abspd. web view- solve one step word problems ... we then talk about what it would mean if you...

ASE MA 03: Algebraic Functions and Modeling

Steve [email protected]

abspd.appstate.edu

Today’s Quote“I’ve come to a frightening conclusion that I am the decisive element in the classroom. It’s my personal approach that creates the climate. It’s my daily mood that makes the weather. As a teacher, I possess a tremendous power to make a student’s life miserable or joyous. I can be a tool of torture or an instrument of inspiration. I can humiliate or humor, hurt or heal. In all situations, it is my response that decides whether a crisis will be escalated or deescalated and a student humanized or dehumanized.”

- Adapted from Haim Ginott

Where Can I Find this Packet?It’s OK to write in this packet! You can find everything from this workshop at abspd.appstate.edu Look under teaching resources, adult secondary resources, math.

Agenda8:30 – 10:00 Math Research Says . . .

Concrete Functions

10:00 – 10:15 Break

10:15 – 11:45 Real Life FunctionsFunctions in Four Forms

11:45 – 12:45 Lunch

12:45 – 2:00 Math Modeling

2:00 – 2:15 Break

2:15 – 4:00 You Can Learn Math!

This course is funded by:

Research Says . . . - Teach math from concrete to representational to abstract (CRA)

Concrete Representational

Volume = length x width x height

Volume = 3 x 3 x 3

Abstract

- Teach math using a problem solving approach with real world applications

Today’s high school equivalency tests (GED, TASC, HiSET) focus on reasoning NOT computation and use real life (home/workplace) scenarios

Teach students a problem solving method that works for any type of problem (see the UPS check method on pages 22 – 23.

- Learning is social. Have students work together and talk about what they are doing.

“The one who does the talking does the learning” Lev Vygotsky

“The best way to learn something is to teach it” Patricia Wolfe (Brain Matters)

- The instructor acts as a facilitator. This means the instructor:

works to get students to do as much math as possible

teaches with questions instead of always telling students what to do

realizes students’ wrong answers are the best opportunity for learning

avoids telling students they are right/wrong and gets them to think as long as possible

- Our goal should be to produce students who can, without our help, know when and how to use particular strategies for problem solving (develop their metacognitive skills)

When we tell students every move to make, they develop “learned helplessness.”

What is our end goal: A student who is dependent on us or an independent thinker who can survive and thrive on their own?

Sources: Education Alliance, NIFL TEAL, US Dept. of Ed

| Algebraic Functions and Modeling

Algebra: Gateway or Gatekeeper Skill for Postsecondary?

“Most important for success in college math is a thorough understanding of the basic concepts, principles, and techniques of algebra. This is different than simply having been exposed to these ideas.

“Much of the subsequent mathematics they will encounter draw upon or utilize these principles. In addition, having learned these elements of mathematical thinking at a deep level, they understand what it means to understand mathematical concepts deeply and are more likely to do so in subsequent areas of mathematical study.

“College-ready students possess more than a formulaic understanding of mathematics. They have the ability to apply conceptual understandings in order to extract a problem from a context, usemathematics to solve the problem, and then interpret the solution back into the context. They know when and how to estimate to determine the reasonableness of answers and can use a calculator appropriately as a tool, not a crutch.”

Source: Conley (2010)


LOK (Levels of Knowing) and DOK (Depth of Knowledge) For planning math units, lessons, and/or activities at all levels.

How can you effectively move your learners through the levels of knowing? Which depth of knowledge level is appropriate for each student and/or class?

