warm up

Holt Algebra 1

4-1 Graphing Relationships

Warm UpState whether each word or phrase represents an amount that is increasing, decreasing, or constant.

1. stays the same2. rises3. drops4. slows down

constant

decreasing

decreasing

increasing

Holt Algebra 1


Match simple graphs with situations.Graph a relationship.

Objectives

Holt Algebra 1


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.

Holt Algebra 1

4-1 Graphing RelationshipsExample 1: Relating Graphs to Situations

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.

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

Holt Algebra 1


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

Key Words Segment Description Graphs…

Each day several leaves fallWind 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.

Example 1 Continued

Holt Algebra 1

4-1 Graphing RelationshipsCheck It Out! Example 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.

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

Holt Algebra 1


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

Key Words Segment Description Graphs…Increased steadilyRemained constantIncreased slightly before dropping sharply

Slanting upward Graph C

Horizontal Graphs 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

Holt Algebra 1

4-1 Graphing RelationshipsAs 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.

Holt Algebra 1

4-1 Graphing RelationshipsExample 2A: Sketching Graphs for Situations

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.Sp

eed

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

Holt Algebra 1


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

Helpful Hint

Holt Algebra 1

4-1 Graphing RelationshipsExample 2B: 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.

Holt Algebra 1

4-1 Graphing RelationshipsCheck It Out! Example 2a

Sketch 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 score is separate, the graph consists of distinct units.

Holt Algebra 1


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.

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.

Holt Algebra 1


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.

Holt Algebra 1

4-1 Graphing RelationshipsExample 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 declines,• remains constant,• and then declines to zero.

Possible Situation:

Holt Algebra 1

4-1 Graphing RelationshipsCheck 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.

Holt Algebra 1

4-2 Relations and FunctionsWarm-Up

1. Write a possible situation for the given graph.

2. A pet store is selling puppies for $50 each. It has 8 puppies to sell. Sketch a graph for this situation.

Holt Algebra 1

4-2 Relations and FunctionsWarm-Up

Write a possible situation for the given graph.

1. 2.

Holt Algebra 1

4-2 Relations and Functions

Identify functions.Find the domain and range of relations and functions.

Objectives

Holt Algebra 1


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.

Holt Algebra 1

4-2 Relations and Functions Example 1: Showing Multiple Representations of

RelationsExpress 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

4

6

3

7

8

x yTable

Holt Algebra 1

4-2 Relations and Functions Example 1 Continued

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

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

Graph

Holt Algebra 1


2

6

4

3

8

7

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

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

Holt Algebra 1


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}.

Holt Algebra 1

4-2 Relations and Functions Example 2: Finding the Domain and Range of a Relation

Give the domain and range of the relation.

Domain: 1 ≤ x ≤ 5Range: 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.

Holt Algebra 1

4-2 Relations and FunctionsCheck It Out! Example 2a


–4

–1

01

2

6

5

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.

Holt Algebra 1

4-2 Relations and FunctionsCheck It Out! Example 2b


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.

Holt Algebra 1


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

Holt Algebra 1

4-2 Relations and Functions 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.

Holt Algebra 1


–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.

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

Holt Algebra 1

4-2 Relations and Functions Example 3C: Identifying Functions

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

Draw in lines to see the domain and range values

D: –5 ≤ x ≤ 3 R: –2 ≤ y ≤ 1

RangeDomain

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

Holt Algebra 1


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

Check It Out! Example 3

a. {(8, 2), (–4, 1), (–6, 2),(1, 9)}

b.

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}

Holt Algebra 1

4-3 Writing FunctionsWarm-Up

1. Express the relation {(–2, 5), (–1, 4), (1, 3), (2, 4)} as a table, as a graph, and as a mapping diagram.

2. Give the domain and range of the relation.

Holt Algebra 1

4-3 Writing Functions

Identify independent and dependent variables.Write an equation in function notation and evaluate a function for given input values.

Objectives

Holt Algebra 1

4-3 Writing Functions Example 1: Using a Table to Write an Equation

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 and

Holt Algebra 1

4-3 Writing FunctionsExample 1 Continued

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 The value of y is one-fifth of x.

Holt Algebra 1

4-3 Writing FunctionsCheck It Out! Example 1

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 and 1 + 2 = 3

Holt Algebra 1


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.

