Linear, Quadratic , Exponential , and Absolute Value Functions · 2015-02-19 · Day 2 NonLinear Functions_Tables.notebook 12 February 19, 2015 Linear, Quadratic & Exponential Functions

Day 2 NonLinear Functions_Tables.notebook

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February 19, 2015

Linear, Quadratic , Exponential , and Absolute Value Functions

Linear Quadratic Exponential Absolute ValueY = mx + b y = ax2 + bx + c y = a ∙ bx y = |x|


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February 19, 2015

What type of graph am I?


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February 19, 2015

Linear Exponential Quadratic

What can you tell me about each type of graph?


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February 19, 2015

Topic 3: Classify Equations I Can: Classify equations as linear, exponential, quadratic, absolute

value or none of these.

What have we already learned about linear equations? How many different types of linear equations can you write?


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February 19, 2015

LINEAR QUADRATIC EXPONENTIAL

f(x) = x2 5

f(x) = 4x + 3

y= 20(3.02x)

ABS VALUE

y = x

y = 2(x + 1) 3

4x + 3y = 24|2x| = 105|3 x| = 25

y 4 = 2(x + 1)2


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February 19, 2015


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February 19, 2015

LINEAR QUADRATIC EXPONENTIAL ABS VALUE

How can we distinguish each type of equation?


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February 19, 2015


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February 19, 2015

Linear, Quadratic & Exponential Functions

What type of function?

(move this box)


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February 19, 2015

Linear, Quadratic & Exponential FunctionsWhat type of function?

(move this box)


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February 19, 2015

Linear, Quadratic & Exponential FunctionsWhat type of function?

(move this box)


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February 19, 2015

Linear, Quadratic & Exponential Functions

In the real world, people often gather data and then must decide what kind of relationship (if any) they

think best describes their data.

You may be able to use the graph of data points to determine a model for the data.


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February 19, 2015

Graphing Data to Choose a Model

Plot the data points and connect them.The data appear to be exponential

Graph each data set. Which kind of model best describes the data?

11 2

3


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February 19, 2015


Plot the data points and connect them.The data appear to be linear


2

3

22


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February 19, 2015


Plot the data points and connect them.The data appear to be exponential


2

3

33


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February 19, 2015


Plot the data points and connect them.The data appear to be quadratic


2

3

44


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February 19, 2015

Choose a Model from Table

Another way to decide which kind of relationship (if any) best describes a data set is to use

patterns.

We can use what we know about arithmetic sequences (common difference) and geometric sequences (common ratio) to find a pattern and

write an equation to model the data.


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February 19, 2015

Topic 1: Classify TablesI Can: Classify a consistent table as a quadratic, exponential, absolute value, or other function. I can also identify the rate of change.

first differences that have a constant ratio.


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February 19, 2015


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February 19, 2015

51.0052.0054.0058.00

1

24


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For every constantchange of +1there is a constant rate of change for the Second Difference. Quadratics have constant SECOND DIFFERENCE.


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February 19, 2015

Determine the type of function:

If the pattern continues, when would the oven be 750?


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February 19, 2015

Classify the function from the table:

x y

0 9

1 7

2 5

3 3

4 5

5 7

What is the y-value when x = 8?


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February 19, 2015


x y

0 0

1 1

2 3

3 7

4 15

5 31

What is the y-value when x = -1?


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February 19, 2015


x y

1 0

0 1

1 0

2 3

3 8

4 15

What is the y-value when x = -2?


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February 19, 2015

Attachments

ComparingLinear&ExponentialFunctionsTeacher.pdf

ComparingLinear&ExponentialFunctionsStudent.doc

ComparingLinear&ExponentialFunctionsStudent.pdf

Comparing_Linear_and_Exponential_Functions.tns

﴾Activity Day 4﴿ TypesFunctionExploration linear and quadratic from I drive.doc

﴾Activity Day 4﴿ TypesFunctionExploration cubic and absolute value from I drive.doc

graphingfunctions.tns

graphingfunctionsteacherguide.doc

graphingfunctionsteacherguide.pdf

Comparing Linear and Exponential Functions TEACHER NOTES MATH NSPIRED

©2011 Texas Instruments Incorporated 1 education.ti.com

Math Objectives

Students will use a table and a graph to compare the changes in

linear and exponential expressions as x increases.

