

Example 1 ContinuedStep 2 Determine if one relationship works for the remaining values.

Step 3 Write an equation.

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

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

15 – 4 ≠ 3 and

20 – 4 ≠ 4 and

10 – 4 ≠ 2 and

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

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

1 2 3 4

3 6 9 12

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

1 • 3 = 3 or 1 + 2 = 3

y = 3x

Step 2 Determine if one relationship works for the remaining 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.

The value of y is 3 times x.

The equation in Example 1 describes a function because for each x-value (input), there is only one y-value (output).

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.

Example 2A: Identifying Independent and Dependent Variables

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 paint

Independent: measurement of the room

in the situation.The height of a candle decreases 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.

Example 2B: Identifying Independent and Dependent Variables

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 medication

Independent: weight of animal

in the situation.

Example 2C: Identifying Independent and Dependent Variables

Helpful Hint

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

Independent

Variable

Dependent

Variablex-values y-values

Domain Range

Input Output

x f(x)

Check It Out! Example 2a

A company charges $10 per hour to rent a jackhammer.

Identify the independent and dependent variable in the situation.

The cost to rent a jackhammer depends on the length of time it is rented.

Dependent variable: cost

Independent variable: time

Identify the independent and dependent variable in the situation.

Apples cost $0.99 per pound.

The cost of apples depends on the number of pounds bought.

Dependent variable: cost

Independent variable: pounds

An algebraic expression that defines a function is a function rule. Suppose Tasha earns $5 for each hour she baby-sits. Then 5 • x is a function rule that models her earnings.

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.

The dependent variable is a function of the independent variable.

y is a function of x.

y = f (x)

y = f(x)

Identify the independent and dependent variables. Write an equation 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.

Example 3A: Writing Functions

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

Dependent: fee

Independent: hoursLet h represent the number of hours of tutoring.

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.

Example 3B: Writing FunctionsIdentify the independent and dependent variables. Write an equation in function notation for the situation.

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

Dependent: total cost

Independent: number of months Let m represent the number of months

Check It Out! Example 3aIdentify the independent and dependent variables. Write an equation in function notation for the situation.

Steven buys lettuce that costs $1.69/lb.

The function for the total cost of the lettuce is f(x) = 1.69x.

The total cost depends on how many pounds of lettuce Steven buys.

Independent: poundsLet x represent the number of pounds Steven buys.

Check It Out! Example 3bIdentify the independent and dependent variables. Write an equation in function notation for the situation.

An amusement park charges a $6.00 parking fee plus $29.99 per person.

The function for the total park cost is

f(x) = 29.99x + 6.

The total cost depends on the number of persons in the car, plus $6.

Independent: number of persons in the carLet x represent the number of persons in the car.

You can think of a function as an input-output machine. For Tasha’s earnings, f(x) = 5x. If you input a value x, the output is 5x.

function

f(x)=5x

output

Example 4A: Evaluating Functions

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 Substitute

7 for x.

f(x) = 3(x) + 2

Simplify.

f(x) = 3(x) + 2

f(–4) = 3(–4) + 2 Substitute

–4 for x.Simplify.= –12 + 2

= –10

Example 4B: Evaluating Functions

For g(t) = 1.5t – 5, find g(t) when t = 6 and when t = –2.

g(t) = 1.5t – 5 g(t) = 1.5t – 5

g(6) = 1.5(6) – 5

= 9 – 5

g(–2) = 1.5(–2) – 5

= –3 – 5

= –8

Example 4C: Evaluating Functions

For , find h(r) when r = 600 and

when r = –12.

= 202 = –2

For h(c) = 2c – 1, find h(c) when c = 1 and when c = –3.

h(c) = 2c – 1

h(1) = 2(1) – 1

= 2 – 1

h(c) = 2c – 1

h(–3) = 2(–3) – 1

= –6 – 1

= –7

For g(t) = , find g(t) when t = –24 and when

t = 400.

= –5 = 101

When a function describes a real-world situation, every real number is not always reasonable for the domain and range. For example, a number representing the length of an object cannot be negative, and only whole numbers can represent a number of people.