Levels ofKnowing

Depth of Knowledge Level 1Recall

Depth of Knowledge Level 2Skill / concept

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INTUITIVE Connect to what

the learner knows

- Recall or recognition of a fact, term, information, or definition

- Recall and use a simple procedure

- Discuss use of mathematical operations in daily life - Describe an existing understanding of a skill and/or concept

CONCRETEHands on

- Recognize an object

- Measure an object

- Organize or classify objects - Make observations - Collect data - Perform an experiment

PICTORIAL A Visual

representation

- Identify and/or recognize an image and/or parts of an image - Recognize a pattern of images - Visualize and or draw a representation

- Organize or classify images - Extend a pattern of images - Estimate using a number line - Display data using a table, line graph or bar graph

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ABSTRACT Numbers and

symbols

- Identify and/or recognize a symbol - Recognize a pattern of symbols - Perform basic computations and/or procedures - Identify components of a formula - Use a formula

- Organize or classify symbols

- Extend a pattern of symbols - Display data using pie charts

APPLICATION Math in a context

- Retrieve information from a graph

- Solve one step word problems

- Make decisions as to how to approach the problem - Apply a skill or concept - Use given information - Estimate to solve a problem - Compare data - Multi-step word problems - Determine probability

COMMUNICATION

Ask questions of learners and provide time for them to talk

about math to you and each other.

- Describe an object, image, symbol, or situation

- Explain or interpret a concept or situation - Demonstrate conceptual knowledge through models and explanations - Explain relationships, examples, and non-examples

Source: Lindsey Cermak and Amy Vickers, 2013 based on Levels of Knowing Mathematics (Mahesh Sharma) and Depth of Knowledge (Norman Webb).


Teaching Functions Using Concrete Examples and Certainty Often, a function is defined like this: “A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.” This definition confuses students. Here are some concrete examples of functions:

A Relationship That’s Certain and Unique If we would like students to experience the need for the certainty functions offer us, it’s helpful to put students in a place to experience the uncertainty of non-functional relationships first. Put the letters A, B, C, and D on a wall, spaced evenly apart. Ask every student to stand up and use these examples:

AgeIf you are 20 or under, stand under AIf you are 21 to 29, stand under BIf you are 30 to 39, stand under CIf you are 40 or older, stand under D

TransportationIf you walked to class today, stand under AIf you took the bus to class today, stand under BIf you drove to class today, stand under CIf you came some other way, stand under D

DurationIf it took 10 minutes or less to get to class, stand under AIf it took 11 to 20 minutes to get to class, stand under BIf it took 21 to 30 minutes to get to class, stand under CIf it took more than 31 minutes to get to class, stand under D

ClothesIf you’re wearing blue, stand under A.If you’re wearing red, stand under B.If you’re wearing black, stand under C.If you’re wearing white, stand under D.

BirthdayIf you were born in January, stand under A.If you were born in February, stand under B.If you were born in March, stand under C.If you were born in April, stand under D.

See how these last two examples generate a lack of certainty. Students were lulled by the first examples and may now feel a headache.

“I’m wearing white and red, so where do I go?” “I was born in August. There’s no place for me to stand.”

Now we gather back together and apply formal language to the concepts we’ve just felt. “Mathematicians call these three relationships ‘functions.’ Here’s why. Why do you think these relationships aren’t functions?” Invite students to interrogate the concept of a function in different contexts. Try to keep the focus on certainty – can you predict the output for any input with certainty?


Pepsi Machine: A Relationship That’s UniqueWe talk about the buttons on the machine and how the first button is Pepsi® and so is the second. Then the third is Diet Pepsi® and the fourth is Mountain Dew® or whatever. The buttons are the input and the sodas are the outputs. The idea of one input giving only one output is clear and students also learn quickly that two buttons (inputs) can produce the same soda (output) and that it is still a properly working soda machine (function). We then talk about what it would mean if you pushed the Mountain Dew® button and sometimes got a Mountain Dew®, sometimes got a Sierra Mist®, and sometimes got a Dr. Pepper®. This then becomes a reference point for whenever we discuss if a relation is a function or not. “Remember the Pepsi® machine.”

Function VideoGoogle: Meat-A-Morphosis: An Introduction to Functions for a video that explains functions using a function machine. There is a viewing guide for this video at: www.abspd.appstate.edu/node/385

Function Machine Use a cardboard box that has holes cut in each end. Label it “The Incredible Function Machine.” Have cards prepared where you have different inputs and outputs. Have students guess if the machine is working properly based on whether each input has exactly one unique output.