The value of y is 3 times x.

Holt Algebra 1


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

Holt Algebra 1


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.

Holt Algebra 1

4-3 Writing Functions Example 2A: Identifying Independent and Dependent

VariablesIdentify the independent and dependent variablesin 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

Holt Algebra 1


Identify the independent and dependent variablesin 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.

Example 2B: Identifying Independent and Dependent Variables

Holt Algebra 1


A veterinarian must weight 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

Identify the independent and dependent variablesin the situation.

Example 2C: Identifying Independent and Dependent Variables

Holt Algebra 1


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

IndependentVariable

DependentVariable

x-values y-valuesDomain RangeInput Output

x f(x)

Holt Algebra 1

4-3 Writing FunctionsCheck 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: costIndependent variable: time

Holt Algebra 1


Identify the independent and dependent variable in the situation.

Check It Out! Example 2b

Camryn buys p pounds of apples at $0.99 per pound.

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

Dependent variable: costIndependent variable: pounds

Holt Algebra 1


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.

Holt Algebra 1


The dependent variable is a function of the independent variable.

y is a function of x.

y = f (x)

y = f(x)

Holt Algebra 1


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.

Example 3A: Writing Functions

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

Dependent: chargesIndependent: hours

Let h represent the number of hours of tutoring.

Holt Algebra 1


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 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

Holt Algebra 1

4-3 Writing FunctionsCheck It Out! Example 3a

Identify the independent and dependent variables. Write a rule in function notation for the situation. Steven buys lettuce that costs $1.69/lb.

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

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

Dependent: total costIndependent: pounds

Let x represent the number of pounds Steven bought.

Holt Algebra 1

4-3 Writing FunctionsCheck It Out! Example 3b

Identify the independent and dependent variables. Write a rule 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.

Dependent: total costIndependent: number of persons in the car

Let x represent the number of persons in the car.

Holt Algebra 1


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

input

10

x

functionf(x)=5x

output

5x

6

30

2

Holt Algebra 1

4-3 Writing Functions 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

= 23Simplify.

f(x) = 3(x) + 2

f(–4) = 3(–4) + 2 Substitute –4 for x.Simplify.= –12 + 2

= –10

Holt Algebra 1

4-3 Writing Functions Example 4C: Evaluating Functions

Evaluate the function for the given input values.

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

= 202 = –2

Holt Algebra 1


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.

Holt Algebra 1

4-3 Writing Functions Example 5: Finding the Reasonable Range and

Domain of a Function

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}.

Holt Algebra 1

4-3 Writing Functions Example 5 Continued

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

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

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

Holt Algebra 1

4-3 Writing FunctionsCheck It Out! Example 5

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 used is 500 times the setting #.

wattsf(x) = 500 • x

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

Holt Algebra 1


x f(x)

0 1 2 3500(0) =

0500(1) =

500500(2) =

1,000500(3) =

1,500

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

Check It Out! Example 5

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

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

Holt Algebra 1

4-4 Graphing FunctionsWarm-Up

Identify the independent and dependent variables. Write a rule in function notation for each situation.1. A buffet charges $8.95 per person.2. A moving company charges $130 for weekly truck

rental plus $1.50 per mile.Evaluate each function for the given input values.

3. For g(t) = , find g(t) when t=20 and t=–12.

4. A theater can be rented for exactly 2, 3, or 4 hours. The cost is a $100 deposit plus $200 per hour.

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

Holt Algebra 1

4-4 Graphing Functions

Graph functions given a limited domain.Graph functions given a domain of all real numbers.

Objectives

Holt Algebra 1


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.You can graph a function by finding ordered pairs that satisfy the function.

Holt Algebra 1

4-4 Graphing Functions Example 1A: Graphing Functions Given a Domain

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

Subtract x from both sides.

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

Simplify.

x – 3y = –6

Holt Algebra 1

4-4 Graphing Functions 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.

Holt Algebra 1


••

••

y

x

Step 3 Graph the ordered pairs.

Example 1A ContinuedGraph the function for the given domain.