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

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

Joe has enough money to buy 1, 2, or 3 DVDs at $15.00 each, if he buys any at all.

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 {0, 1, 2, 3}.

Example 5 Continued

Substitute the domain values into the function rule to find the range values.

A reasonable range for this situation is {$0, $15, $30, $45}.

x 1 2 3

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

15(0) = 0

The settings on a space heater are the whole numbers from 0 to 3. The total number of watts used for each setting is 500 times the setting number. Write a function to describe the number of watts used for each setting. Find the reasonable domain and range for the function.

Number of

watts used

is 500 times the setting #.watts

f(x) = 500 • x

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

0 1 2 3

500(0) =

500(1) =

500(2) =

500(3) = 1,500

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

Substitute these values into the function rule to find the range values.

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

4-4 Graphing Functions

Graph functions given a limited domain.

Graph functions given a domain of all real numbers.

Objectives

Scientists can use a function to make conclusions about the rising sea level.

Sea level is rising at an approximate rate of 2.5 millimeters per year. If this rate continues, the function y = 2.5x can describe how many millimeters y sea level will rise in the next x years.

One way to understand functions such as the one above is to graph them. You can graph a function by finding ordered pairs that satisfy the function.

Example 1A: Graphing Solutions Given a DomainGraph 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

Subtract x from both sides.

Since y is multiplied by –3, divide both sides by –3.

Simplify.

x – 3y = –6

Example 1A Continued

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.

••

Step 3 Graph the ordered pairs.

Example 1A ContinuedGraph the function for the given domain.

f(x) = x2 – 3; D: {–2, –1, 0, 1, 2}

Example 1B: Graphing Solutions Given a Domain

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

f(x) = x2 – 3 (x, f(x))x

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)

f(x) = x2 – 3; D: {–2, –1, 0, 1, 2}

Example 1B Continued

–2x + y = 3; D: {–5, –3, 1, 4}

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

–2x + y = 3

+2x +2x

y = 2x + 3

Add 2x to both sides.

Check It Out! Example 1a ContinuedGraph the function for the given domain.

–2x + y = 3; D: {–5, –3, 1, 4}

y = 2x + 3 x (x, y)

y = 2(–5) + 3 = –7–5 (–5, –7)

y = 2(1) + 3 = 51 (1, 5)

y = 2(4) + 3 = 11 4 (4, 11)

y = 2(–3) + 3 = –3–3 (–3, –3)

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

Check It Out! Example 1a Continued

–2x + y = 3; D: {–5, –3, 1, 4}

f(x) = x2 + 2; D: {–3, –1, 0, 1, 3}

f(x) = x2 + 2 x (x, f(x))

f(x) = (–32) + 2= 11–3 (–3, 11)

f(x) = 02 + 2= 20 (0, 2)

f(x) = 12 + 2=3 1 (1, 3)

f(x) = (–12 ) + 2= 3–1 (–1, 3)

3 f(x) = 32 + 2=11 (3, 11)

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

f(x) = x2 + 2; D: {–3, –1, 0, 1, 3}

If the domain of a function is all real numbers, any number can be used as an input value. This process will produce an infinite number of ordered pairs that satisfy the function. Therefore, arrowheads are drawn at both “ends” of a smooth line or curve to represent the infinite number of ordered pairs. If a domain is not given, assume that the domain is all real numbers.

Graphing Functions Using a Domain of All Real Numbers

Step 1 Use the function to generate ordered pairs by choosing several values for x.

Step 2

Step 3

Plot enough points to see a pattern for the graph.

Connect the points with a line or smooth curve.

x –3x + 2 = y (x, y)

Example 2A: Graphing Functions 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)

Step 2 Plot enough points to see a pattern.

Graph the function –3x + 2 = y.

Step 3 The ordered pairs appear to form a line. Draw a line through all the points to show all the ordered pairs that satisfy the function. Draw

arrowheads on both “ends” of the line.

Graph the function –3x + 2 = y.