Guess My RuleHave students play Guess My Rule. This builds on the concept of a function machine but is more abstract. Students will study the inputs and outputs and eventually work on creating equations like y = x + 4 to describe the relationship between the input and output.

Hourly WageI offer students an example like getting paid an hourly wage of $20, working 40 hours in week 1 and grossing $800, working 40 hours in week 2 and grossing $800, and then working 40 hours in week 3 and grossing only $600. When they go to the boss to complain, they’re told, “Oh, well, we had a bad week. So we can only afford to pay you $15 an hour this time.” Naturally, this would be an outrageous violation of a guarantee.

This Builds to the Definition . . . A function is a certain number relationship where each input value gives back one unique output value.

Source: A relationship that’s certain, Pepsi® machine, and hourly wage from dy/dan blog


In Out1 52 63 74 8

http://www.abspd.appstate.edu/node/385

Using a Function Machine to Explain Vocabulary

In Out

xf(x)

f(x) is the same as y

Independent Variable

DependentVariable

Domain Range

d W


For the formula:

W = .44 d

Real Life FunctionsAnother way to develop algebraic thinking in adult learners and answer the, “When are we ever gonna use this?” question is to have students explore situations where linear functions are found. Once students have brainstormed and identified areas where functions are found in real life, they should represent the function in words, as a graph, in an equation, and as a table.

One idea to get you started: If gas costs $2.25 a gallon and I buy none, it costs $0. If I buy 1 gallon it costs $2.25, 2 gallons cost $4.50, 3 gallons cost $6.75 etc.

The follow up activity to this is a gallery walk where students look at other students work posted around the classroom.

Adapted from Life Functions by Donna Parrish in The Math Practitioner, Spring 2013.

The Vertical Line TestA function will pass the vertical line test. If we draw a straight up and down line (vertical) on a function’s graph, the line will only cross a function one. This happens since each x value has only one y value.

To demonstrate this in class, have students form graphs with their arms. Then use a rope or a ruler to show that the vertical line test works.


Showing Functions in Four Ways: Table/Equation/Words/GraphTableThis table shows a function since each input (x) has only one output (y):

X Y0 - 101 - 72 - 43 - 1

EquationThe equation y = 3x – 10 shows a function since there can be only one output for every input. As we plug in different input values for x, we will always get only one output value for y. Putting in 0 for x will give us - 10 for y, putting in 1 for x will give us - 7 for y, and this pattern will continue.

WordsWe can also describe a function in words: y is equal to three times x minus ten. A real life example could be: Keisha sells programs at football games. She charges $3 for each program. Her cost to produce all the programs is $10. Her profit is described by the function, profit (y) is equal to three times the number of programs she sells (x) minus the 10 dollar production cost.

GraphThe graph for the function y = 3x -10 is shown here:


Activities that Show Functions in Four FormsFunction MatchingThis activity gives students practice in seeing four different representations of a function: words, graphs, equations, and tables. You will be given 8 functions and must match the cards representing the function in words, graphs, equations, and tables. The master for this activity is on the abspd.appstate.edu website at www.abspd.appstate.edu/node/385

Written Description

Table Graph Equation

y is 3 less than twice a number x y = 2x – 3

Adapted from http://mrsreillyblog.wordpress.com

Function Foldables This activity takes a minimum of prep time and all you need are a few sheets of paper. Cut the paper in half and have students fold it in fourths. Give students a function equation and have them show it in words, graphs, and as a table. You could also vary the activity and have students start with either a function in words/graphs/tables and show the other forms.

Adapted from Math Activities for Migrating to the Common Core Classroom (Donna Ware)


x y-4 -11-2 -70 -32 14 5

y

x


Functions and Rate of ChangeTo introduce rate of change, I have students first watch this YouTube video (Google: 2010 NFL Draft Combine 40 Yard Dash Funnies, go to the 1:09 mark) and ask the question: Who has the faster rate of change: the guy in the suit, the guy in red, or in the guy in green? How do we know?