Holt Algebra 1


Graph the function for the given domain.f(x) = x2 – 3; D: {–2, –1, 0, 1, 2}

Example 1B: Graphing Functions 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–2

–1012

f(x) = (–2)2 – 3 = 1f(x) = (–1)2 – 3 = –2

f(x) = 02 – 3 = –3f(x) = 12 – 3 = –2f(x) = 22 – 3 = 1

(–2, 1)

(–1, –2)(0, –3)(1, –2)(2, 1)

Holt Algebra 1


••

•

•

•

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}

Example 1B Continued

Holt Algebra 1


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.

Holt Algebra 1


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.

Holt Algebra 1


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)

Holt Algebra 1


Step 2 Plot enough points to see a pattern.

Example 2A Continued Graph the function –3x + 2 = y.

Holt Algebra 1


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.

Example 2A Continued Graph the function –3x + 2 = y.

Holt Algebra 1

4-4 Graphing Functions 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)

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

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))

Holt Algebra 1


Step 2 Plot enough points to see a pattern.

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

Holt Algebra 1


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

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

Holt Algebra 1

4-4 Graphing Functions 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

Holt Algebra 1


6 2 + 46 6

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.

6

Use a graph of the function to find the value of f(x) when x = –4. Check your answer. f(–4) = 6

Holt Algebra 1

4-4 Graphing FunctionsCheck It Out! Example 3

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

Holt Algebra 1


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.

Holt Algebra 1

4-4 Graphing FunctionsCheck It Out! Example 4

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.

Holt Algebra 1

4-4 Graphing FunctionsCheck It Out! Example 4 Continued

Understand the Problem1

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.

Holt Algebra 1


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.

2 Make a PlanCheck It Out! Example 4 Continued

Holt Algebra 1


Solve3

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)


Holt Algebra 1


Solve3

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.

Holt Algebra 1

4-5 Scatter Plots and Trend LinesWarm-Up

1. Graph the function for the given domain.3x + y = 4D: {–1, 0, 1, 2}

2. Graph the function y = |x + 3|.

3. The function y = 3x describes the distance (in inches) a giant tortoise walks in x seconds. Graph the function. Use the graph to estimate how many inches the tortoise will walk in 5.5 seconds.

Holt Algebra 1

4-5 Scatter Plots and Trend Lines

Create and interpret scatter plots.Use trend lines to make predictions.

Objectives

Holt Algebra 1


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 graph some types of data.

Holt Algebra 1

4-5 Scatter Plots and Trend LinesExample 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.

Holt Algebra 1

4-5 Scatter Plots and Trend LinesCheck It Out! Example 1

The 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.Game 1 2 3 4Score 6 21 46 34

Holt Algebra 1


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.

Holt Algebra 1


Holt Algebra 1


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 visitor increased.

Holt Algebra 1


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.

Holt Algebra 1

4-5 Scatter Plots and Trend Lines 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.

Holt Algebra 1


the number of people in an audience and ticket sales

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

Example 3B: Identifying Correlations


Holt Algebra 1


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.

Example 3C: Identifying Correlations


Holt Algebra 1

4-5 Scatter Plots and Trend LinesCheck It Out! Example 3a

Identify 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.

Holt Algebra 1


the number of members in a family and the size of the family’s grocery billYou would expect to see 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.

Holt Algebra 1


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.

Holt Algebra 1


Graph A Graph B Graph C

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 age of a car and the amount of money spent each year on repairs. Explain.

Holt Algebra 1


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.

Holt Algebra 1


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.

Holt Algebra 1

4-5 Scatter Plots and Trend Lines Example 5: Fund-Raising Application

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.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.

Holt Algebra 1


Based 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.

Holt Algebra 1

4-6 Arithmetic SequencesWarm-Up

For Items 1 and 2, identify the correlation you would expect to see between each pair of data sets. Explain.

1. The outside temperature in the summer and the cost of the electric bill

2. The price of a car and the number of passengers it seats

3. The scatter plot shows the number of orders placed for flowers before Valentine’s Day at one shop. Based on this relationship, predict the number of flower orders placed on February 12.

Holt Algebra 1

4-6 Arithmetic Sequences

Recognize and extend an arithmetic sequence.Find a given term of an arithmetic sequence.

Objectives

Holt Algebra 1


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.

Holt Algebra 1


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)

Holt Algebra 1

4-6 Arithmetic SequencesExample 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.