Example 2B: Graphing Functions

g(x) = |–2| + 2= 4–2 (–2, 4)

g(x) = |1| + 2= 31 (1, 3)

0 g(x) = |0| + 2= 2 (0, 2)

g(x) = |–1| + 2= 3–1 (–1, 3)

g(x) = |2| + 2= 42 (2, 4)

g(x) = |3| + 2= 53 (3, 5)

Graph the function g(x) = |x| + 2.

x g(x) = |x| + 2 (x, g(x))

Example 2B ContinuedGraph the function g(x) = |x| + 2.

Step 3 The ordered pairs appear to form a v-shaped graph. Draw lines through all the points to show all the ordered pairs that satisfy the function. Draw arrowheads on the “ends” of the “V”.

Check If the graph is correct, any point on it will satisfy the function. Choose an ordered pair on the graph that was not in your table. (4, 6) is on the graph. Check whether it satisfies g(x)= |x| + 2.

g(x) = |x| + 2

6 |4| + 26 4 + 26 6

Substitute the values for x and y into the function. Simplify.

The ordered pair (4, 6) satisfies the function.

Check It Out! Example 2aGraph the function f(x) = 3x – 2.

f(x) = 3(–2) – 2 = –8–2 (–2, –8)

f(x) = 3(1) – 2 = 11 (1, 1)

0 f(x) = 3(0) – 2 = –2 (0, –2)

f(x) = 3(–1) – 2 = –5–1 (–1, –5)

f(x) = 3(2) – 2 = 42 (2, 4)

f(x) = 3(3) – 2 = 73 (3, 7)

x f(x) = 3x – 2 (x, f(x))

Graph the function f(x) = 3x – 2.

Step 3 The ordered pairs appear to form a line. Draw a line through all the points to show all the ordered pairs that satisfy the function. Draw arrowheads on both “ends” of the line.

Check It Out! Example 2a ContinuedGraph the function f(x) = 3x – 2.

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)

y = |–1 – 1| = 2–1 (–1, 2)

y = |2 – 1| = 12 (2, 1)

Graph the function y = |x – 1|.Check It Out! Example 2b Continued

Step 3 The ordered pairs appear to form a v-shaped graph. Draw lines through the points to show all the ordered pairs that satisfy the function. Draw arrowheads on both “ends” of the “V”.

Check If the graph is correct, any point on the graph will satisfy the function. Choose an ordered pair on the graph that is not in your table. (3, 2) is on the graph. Check whether it satisfies y = |x – 1|.

y = |x – 1|

2 |3 – 1|2 |2|

Substitute the values for x and y into the function. Simplify.

Example 3: Finding Values Using Graphs

Use a graph of the function to find

the value of f(x) when x = –4. Check your

answer.

Locate –4 on the x-axis. Move up to the graph of the function. Then move right to the y-axis to find the corresponding value of y.

f(–4) = 6

6 2 + 4

Check Use substitution.

Example 3 Continued

Substitute the values for x and y into the function.

Simplify.

The ordered pair (–4, 6) satisfies the function.

Use a graph of the function to find

the value of f(x) when x = –4. Check your

answer.

f(–4) = 6

Use the graph of to find the value of x

when f(x) = 3. Check your answer.

Locate 3 on the y-axis. Move right to the graph of the function. Then move down to the x-axis to find the corresponding value of x.

f(3) = 3

3 1 + 2

Check Use substitution.

Substitute the values for x and y into the function.

Simplify.

Use the graph of to find the value of x

when f(x) = 3. Check your answer. f(3) = 3

Recall that in real-world situations you may have to limit the domain to make answers reasonable. For example, quantities such as time, distance, and number of people can be represented using only nonnegative values. When both the domain and the range are limited to nonnegative values, the function is graphed only in Quadrant I.

Example 4: Problem-Solving Application

A mouse can run 3.5 meters per second. The function y = 3.5x describes the distance in meters the mouse can run in x seconds. Graph the function. Use the graph to estimate how many meters a mouse can run in 2.5 seconds.