Students will recognize that as x increases, a linear expression

increases at a constant rate (additively) while an exponential

function increases multiplicatively.

Students will recognize that an exponential function with a

positive base will never be less than or equal to 0, but will get

smaller and smaller as x decreases.

Students will determine whether a graph represents a linear or an

exponential function.

Students will use appropriate tools strategically (CCSS

Mathematical Practice).

Students will construct viable arguments and critique the

reasoning of others (CCSS Mathematical Practice).

Vocabulary

exponential function

About the Lesson

This lesson involves moving a point that changes the value of x

and observing and comparing the values of a linear expression

and an exponential expression.

As a result, students will:

Compare linear and exponential expressions.

Compare linear and exponential functions.

TI-Nspire™ Navigator™ System

Use Screen Capture to compare linear and exponential

expressions.

Use a Notes page and Screen Capture to compare and contrast

linear and exponential functions.

Use Quick Polls to assess students’ understanding throughout

the lesson.

Use Teacher Edition computer software to review student

documents.

TI-Nspire™ Technology Skills:

Download a TI-Nspire

document

Open a document

Move between pages

Grab and drag a point

Tech Tips:

Make sure the font size on

your TI-Nspire handheld is set

to Medium.

You can hide the entry line by

pressing / G.

Lesson Materials: Student Activity

Comparing_Linear_and_ Exponential_Functions_Student

.pdf

Comparing_Linear_and_

Exponential_Functions_Student

.doc

TI-Nspire document

Comparing_Linear_and_

Exponential_Functions.tns



Discussion Points and Possible Answers

Tech Tip: If students experience difficulty dragging a point, check to make

sure that they have moved the cursor until it becomes a hand (÷) getting

ready to grab the point. Also, be sure that the word point appears. Then

press / x to grab the point and close the hand ({).

Teacher Note: This lesson can be used to probe more deeply into the

behavior of exponential functions by changing the base in the .tns

document, using numbers such as 2 or 0.5 for the base.

TI-Nspire Navigator Opportunity

Use Screen Capture to determine whether or not students are experiencing difficulty using

the .tns file. Use Live Presenter to demonstrate the correct procedure for using the file.

You may want to take a Quick Poll to see if most of the students are obtaining the correct

answer to questions 2 through 4. This will enable you to either stop and clear up any

misunderstandings, or continue with the lesson.

Move to page 1.2.

1. Grab and drag the point to change the value of x. Complete

the table below. Which column is growing faster?

Answer: The 3x column is growing faster.

x 3x 3x

0 0 1

1 3 3

2 6 9

3 9 27

4 12 81

5 15 243



2. a. As x increases from 2 to 3, how does the value of 3x change?

Answer: The value of 3x increases by 3.

b. As x increases by 1, describe the pattern in the numbers in the 3x column of the table.

Answer: The numbers increase by 3 each time.

Teacher Tip: At this point, check for student understanding of repeated

addition of 3.

c. As x increases from 2 to 3, how does the value of 3x change?

Answer: It triples; it increases 3 times as much.

d. As x increases from 3 to 4, how does the value of 3x change?

Answer: It triples; it increases 3 times as much.

e. As x increases by 1, describe the pattern in the numbers in the 3x column of the table.

Answer: The numbers are being multiplied by 3. The values triple.

Teacher Tip: Since the rate of change for 3x is constant, students might

initially examine the values of 3x in terms of rate of change. For instance, a

student could respond "the value of 3x increases by 18." In this case, you

might ask the student if this pattern holds true for all changes in the value

of 3x. Since it does not, encourage the student to search for another

pattern in the table.