9, 13, 17, 21,…+4 +4 +4

Holt Algebra 1


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.”

Holt Algebra 1

4-6 Arithmetic Sequences Example 1B: 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.

The difference between successive terms is not the same.

This sequence is not an arithmetic sequence.

10, 8, 5, 1,…–2 –3 –4

Holt Algebra 1

4-6 Arithmetic SequencesCheck It Out! Example 1a

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.

Holt Algebra 1


4, 1, –2, –5,…


4, 1, –2, –5,… –3 –3 –3


Check It Out! Example 1d

Holt Algebra 1



Check It Out! Example 1c

–4, –2, 1, 5,…


–4, –2, 1, 5,…+2 +3 +4

The difference between successive terms is not the same.

This sequence is not an arithmetic sequence.

Holt Algebra 1

4-6 Arithmetic SequencesThe 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… Terma1 a2 a3 a4 an

Holt Algebra 1


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.

Holt Algebra 1


Holt Algebra 1

4-6 Arithmetic Sequences Example 2A: Finding the nth Term of an Arithmetic

SequenceFind 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)da16 = 4 + (16 – 1)(4)

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

Holt Algebra 1

4-6 Arithmetic SequencesExample 2B: Finding the nth Term of an Arithmetic

SequenceFind 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 + (–48)= –53

Holt Algebra 1

4-6 Arithmetic SequencesCheck It Out! Example 2a

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.an = a1 + (n – 1)d

a60 = 11 + (60 – 1)(–6)= 11 + (59)(–6)= 11 + (–354)= –343

Holt Algebra 1

4-6 Arithmetic Sequences Example 3: Application

A bag of cat food weighs 18 pounds. Each day, the cats are feed 0.5 pound of food. How much does the bag of cat food weigh after 30 days?Step 1 Determine whether the situation appears to be

arithmetic. The sequence for the situation is arithmetic because the cat food decreases by 0.5 pound each day.

Step 2 Find d, a1, and n.Since the weight of the bag decrease by 0.5 pound each day, d = –0.5. Since the bag weighs 18 pounds to start, a1 = 18.

Since you want to find the weight of the bag after 30 days, you will need to find the 31st term of the sequence so n = 31.

Holt Algebra 1

4-6 Arithmetic Sequences Example 3 Continued

Step 3 Find the amount of cat food remaining for an.

There will be 3 pounds of cat food remaining after 30 days.

an = a1 + (n – 1)d

a31 = 18 + (31 – 1)(–0.5)

= 18 + (30)(–0.5)

= 18 + (–15) = 3

Holt Algebra 1

4-6 Arithmetic SequencesCheck It Out! Example 3

Each time a truck stops, it drops off 250 pounds of cargo. It started with a load of 2000 pounds. How much does the load weigh after the 5th stop?

Step 1 Determine whether the situation appears to be arithmetic. The sequence for the situation is arithmetic because the load is decreased by 250 pounds at each stop.

Step 2 Find d,a1, and n.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 5th stop, you will need to find the 6th term of the sequence, so n = 6.

Holt Algebra 1


Step 3 Find the amount of cargo remaining for an.

There will be 750 pounds of cargo remaining after 5 stops.

Write the rule to find the nth term.

Simplify the expression in parenthesis.

Multiply.Add.

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) = 750

Holt Algebra 1

4-6 Arithmetic SequencesLesson Quiz: Part I

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

1. 3, 9, 27, 81,… not arithmetic

2. 5, 6.5, 8, 9.5,… arithmetic; 1.5; 11, 12.5, 14

Holt Algebra 1

4-6 Arithmetic SequencesLesson Quiz: Part II

Find the indicated term of each arithmetic sequence.3. 23rd term: –4, –7, –10, –13, … 4. 40th term: 2, 7, 12, 17, … 5. 7th term: a1 = –12, d = 26. 34th term: a1 = 3.2, d = 2.67. Zelle has knitted 61 rows of a scarf. Each

day she adds 17 more rows. How many rows total has Zelle knitted 16 days later?

–701970

89

333 rows

warm up

Documents

correct graph

discrete graphsthe graph

continuous graphs

graphing relationshipsstep

graphing relationshipsexample

continuedholt algebra

sketching graphs

downward graphs b