Example 4 Continued

Understand the Problem11

The answer is a graph that can be used to find the value of y when x is 2.5.

List the important information:

• The function y = 3.5x describes how many meters the mouse can run.

Think: What values should I use to graph this function? Both the number of seconds the mouse runs and the distance the mouse runs cannot be negative. Use only nonnegative values for both the domain and the range. The function will be graphed in Quadrant I.

22 Make a Plan

Example 4 Continued

Solve33

Choose several nonnegative values of x to find values of y.

y = 3.5x x (x, y)

y = 3.5(1) = 3.5 1 (1, 3.5)

y = 3.5(2) = 72 (2, 7)

3 y = 3.5(3) = 10.5 (3, 10.5)

y = 3.5(0) = 0 0 (0, 0)

Example 4 Continued

Graph the ordered pairs.

Draw a line through the points to show all the ordered pairs that satisfy this function.

Use the graph to estimate the y-value when x is 2.5.

A mouse can run about 8.75 meters in 2.5 seconds.

Solve33

Example 4 Continued

Look Back44

As time increases, the distance traveled also increases, so the graph is reasonable. When x is between 2 and 3, y is between 7 and 10.5. Since 2.5 is between 2 and 3, it is reasonable to estimate y to be 8.75 when x is 2.5.

Example 4 Continued

The fastest recorded Hawaiian lava flow moved at an average speed of 6 miles per hour. The function y = 6x describes the distance y the lava moved on average in x hours. Graph the function. Use the graph to estimate how many miles the lava moved after 5.5 hours.

Understand the Problem11

The answer is a graph that can be used to find the value of y when x is 5.5.

List the important information:

• The function y = 6x describes how many miles the lava can flow.

Think: What values should I use to graph this function? Both the speed of the lava and the number of hours it flows cannot be negative. Use only nonnegative values for both the domain and the range. The function will be graphed in Quadrant I.

22 Make a Plan

Solve33

Choose several nonnegative values of x to find values of y.

y = 6x x (x, y)

y = 6(1) = 6 1 (1, 6)

y = 6(3) = 183 (3, 18)

5 y = 6(5) = 30 (5, 30)

Solve33

Draw a line through the points to show all the ordered pairs that satisfy this function.

Use the graph to estimate the y-value when x is 5.5.

The lava will travel about 32.5 meters in 5.5 seconds.

Graph the ordered pairs.

Look Back44

As the amount of time increases, the distance traveled by the lava also increases, so the graph is reasonable. When x is between 5 and 6, y is between 30 and 36. Since 5.5 is between 5 and 6, it is reasonable to estimate y to be 32.5 when x is 5.5.

4-5 Scatter Plots and Trend Lines

Create and interpret scatter plots.

Use trend lines to make predictions.

Objectives

scatter plotcorrelationpositive correlation negative correlationno correlationtrend line

Vocabulary

In this chapter you have examined relationships between sets of ordered pairs or data. Displaying data visually can help you see relationships. A scatter plot is a graph with points plotted to show a possible relationship between two sets of data. A scatter plot is an effective way to display some types of data.

Example 1: Graphing a Scatter Plot from Given DataThe table shows the number of cookies in a jar from the time since they were baked. Graph a scatter plot using the given data.

Use the table to make ordered pairs for the scatter plot.

The x-value represents the time since the cookies were baked and the y-value represents the number of cookies left in the jar.

Plot the ordered pairs.

Check It Out! Example 1The table shows the number of points scored by a high school football team in the first four games of a season. Graph a scatter plot using the given data.

Use the table to make ordered pairs for the scatter plot.

The x-value represents the individual games and the y-value represents the points scored in each game.

Plot the ordered pairs.

Game 1 2 3 4

Score 6 21 46 34

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.

Example 2: Describing Correlations from Scatter Plots

Describe the correlation illustrated by the scatter plot.

There is a positive correlation between the two data sets.

As the average daily temperature increased, the number of visitors increased.

Describe the correlation illustrated by the scatter plot.

There is a positive correlation between the two data sets.

