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.
1.
2.
3.
4.
5.
, complete the
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. 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.
2. 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.
3. 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.)
4. Distribute Exponential Graph Characteristics worksheet. Assign students in pairs to have them work through
the activity (Growth = Form A, Decay = Form B).
5. 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:
Teacher(s):
Start/end times:
and MP6.)
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
To be determined by teacher.
Computer
LCD
Exponential Graphs Characteristics worksheet
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
x
 Type the parent function y  2 into Y1, the type each transformed equation into Y2 and compare.
 Complete table of values for each and sketch the graph. Sketch the parent function on each graph.
y  2x
x
-2
-1
0
1
2
3
y  3(2)x
y
x
-2
-1
0
1
2
3
y
y  2(2)x
y  2x  3
x
-2
-1
0
1
2
3
x
-2
-1
0
1
2
3
y
y  2x4
y  2x  1
x
-2
-1
0
1
2
3
y
y
x
-2
-1
0
1
2
3
y  2x
y  3(2) x
y  2(2) x
y
y  2x  3
y  2x 1
y  2x4
Range
Horizontal
Asymptote
y-intercept
Rate of
Change
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  2x
y  3(2) x
y  2(2) x
y  2x  3
y  2x 1
y  2x4
Similarities
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-Formx B
 3  into Y , the type each transformed equation into Y and compare.
 Type the parent function

y 
4 
1
2
Complete a table of values for each and sketch the graph. Sketch the parent function on each graph.
x


x
 3
y   
 4
x
-2
-1
0
1
2
3

y

x
 3
y  2  
4 
x
y
-2
-1
0
1
2
3
 3
y  3  
 4
x
-2
-1
0
1
2
3
x
-2
-1
0
1
2
3
 3
y     1
4 
x
-2
-1
0
1
2
3
x
 3
y  3
4 
x

y
 3
y   
4 
y

x
-2
-1
0
1
2
3
y
x4
y





3
y   
 4 
x
3
y  3 
 4 
x
3
y  2 
 4 
x
x
3
y    3
 4 
x
3
y    1
 4 
3
y   
 4 
Range
Horizontal
Asymptote
y-intercept
Rate of
Change
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.
x4
Compare each 
function to
x
3
y   
 4 
3
y  3 
 4 
x

3
y  2 
 4 
x

x
3
y    3
 4 

x
3
y    1
 4 

3
y   
 4 
x4



Similarities



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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