I then use the handout about real life functions to further this concept. (Find this handout at: www.abspd.appstate.edu/node/385 ) We talk about rate of change and starting amount and identify them using the Towing Service example. We develop an equation that describes the Towing Service example. I then have students work together on the other examples (T Shirt Shop, Plumber, and Cell Phone Charges). We go over the answers for these examples and focus on rate of change and starting amount. (We will later call them by the names that math people use: slope and y intercept.)

I then ask students to develop a rule so we could look at which rate of change was greater in a graph, equation, or table. They will eventually find that the greater the slope, the greater the rate of change is.

Functions Humor!



Function NotationName ofthe function The function

f (x) = 3x + 4

Tells what numberto plug in the function

f(x) is the same as y. f(x) = 3(x) + 4 is the same as y = 3x + 4. We use f(x) since it makes it easier to put in different inputs for x:

For Example:

f(x) = 3(x) + 4

f(0) = 3(0) + 4

f(0) = 4

f(x) = 3(x) + 4

f(1) = 3(1) + 4

f(1) = 7

f(x) = 3(x) + 4

f(2) = 3(2) + 4

f(2) = 10

If: f(x) = 2x + 5 g(x) = 3x2 – 2x + 6 h(x) = - 5x2 + 7x

What is:

1. f(0) 2. f(10) 3. f(-3) 4. f(-9) 5. g(-1)

6. g(2) 7. g(6) 8. h(1) 9. h(-2) 10. h(3)


2

10

3x + 4

Answers begin on pg. 27Linear, Quadratic, and Exponential Function Graphs

Linear Quadratic ExponentialDescription: Straight line U shape (parabola) that

opens up or downGrows/Shrinks fast and levels off on 1 side

Graph looks like:

Equation: y = 2x y = 3x2 - 8 y = 2x

How to tell from the equation what the graph looks like:

The x term is to the first power (x)

The x term is to the second power (x2)

The x term is the exponent (2x)

Graph specific information:

In the equation y = mx + b, if m is negative, the line decreases (goes down) as we look from left to right: y = - x + 5

If m is positive, the line increases (goes up) as we look from left to right: y = .5x + 2

If the number with the x2 is negative, the U shape opens down. If the number with the x2 is positive, the U shape opens up:

In the equation y = ax , if a is bigger than 1 the curve will grow fast. If a is between 0 and 1, the curve will shrink fast.

y = 2 x


y = 12

x

Finding the x and y Intercepts in a Linear FunctionThe x intercept is where a function crosses the x axis and the y intercept is where a function crosses the y axis. In this graph, the x intercept is -2 and the y intercept is 2.

x InterceptTo find the x intercept in a linear function equation, we are going to set y = 0 and solve for x

y = x + 6

0 = x + 6

- 6 = x

The x intercept is -6

y InterceptTo find the y intercept in a linear function equation, we’ll set x = 0 and solve for y:

y = x + 6

y = 0 + 6

y = 6

The y intercept is 6

Quadratic Functions – InterceptsThe x intercepts in a quadratic function are the places where the graph crosses the x axis (where y = 0). The y intercept is the place where the graph crosses the y axis (where x = 0).

For this graph, the x intercepts are -1 and 3. The y intercept is -3.


Quadratic Functions: Maximums and MinimumsGraphs of quadratic functions either have maximums or minimums. The maximum is the highest point on the upside down U shape (or parabola) where we have the largest y value. It is also called the vertex.

The minimum is the lowest point on the upward facing U shape (or parabola) where we have the smallest y value. It is also called the vertex. Maximums and minimums are expressed by their y values.

If we compare graphs to mountains and valleys, think about the maximum as the mountaintop. If we compare minimums to valleys, think about the minimum as the lowest point in the valley.

In the examples to the left, the parabola with the upside down U has a maximum. The y value for this maximum is 0.

The U shaped parabola shows a minimum and has a y value of 2. An easy way to find the y value is to draw a straight line from the vertex to the y axis and see where it crosses.