As the years passed, the number of participants in the snowboarding competition increased.

Example 3A: Identifying Correlations

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.

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.

Example 3B: Identifying Correlations

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

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

Example 3C: Identifying Correlations

Check It Out! Example 3aIdentify the type of correlation you would expect to see between the pair of data sets. Explain.

the temperature in Houston and the number of cars sold in Boston

You would except to see no correlation. The temperature in Houston has nothing to do with the number of cars sold in Boston.

the number of members in a family and the size of the family’s grocery bill

You would expect to see a positive correlation. As the number of members in a family increases, the size of the grocery bill increases.

Check It Out! Example 3bIdentify the type of correlation you would expect to see between the pair of data sets. Explain.

the number of times you sharpen your pencil and the length of your pencil

You would expect to see a negative correlation. As the number of times you sharpen your pencil increases, the length of your pencil decreases.

Check It Out! Example 3cIdentify the type of correlation you would expect to see between the pair of data sets. Explain.

Example 4: Matching Scatter Plots to Situations

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.

Graph A Graph B Graph C

Example 4 ContinuedChoose 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.

Graph A

The age of the car cannot be negative.

Example 4 Continued

Graph B

This graph shows all positive values and a positive correlation, so it could represent the data set.

Example 4 Continued

Graph C

There will be a positive correlation between the amount spent on repairs and the age of the car.

Example 4 Continued

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 number of minutes since a pie has been taken out of the oven and the temperature of the pie. Explain.

Graph A

The pie is cooling steadily after it is taken from the oven.

Graph B

The pie has started cooling before it is taken from the oven.

Check It Out! Example 4 ContinuedChoose 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 C

The temperature of the pie is increasing after it is taken from the oven.

Check It Out! Example 4Choose 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.

You can graph a function on a scatter plot to help show a relationship in the data. Sometimes the function is a straight line. This line, called a trend line, helps show the correlation between data sets more clearly. It can also be helpful when making predictions based on the data.

Example 5: Fund-Raising ApplicationThe 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.Draw a line that has about the same number of points above and below it. Your line may or may not go through data points.

Find the point on the line whose x-value is 150. The corresponding y-value is 750.

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

Check It Out! Example 5Based on the trend line, predict how many wrapping paper rolls need to be sold to raise $500.

Find the point on the line whose y-value is 500. The corresponding x-value is about 75.

Based on the data, about 75 wrapping paper rolls is a reasonable prediction of how many rolls need to be sold to raise $500.

4-6 Arithmetic Sequences

Recognize and extend an arithmetic sequence.

Find a given term of an arithmetic sequence.

Objectives

sequencetermarithmetic sequencecommon difference

Vocabulary

During a thunderstorm, you can estimate your distance from a lightning strike by counting the number of seconds from the time you see the lightning until you hear the thunder.

When you list the times and distances in order, each list forms a sequence. A sequence is a list of numbers that often forms a pattern. Each number in a sequence is a term.

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

In the distance sequence, each distance is 0.2 mi greater than the previous distance. 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. The distances in the table form an arithmetic sequence with d = 0.2.

Time (s)

Distance (mi)

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.

To find a term in an arithmetic sequence, add d to the previous term.

Example 1A: Identifying Arithmetic Sequences

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.

You add 4 to each term to find the next term. The common difference is 4.

9, 13, 17, 21,…

+4 +4 +4

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

9, 13, 17, 21,

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

9, 13, 17, 21,…

25, 29, 33,… an = an-1 + d

Reading Math

The three dots at the end of a sequence are called an ellipsis. They mean that the sequence continues and can read as “and so on.”

Example 1B: Identifying Arithmetic Sequences

10, 8, 5, 1,…

Find the difference between successive terms.

The difference between successive terms is not the same.

This sequence is not an arithmetic sequence.

10, 8, 5, 1,…

–2 –3 –4

Check It Out! Example 1aDetermine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms.

You add to each term to find

the next term. The common

difference is .

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

The sequence appears to be

an arithmetic sequence with

a common difference of .