3. Complete the bottom row of the table for x = 6. How did you determine the values for 3x and 3x?

Answer: Students might say that they added 3 to 15 (previous row) to get 18 and multiplied

243 by 3 to get 729, or any other acceptable method.

x 3x 3x

6 18 729



4. Why are the values for 3x increasing faster than the values for 3x?

Answer: The values of 3x are increasing faster than 3x because you multiply the previous

number by 3 instead of adding 3 to the previous number. When a whole number greater than

1 is repeatedly multiplied by 3, the result gets greater faster than when you repeatedly add 3.

For example, if the whole number were 2, 2 ∙ 3 = 6 while 2 + 3 = 5. The product is greater at

the beginning, and the sum will never catch up. 2 ∙ 3 ∙ 3 = 18 while 2 + 3 + 3 = 8.

Teacher Tip: While multiplying whole numbers greater than 1 by a positive

integer greater than 1 makes the product increase, students should

recognize that when a fraction between 0 and 1 is multiplied by a constant

multiplier greater than one, the results get smaller and smaller. For

example, 1/3, 1/9, 1/27, and so on.

You might want to have students reflect on how multiplication works as

repeated addition, that is 3 ∙ 2 means two 3s or 3 + 3. Thus, comparing 3x

to 3x going from x = 5 to x = 6 means for 3x you have five 3s or 3 + 3 + 3 +

3 + 3 and the next term would have six 3s or (3 + 3 + 3 + 3 + 3) + 3 where

you added a 3. With 35, the next term would be found by multiplying 3

5 by 3

or adding 35 three times: 3 ∙ 3

5 = (3

5 + 3

5 + 3

5). Two 3

5s were actually

added to the previous term.

5. The function f(x) = 3x is called an exponential function, while the function f(x) = 3x is a linear

function. Describe the difference in the two functions.

Answer: A linear function has the variable as a factor in defining the function. In an exponential

function, the variable is part of the exponent.



Move to page 2.1.

6. Drag the point to the right to produce two graphs—one solid, one

dashed. Use the information from the table in question 1 to

identify which graph represents an exponential function and

which graph represents a linear function. Justify your answer.

Answer: The dashed graph remains closer to the x-axis and is f(x) = 3x because it is

increasing at a slower rate than the graph f(x) = 3x. The graph of f(x) = 3x increases at a

constant rate, 3 units vertically for every 1 unit horizontally. The solid graph, f(x) = 3x,

increases at an increasing rate.

7. How do the graphs of f(x) = 3x and f(x) = 3x support your response to question 4?

Answer: When comparing the y-values for f(x) =3x, each time x increases by 1 unit, the y-value

increases by 3 units. For f(x) = 3x, each time x increases by 1 unit, the new y-value is 3 times the

previous y-value.

8. Aaron says that the values of f(x) = 5x will increase faster than the values of the linear

function f(x) = 5x. Do you agree or disagree? Justify your answer.

Answer: I agree with Aaron because for f(x) = 5x, the y-values will be multiplied by 5 every

time the x-value is increased by 1. For f(x) =5x, 5 will be added to the previous y-value each

time the x-value is increased by 1.

TI-Nspire Navigator Opportunity

Use Quick Polls to determine the number of students agreeing with the statement in question 8.

Teacher Tip: This might be a good time to ask students to give you

examples of other linear or exponential functions.



TI-Nspire Navigator Extension Opportunity

Have students press / ~ and choose Add Notes to add a new notes page to the file. Have

students compare and contrast linear and exponential functions on the page. Capture students’

screens and discuss their responses.

Wrap Up

Upon completion of the discussion, the teacher should ensure that students understand:

Expressions of the form 3x increase by repeated addition.

Expressions of the form 3x increase by repeated multiplication.

Graphs of linear functions increase at a constant rate.

Graphs of exponential functions of the form y = bx, where b is greater than 1 increase faster

than graphs of linear functions of the form y = bx.

Exponential functions of the form y = bx, where b is greater than 0 will never have values for

f(x) that are 0 or negative.

Extension: Trying Other Bases

Have students press / G to show the function entry line on page 2.1. Then press the £ on the Touchpad twice to move to f1(x) and press the ¡ until the cursor is between the base and the exponent. Press .and change the base from 3 to 5. Press ·.