Do these parabolas show a maximum or minimum and what are their values?1. 2.

3. 4.


Answers begin on pg. 27Quadratic Functions – SymmetryThe axis of symmetry (sometimes called line of symmetry) is the straight up and down line through the vertex on the graph of a quadratic function that cuts the U shape in half.

1. 2.


What is the line of symmetry for this function? What is the line of symmetry for this function?

Answers begin on pg. 27Increasing/Decreasing/Constant We always read graphs like we read a book from left to right. Look to see where the graph is going up (increasing), going down (decreasing), or staying the same (constant). Picture a roller coaster or a car riding the function to help decide when the function increases, decreases, or stays the same.

Where are these functions g(x) increasing, decreasing, and remaining constant?

1. 2. | Algebraic Functions and Modeling

Where are the graphs of these functions increasing, decreasing, and remaining constant?

3. 4.

5. 6.

Answers begin on pg. 27


A New Way to View Math Related StressView the TED talk, “How to Make Stress Your Friend.” (Find it by Googling: How to make stress your friend ted.) Teach students that:

1. How we think and act can transform stress experiences. When we choose to view stress responses (pounding heart, faster breathing) as our body’s way of gearing up to help us successfully face a challenge, the harmful effects of stress end. Instead of being viewed as negative, our pounding heart is preparing us for action. The increased breathing is bringing more oxygen to our brain.

2. When we choose to connect with others under stress, we gain the strength to meet your challenges. So, create community in the classroom so students can share stressful experiences.

Involve Your Students: Teaching with Questions Video1. How does this instructor present new material?

2. How does the instructor make his students curious about new material?

3. What did he mean by, “Sometimes the best thing I can do is walk away”?

4. What are the benefits of disagreements in the classroom?

5. What does the instructor mean when he says, “Wrong answers provide the best opportunity for learning?

Sample Classroom Questions to Help Students Understand Math Reasoning

What does this mean? What are you doing here? (indicating something on student work) Tell me where you’re getting each of your numbers from here. Why did you decide to…? I don’t understand. Could you show me an example of what you mean?


So what are you going to try next? What are you thinking about? Is there another idea you might try? Why did you decide to begin with…? Do you have any ideas about how you might figure out…? You just wrote down ____. Tell me how you got that. What are you doing there with those numbers? Do you agree with ______’s answer? Why or why not? Is ______ always true, sometimes true, or never true?

Source: https://youtu.be/hv7MBSEcpW8 Questions adapted from GED Testing Service

Math Modeling: CommutingMath modeling is a fancy way of taking a real life situation, showing it as a mathematical equation, and then solving it.

How long does it take you to get to work, school, or the store? That can depend on several factors such as whether you drive, take the bus or walk. It can also depend on other factors like the weather. First, brainstorm a list of factors that will affect your commute. Then assign a time value to each one. Finally, develop an equation that shows all the factors. C will be the total time it takes to commute to work.

Normal Commute Time

Weather

______ minutes W = weather3W = 3 minutes delay for bad weatherW = 0 Sunny dayW = 1 Rainy dayW = 3 Snowy dayW = 4 Foggy day

Equation: C =


Answers begin on pg. 27Modeling: Window Washing at the Luxor HotelDan Meyer’s blog (search for it by Googling: dy/dan) has a number of creative math modeling activities in his “Math in 3 Acts” section.

How long would it take to wash the windows at the Luxor Hotel in Las Vegas?

First show students some pictures of the hotel. Then ask them what information they would like to have in order to solve the problem. (“The formulation of the problem is often more essential than its solution.” – Albert Einstein)

Give the students this information after they brainstormed the information they think they will need:

How long would it take to clean one square foot of glass?

Draw a square foot on the board and have a student clean it and see how long it takes

How much glass is there?

The Luxor Hotel has 13 acres of glass

How much is an acre in square feet?

1 acre is 43,560 square feet

How many hours per day will we work? | Algebraic Functions and Modeling

Decide as a class

How many people are in the work crew?