The next three terms are

–4, –2, 1, 5,…

+2 +3 +4

The difference between successive terms is not the same.

This sequence is not an arithmetic sequence.

To find the nth term of an arithmetic sequence when n is a large number, you need an equation or rule. Look for a pattern to find a rule for the sequence below.

1 2 3 4… n Position

The sequence 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… Terma1 a2 a3 a4 an

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.

Example 2A: Finding the nth Term of an Arithmetic Sequence

Find the indicated term of the arithmetic sequence.

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

+4 +4 +4

The common difference is 4.

Step 2 Write a rule to find the 16th term.

The 16th term is 64.

Write a rule to find the nth term.

Simplify the expression in parentheses.Multiply.Add.

Substitute 4 for a1,16 for n, and 4 for d.

an = a1 + (n – 1)d

a16 = 4 + (16 – 1)(4)

= 4 + (15)(4)= 4 + 60= 64

Example 2B: Finding the nth Term of an Arithmetic Sequence

The 25th term: a1 = –5; d = –2

Simplify the expression in parentheses.

Multiply.

The 25th term is –53.

Substitute –5 for a1, 25 for n, and –2 for d.

an = a1 + (n – 1)d

a25 = –5 + (25 – 1)(–2)

= –5 + (24)(–2)

= –5 + (–48)

= –53

Check It Out! Example 2aFind 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

The common difference is –6.

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

The 60th term is –343.

Multiply.Add.

Substitute 11 for a1, 60 for n, and –6 for d.

an = a1 + (n – 1)d

a60 = 11 + (60 – 1)(–6)

= 11 + (59)(–6)

= 11 + (–354)= –343

12th term: a1 = 4.2; d = 1.4

Multiply.

Add.The 12th term is 19.6.

Substitute 4.2 for a1,12 for n, and 1.4 for d.

an = a1 + (n – 1)d

a12 = 4.2 + (12 – 1)(1.4)

= 4.2 + (11)(1.4)

= 4.2 + (15.4)

= 19.6

Example 3: ApplicationA bag of cat food weighs 18 pounds at the beginning of day 1. Each day, the cats are fed 0.5 pound of food. How much does the bag of cat food weigh at the beginning of day 30?

Notice that the sequence for the situation is arithmetic with d = –0.5 because the amount of cat food decreases by 0.5 pound each day.

Since the bag weighs 18 pounds to start, a1 = 18.

Since you want to find the weight of the bag on day 30, you will need to find the 30th term of the sequence, so n = 30.

A bag of cat food weighs 18 pounds at the beginning of day 1. Each day, the cats are fed 0.5 pound of food. How much does the bag of cat food weigh at the beginning of day 30?

Example 3 Continued

There will be 3.5 pounds of cat food remaining at the beginning of day 30.

Write the rule to find the nth term.

Multiply.

Substitute 18 for a1, –0.5 for d, and 30 for n.

an = a1 + (n – 1)d

a31 = 18 + (30 – 1)(–0.5)

= 18 + (29)(–0.5)

= 18 + (–14.5)

Check It Out! Example 3Each time a truck stops, it drops off 250 pounds of cargo. After stop 1, its cargo weighed 2000 pounds. How much does the load weigh after stop 6?

Notice that the sequence for the situation is arithmetic because the load decreases by 250 pounds at each stop.

Since the load will be decreasing by 250 pounds at each stop, d = –250.

Since the load is 2000 pounds, a1 = 2000.

Since you want to find the load after the 6th stop, you will

need to find the 6th term of the sequence, so n = 6.

There will be 750 pounds of cargo remaining after stop 6.

Write the rule to find the nth term.

Multiply.

Substitute 2000 for a1, –250 for d, and 6 for n.

an = a1 + (n – 1)d

a6 = 2000 + (6 – 1)(–250)

= 2000 + (5)(–250)

= 2000 + (–1250)

Each time a truck stops, it drops off 250 pounds of cargo. After stop 1, its cargo weighed 2000 pounds. How much does the load weigh after stop 6?

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