Have students press / G again and press the £ on the Touchpad once to move to f2(x). Move the

cursor until it is to the right of 3 and press .. Change the 3 to a 5. Press ·.

TI-Nspire Navigator Extension Opportunity

Students then drag the point on the arrow to the right to see the two graphs. Use Screen Capture to

view the screens. Was Aaron correct?

You might want to have different groups of students change the coefficient of the linear equation and

the base on the exponential equation to other numbers greater than 1 and use Screen Capture to

compare the results. Numbers between 0 and 1 can be used. Have students press Menu > Window /

Zoom > Zoom – Out > · before moving the point on the arrow to the left.

SMART Notebook

Comparing Linear and Exponential FunctionsName

Student ActivityClass

Comparing Linear and Exponential RelationsStudent Activity

Open the TI-Nspire document Comparing_Linear_and_Exponential_Functions.tns.

In this activity, you will explore the values of the expressions 3x and 3x as x changes from 0 to 5. You will compare the two expressions by investigating patterns in how their values change both in a table and graphically.

Move to page 1.2.

Press / ¢ and / ¡ to navigate through the lesson.

1.Grab and drag the point to change the value of x. Complete the table below. Which column is growing faster?

x

3x

3x

0

1

2

3

4

5

2.a.As x increases from 2 to 3, how does the value of 3x change?

b.As x increases by 1, describe the pattern in the numbers in the 3x column of the table.

c.As x increases from 2 to 3, how does the value of 3x change?

d.As x increases from 3 to 4, how does the value of 3x change?

e.As x increases by 1, describe the pattern in the numbers in the 3x column of the table.

3.Complete the bottom row of the table for x = 6. How did you determine the values for 3x and 3x?

4.Why are the values for 3x increasing faster than the values for 3x?

5.The function f(x) = 3x is called an exponential function, while the function f(x) = 3x is a linear function. Describe the differences in the two functions.

Move to page 2.1.

Press / ¢ and / ¡ to navigate through the lesson.

6.Drag the point to the right to produce two graphs—one solid, one dashed. Use the information from the table in question 1 to identify which graph represents an exponential function and which graph represents a linear function. Justify your answer.

7.How do the graphs of f(x) = 3x and f(x) = 3x support your response to question 4?

8.Aaron says that the values of f(x) = 5x will increase faster than the values of the linear function f(x) = 5x. Do you agree or disagree? Justify your answer.

©2011 Texas Instruments Incorporated1education.ti.com

©2011 Texas Instruments Incorporated2education.ti.com

SMART Notebook

Comparing Linear and Exponential Functions Name Student Activity Class


Open the TI-Nspire document

Comparing_Linear_and_Exponential_Functions.tns.

In this activity, you will explore the values of the expressions 3x and 3x

as x changes from 0 to 5. You will compare the two expressions by

investigating patterns in how their values change both in a table and

graphically.

Move to page 1.2.

Press / ¢ and / ¡ to

navigate through the lesson.

1. Grab and drag the point to change the value of x. Complete the table below. Which column is growing

faster?

x 3x 3x

0

1

2

3

4

5

2. a. As x increases from 2 to 3, how does the value of 3x change?

b. As x increases by 1, describe the pattern in the numbers in the 3x column of the table.

c. As x increases from 2 to 3, how does the value of 3x change?

d. As x increases from 3 to 4, how does the value of 3x change?

e. As x increases by 1, describe the pattern in the numbers in the 3x column of the table.

Comparing Linear and Exponential Relations Student Activity


3. Complete the bottom row of the table for x = 6. How did you determine the values for 3x and

3x?

4. Why are the values for 3x increasing faster than the values for 3x?

5. The function f(x) = 3x is called an exponential function, while the function f(x) = 3x is a linear

function. Describe the differences in the two functions.

Move to page 2.1.

Press / ¢ and / ¡ to

navigate through the lesson.

6. Drag the point to the right to produce two graphs—one solid, one dashed. Use the information

from the table in question 1 to identify which graph represents an exponential function and

which graph represents a linear function. Justify your answer.

