Exponential graphs - characteristics

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Lesson Title: Exponential Graph Characteristics Date: Teacher(s): Course: Algebra II, Unit 2 Start/end times: Lesson Standards/Objective(s): What mathematical skill(s) and understanding(s) will be developed? Which

Mathematical Practices do you expect students to engage in during the lesson?

F.IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

MP1: Make sense of problems and persevere in solving them. MP2: Reason abstractly and quantitatively. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for and make use of structure. MP8: Look for and express regularity in repeated reasoning.

Algebra II, Unit 2

Lesson Launch Notes: Exactly how will you use the

first five minutes of the lesson? x y

1 64 2 32 3 16 4 8 5 4 Use the sequence above to answer the following questions: 1.

What type of sequence is this? What pattern can you identify? 2.

Looking at the graph of the first

10

terms, what do you observe about the graph? 3.

If this same pattern began with the number

eight

, what would be the first

four

terms? The first

10

? 4.

Graph the terms of both patterns on the same grid in two different colors. Compare the two graphs. (Look for evidence of MP1 and MP8.) Lesson Closure Notes: Exactly what summary activity,

questions, and discussion will close the lesson and connect big ideas? List the questions. Provide a foreshadowing of tomorrow.

Given the equation, following without a calculator. , complete the 1.

2.

3.

4.

5.

Identify the parent function. Name all transformations. Give the domain, range, and asymptote. What is the

y

-intercept? Sketch the function. (Look for evidence of MP2 and MP7.) Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,

problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic connections to appropriate mathematical practices.

1.

2.

3.

4.

5.

Show Movie clip from IQ about the Zeno’s Paradox (an example of why halving the distance will never make zero), available at http://www.youtube.com/watch?v=YuftF3ZnMZM . Discuss with students how if the pattern of halving the terms from the warm up will never get you to zero, and how that relates to a horizontal asymptote. In small groups, have students complete the Frayer Model for Exponential Functions to revisit their knowledge of Exponential Functions from Algebra I. Then discuss as a whole group. Direct the discussion to the key features of the graph of an exponential function. Ask, “What are the universal characteristics of exponential functions?” (Use this as a formative assessment to check entry points for next activity. It is okay if students do not identify all of the key features at this point.) Distribute

Exponential Graph Characteristics

worksheet. Assign students in pairs to have them work through the activity (Growth = Form A, Decay = Form B). When complete, have students select a new partner with a Form unlike their own. (Look for evidence of MP5 HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand

,

S. (2009).

Accessible mathematics

:

10 instructional shifts that raise student achievement.

Portsmouth

,

NH

:

Heinemann

.

Lesson Title: Exponential Graph Characteristics Course: Algebra II, Unit 2 Date: and MP6.) Teacher(s): Start/end times: 6. Lead a whole-class discussion to summarize key findings and revisit key features of exponential graphs. Have students revisit Frayer Model and make any necessary additions or adjustments. Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I

measure student success? That is, deliberate consideration of what performances will convince you (and any outside observer) that your students have developed a deepened and conceptual understanding.

Students should be able to name all transformations of the exponential graphs without the use of a calculator. They should also be able to give the domain, range and asymptote from the transformations. Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc. Define

horizontal asymptote

and relate it to the exponential graphs’

range

value. The warm up will lead you to the first example of horizontal asymptote. Show how the

y

-intercept is impacted by

a

,

b

and

k

(given ) Resources: What materials or resources are essential

for students to successfully complete the lesson tasks or activities?

Homework: Exactly what follow-up homework tasks,

problems, and/or exercises will be assigned upon the completion of the lesson?

Graphing Calculator Computer LCD Exponential Graphs Characteristics worksheet To be determined by teacher. Lesson Reflections: How do you know that you were effective? What questions, connected to the lesson

standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?

Did students make the connection between the horizontal asymptote and the horizontal shift (

k

)? Did students recognize that the

y

-intercept is affected by all transformations? Can the students make the connection between the transformations they see in the equation and what they see on the graph? Will the students be able to write equations from a graph or data?

Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a

Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License

.

HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand

,

S. (2009).

Accessible mathematics

:

10 instructional shifts that raise student achievement.

Portsmouth

,

NH

:

Heinemann

.

Definition Examples Characteristics Non-exampies

Exploring Exponential Graphs-

Form A

Type the parent function

y

 2

x

into Y

1

, the type each transformed equation into Y

2

and compare.

 

Complete table of values for each and sketch the graph. Sketch the parent function on each graph.

y

 2

x x

-2 -1 0 1 2 3

y y

  2(2)

x x

-2 -1 0 1 2 3

y y

 2

x

 1

x

-2 -1 0 1 2 3

y

Range Horizontal Asymptote y-intercept Rate of Change

y

2

x y

3(2)

x y

 

2(2)

x y

 3(2)

x x

-2 -1 0 1 2 3

y y

 2

x

 3

x

-2 -1 0 1 2 3

y x

-2 -1 0 1 2 3

y

 2

x

 4

y y

2

x

3

y

2

x

1

y

2

x

 4 Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License .

Compare each function to

y

2

x

Similarities

y

3(2)

x y

 

2(2)

x y

2

x

3

y

2

x

1

y

2

x

 4

Differences

Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License .

Exploring Exponential Graphs-

 

Type the parent function

Form B

 3 

y

  4 

into Y , the type each transformed equation into Y

1 2

and compare. Complete a table of values for each and sketch the graph. Sketch the parent function on each graph.

  3 

x y

  4 

x

-2 -1 0 1 2 3

y

  3 

x y

  2  4 

x y

-2 -1 0 1 2 3       3 

x y

  4 

x

-2 -1 0 1 2 3  1

y

x

-2 -1 0 1 2 3

y

 

3

 

4



x

Range Horizontal Asymptote y-intercept Rate of Change

y

y

3

 

3

 

4



x

   3 

x y

 3  4 

x

-2 -1 0 1 2 3

y

3

x y

 

4

 

3

x y

-2 -1 0 1 2 3

y

   3 4  

x

 4

y

 

2

3

 

4



x y

 

3

  

4



x

3

y

  

3

  

4



x

1

y

 

3

 

4



x

 4 Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License .

  

Compare each function to

y

 

3

 

4



x

Similarities

y

3

3

 

4



x

y

 

2

3

 

4



x

y

 

3

  

4



x

3

y

  

3

  

4



x

1

y

 

3

 

4



x

 4   

Differences

Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License .

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