Decide as a class

Students will come up with various answers depending on how long it takes to clean one square foot of glass, how big their work crew is, and how many hours per day they work.

Source: dy/dan

Polya Problem Solving Method - UPS ✔

1. Understand the problem

What are you asked to do?

Will a picture or diagram help you understand the problem?

Can you rewrite the problem in your own words?

2. Create a plan

Use a problem solving strategy:

Guess and check Solve an easier problemMake a list ExperimentDraw a picture or diagram Act it outLook for a pattern Work backwardsMake a table Change your viewpointUse a variable

3. Solve

Be patient

Be persistent

Try different strategies

4. Check

Does your answer make sense?

Are all the questions answered? | Algebraic Functions and Modeling

What other ways are there to solve this problem?

What did you learn from solving this problem?

Understand

Plan

Solve


Check

Function Practice Problems1. Sarisa says that the tables below represent data from the same function. Is she right or wrong?

Table Ax f(x)-2 4-1 10 01 12 43 9

Table Bx f(x)4 85 106 127 148 169 18

2. Suppose that the function f(x) = 2x +12 represents the cost to rent x movies a month from an internet movie club. Makayla now has $10. How many more dollars does Makayla need to rent 7 movies next month?

3. The table below shows the cost of a pizza based on the number of toppings.

Number of Toppings (n) Cost (C)1 $122 $13.503 $154 $16.50

Which function represents the cost of a pizza with n toppings?A. c(n) = 12 + 1.5 (n – 1)B. c(n) = 1.5n + 12C. c(n) = 12 + nD. c(n) = 12n

The function f(x) = 0.49x + 44.95 describes the total cost for a company to print flyers, where x is the | Algebraic Functions and Modeling

number of flyers printed.

4. What is the cost to print 1 flyer?

5. $ ____________ is the cost to print 250 flyers

6. Using the following two functions:

f(x) = x2 – 5x + 6

g(x) = - 12 x + 3

Put the following in order from least to greatest:

g(-3)

f(0)

g(0)

f(3)

Least Greatest

7. Which of the following is true for the function f(x) = 3x2 – 4x + 1

A. f(0) > f(1)B. f(0) > f(-1)C. f(1) > f(-1)D. f(1) > f(0)

8. Which table of values corresponds to the function f(x) = 11x – 3

A. x -2 -1 0 1 2 3

f(x) 19 16 13 10 7 4

B. x -2 -1 0 1 2 3

f(x) -25 -22 -19 -16 -13 -10

C. x -2 -1 0 1 2 3

f(x) -25 -14 -3 8 19 30 | Algebraic Functions and Modeling

D.

x -2 -1 0 1 2 3f(x) 19 8 -3 -14 -25 -36

Source: NC Department of Public Instruction and Common Core Achieve Exercise Book, McGraw Hill/Contemporary

Answers begin on pg. 27Using the TI 30 XS to Solve FunctionsPress: Table

Enter the function after the y = The x key is to the left of the 4

Press: Enter

You will see Start = 0. Put in the x value where you want your table to start

Press: Enter, Enter, Enter, Enter (4 times)

A table of x and y values will be displayed!

ResourcesAlgebra Resources

Google: geogebra www.geogebra.org

Geogebra is a free graphing calculator app you can use to model functions.

Google: get the math www.thirteen.org/get-the-math

Get the Math show how algebra is used in the real world in music, sports,fashion, video games, restaurants, and special effects.

Research Base


http://www.thirteen.org/get-the-math

http://www.geogebra.org/

Conley, D. T. (2007). Redefining college readiness. Eugene, OR: Educational Policy Improvement Center.

National Institute for Literacy. (2010). Algebraic thinking in adult education. Washington, DC: Author.

Nonesuch, K. (2006). Changing the way we teach math: a manual for teaching basic math to adults.

Duncan, B.C., Canada: Malaspina University-College.

U.S. Department of Education, Office of Career, Technical, and Adult Education. (2014).

TEAL math works! guide. Washington, DC: Author.