7. How do the graphs of f(x) = 3x and f(x) = 3x support your response to question 4?

8. Aaron says that the values of f(x) = 5x will increase faster than the values of the linear function

f(x) = 5x. Do you agree or disagree? Justify your answer.

SMART Notebook

SMART Notebook

3

4

5

yx

=-

X

Y

-10

-5

0

5

10

25

yx

=-+

X

Y

-4

-2

0

2

4

7

(5)1

4

yx

=--

X

Y

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1

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9

13

3(2)4

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yxx

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X

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_1369126265.unknown

_1369126388.unknown

_1369126506.unknown

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SMART Notebook

34

yx

=-+

X

Y

1

2

3

4

5

3

310

yxx

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X

Y

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-1

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x

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_1355657342.unknown

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SMART Notebook

SMART Notebook

Graphing Functions

Teacher Guide

by: Tina Hill, Daniel Boone High School, Washington County, TN

Activity Overview

This activity may be used as a review of functions. It is set up with self-check answers. Students determine if the graph is a function and, if it is, name the function. The students will also graph various functions and compare/contrast the graphs.

Concepts

· Graphing functions

Tennessee Standards

· Algebra I

· 3102.1.14 Apply graphical transformations that occur when changes are made to coefficients and constants in functions.

· 3102.3.16 Determine if a relation is a function from its graph or from a set of ordered pairs.

· 3102.3.17 Recognize “families” of functions.

· 3102.3.18 Analyze the characteristics of graphs of basic linear relations and linear functions including constant function, direct variation, identity function, vertical lines, absolute value of linear functions. Use technology where appropriate.

· 3102.5.6 Draw qualitative graphs of functions and describe a general trend or shape.

Teacher Preparation

· Load or have the students load the tns file: graphing functions.tns

· There is no student sheet with this activity. The teacher may request answers to the compare and contrast questions. If so, the student may write the answers on paper.

TI Nspire Applications

Graphs & Geometry

Notes

Question/Answer

Problem 1

In problem 1 students name the function.

Students observe the given function. They then decide if the graph is a function using the vertical line test; then classify the function by clicking on the circle of the correct function name.

Problem 2

In problem 2 students graph more than one equation on the same graph of the linear function family. The students then compare and contrast the graphs.

On page 2.2, students graph three linear functions with different slopes and intercepts.

On page 2.3, students’ answers will vary. An example: All three graphs were straight lines but they had different slopes and y-intercepts.

Problem 3

In problem 3 students graph more than one equation on the same graph of the quadratic function family. The students then compare and contrast the graphs.

On page 3.2, students graph quadratic equations with different x-coefficients and y-intercepts.

On page 3.3, students’ answers will vary. An example: All four graphs were parabolas but they had different y-intercepts, different vertices, and different lines of symmetry.

Problem 4

In problem 4 students graph more than one equation on the same graph of the exponential function family. The students then compare and contrast the graphs.

On page 4.2, students graph exponential equations with different exponents and base.

On page 4.3, students’ answers will vary. An example: All four graphs were didn’t touch the x-axis. They crossed the y-axis at different coordinates. They are all increasing.

Problem 5

In problem 5 students graph more than one equation on the same graph of the absolute value function family. The students then compare and contrast the graphs.

On page 5.2, students graph absolute value equations.

On page 5.3, students’ answers will vary. An example: All of the graphs form a “v”. The graph with the negative on the outside of the absolute value made the graph upside-down. The others were shifted.

Problem 6

In problem 6 students graph more than one equation on the same graph of the sinusoidal function family. The students then compare and contrast the graphs.

On page 6.2, students graph sinusoidal equations.

On page 6.3, students’ answers will vary. An example: The graphs have the same wavy pattern but the 2sin(x) is longer.

On page 6.4, students’ answers will vary. An example: The graphs have the same wavy pattern but the 2cos(x) is longer.

On page 6.5, students’ answers will vary. An example: The graphs have the same wavy pattern, but the cos(x) crosses the y-axis at (0, 1) and the sin(x) crosses the y-axis at the origin.