Answer KeyPg. 12

1. f(x) = 2x + 5

f(0) = 2(0) + 5

f(0) = 5

2. f(x) = 2x + 5

f(10) = 2(10) + 5

f(10) = 20 + 5

f(10) = 25

3. f(x) = 2x + 5

f(-3) = 2(-3) + 5

f(-3) = -6 + 5

f(-3) = -1

4. f(x) = 2x + 5

f(-9) = 2(-9) + 5

f(-9) = -18 + 5

f(-9) = -13

5. g(x) = 3x2 – 2x + 6

g(-1) = 3(-1)2 – 2(-1) + 6

g(-1) = 3(1) – (-2) + 6

g(-1) = 3 +2 + 6

g(-1) = 11

6. g(x) = 3x2 – 2x + 6

g(2) = 3(2)2 – 2(2) + 6

g(2) = 3(4) – 4 + 6

g(2) = 12 – 4 + 6

g(2) = 14

7. g(x) = 3x2 – 2x + 6

g(6) = 3(6)2 – 2(6) + 6

g(6) = 3(36) – 12 + 6

g(6) = 108 – 12 + 6

g(6) = 102

8. h(x) = - 5x2 + 7x

h(1) = - 5(1)2 + 7(1)

h(1) = -5(1) + 7

h(1) = -5 + 7

h(1) = 2

9. h(x) = - 5x2 + 7x

h(-2) = - 5(-2)2 + 7(-2)

h(-2) = -5(4) + -14

h(-2) = -20 + -14

h(-2) = -34

10. h(x) = - 5x2 + 7x

h(3) = - 5(3)2 + 7(3)

h(3) = -5(9) + 21

h(3) = -45 + 21

h(3) = -24

Pg. 15

1. Maximum, y = 4 2. Minimum, y = -1 3. Minimum, y = 0 4. Maximum, y = 16

Pg. 16

1. x = 2 2. x = - 3

Pgs. 17 and 18


1. g(x) is increasing: (-∞, -3) U (1, 4)

g(x) is decreasing: (-3, 1)

g(x) is constant: (4, ∞)

2. g(x) is increasing: (-5, -2) U (4, 5)


g(x) is constant: (2, 4)

3. g(x) is increasing: (-2, ∞)

g(x) is decreasing: (-6, -2)

g(x) is constant: (-∞, -6)

4. g(x) is increasing: (4, ∞)

g(x) is decreasing: (-∞, -3)

g(x) is constant: (-3, 4)

5. g(x) is increasing: (1, 3) U (4, ∞)

g(x) is decreasing: (-∞, 1) U (3, 4)

g(x) is constant: none

6. g(x) is increasing: (-5, -2) U (0, 3)


g(x) is constant: (3, 6)

Pg. 20

A possible answer could be:

C = 15 + 3W + 2R + T

15 = 15 minutes normal commute time

W = weather 3W = 3 minutes delay for bad weather

W = 0 Sunny day W = 1 Rainy day W = 3 Snowy day W = 4 Foggy day

R = red light 2R = 2 minutes per red light

T = traffic

T = 0 No traffic T = 5 Moderate traffic T = 10 Heavy traffic

Pgs. 24 and 25

1. Table A is n2 Table B is 2n so they are not the same table

5. f(x) = 0.49x + 44.95 f(250) = 0.49(250) + 44.95 f = 167.45

2. f(x) = 2x +12 f(7) = 2(7) + 12 f(7) = 26

6. f(3) = 0 g(0) = 3


$26 – $10 = $16 g(-3) = 4.5 f(0) = 6

3. With each topping, the cost of the pizza increases by 1.50 so B. c(n) = 1.5n + 12

7. f(1) = 0 f(0) = 1 A. f(0) > f(1) f(-1) = 8

4. f(x) = 0.49x + 44.95 f(1) = 0.49(1) + 44.95 f = .49 + 44.95 f = 45.44

8. f(x) = 11x – 3

f(0) = -3 f(1) = 8 so C


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