On page 6.6, students’ answers will vary. An example: The graphs will have the same wavy pattern and the graphs will move up 4 units, but they will cross the y-axis at different points.

On page 6.7, the students’ will test their prediction.

1

SMART Notebook

Graphing FunctionsTEACHER GUIDE

1

by: Tina Hill, Daniel Boone High School,Washington County, TN

Activity Overview

This activity may be used as a review of functions. It is set up with self-check answers. Students determineif the graph is a function and, if it is, name the function. The students will also graph various functions andcompare/contrast the graphs.

Concepts

Graphing functions

Tennessee Standards

Algebra I

o 3102.1.14 Apply graphical transformations that occur when changes are made to coefficients andconstants in functions.

o 3102.3.16 Determine if a relation is a function from its graph or from a set of ordered pairs.o 3102.3.17 Recognize “families” of functions.o 3102.3.18 Analyze the characteristics of graphs of basic linear relations and linear functions

including constant function, direct variation, identity function, vertical lines, absolutevalue of linear functions. Use technology where appropriate.

o 3102.5.6 Draw qualitative graphs of functions and describe a general trend or shape.

Teacher Preparation

Load or have the students load the tns file: graphing functions.tns There is no student sheet with this activity. The teacher may request answers to the compare and

contrast questions. If so, the student may write the answers on paper.

TI Nspire Applications

Graphs & Geometry

Notes

Question/Answer

Problem 1

In problem 1 students name the function.

Students observe the given function. Theythen decide if the graph is a function using thevertical line test; then classify the function byclicking on the circle of the correct functionname.


2


3

Problem 2

In problem 2 students graph more than one equation on the same graph of the linearfunction family. The students then compare and contrast the graphs.

On page 2.2, students graph three linearfunctions with different slopes and intercepts.

On page 2.3, students’ answers will vary. Anexample: All three graphs were straight lines butthey had different slopes and y-intercepts.

Problem 3

In problem 3 students graph more than one equation on the same graph of the quadraticfunction family. The students then compare and contrast the graphs.


4

On page 3.2, students graph quadraticequations with different x-coefficients and y-intercepts.

On page 3.3, students’ answers will vary. Anexample: All four graphs were parabolas butthey had different y-intercepts, different vertices,and different lines of symmetry.

Problem 4

In problem 4 students graph more than one equation on the same graph of the exponentialfunction family. The students then compare and contrast the graphs.

On page 4.2, students graph exponentialequations with different exponents and base.

On page 4.3, students’ answers will vary. Anexample: All four graphs were didn’t touch the x-axis. They crossed the y-axis at differentcoordinates. They are all increasing.


5

Problem 5

In problem 5 students graph more than one equation on the same graph of the absolute valuefunction family. The students then compare and contrast the graphs.

On page 5.2, students graph absolute valueequations.

On page 5.3, students’ answers will vary. Anexample: All of the graphs form a “v”. The graphwith the negative on the outside of the absolutevalue made the graph upside-down. The otherswere shifted.

Problem 6

In problem 6 students graph more than one equation on the same graph of the sinusoidalfunction family. The students then compare and contrast the graphs.

On page 6.2, students graph sinusoidalequations.


6

On page 6.3, students’ answers will vary. Anexample: The graphs have the same wavypattern but the 2sin(x) is longer.

On page 6.4, students’ answers will vary. Anexample: The graphs have the same wavypattern but the 2cos(x) is longer.

On page 6.5, students’ answers will vary. Anexample: The graphs have the same wavypattern, but the cos(x) crosses the y-axis at (0, 1)and the sin(x) crosses the y-axis at the origin.

On page 6.6, students’ answers will vary. Anexample: The graphs will have the same wavypattern and the graphs will move up 4 units, butthey will cross the y-axis at different points.

On page 6.7, the students’ will test theirprediction.

SMART Notebook

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Linear, Quadratic , Exponential , and Absolute Value Functions · 2015-02-19 · Day 2 NonLinear Functions_Tables.notebook 12 February 19, 2015 Linear, Quadratic & Exponential Functions