9th Grade Unit 3
9th Grade Math Class; Lesson 3
Key Standards addressed in this Lesson: MCC9-12 F.IF4, , MCC9-12 F.IF7a, MCC9-12 F.
IF7e; MCC9-12 F.BF3; MC9-12 F.IF9; LE5
Time allotted for this Lesson: 4 to 5 days
Materials Needed:
Colored pencils
Graph paper
Key Concepts in Standards: Refer to TE
MCC9‐12.F.IF.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. ★ (Focus
on linear and exponential functions.)
Essential Question(s): Refer to TE
How do I analyze and graph exponential functions?
Vocabulary: (Tier) Refer to TE
Tier 1: already knows
Tier 2: needs review
Tier 3: New Vocabulary
Tier 1
Tier 2
Tier 3
Asymptote
Coefficient
Average Rate of Change
Continuous
Slope
Constant Rate of Change
End Behavior
x-intercept
Domain
Interval Notation
y-intercept
Exponential Function
Vertical Transformation
Exponential Model
Horizontal Transformation
Linear Function
Parameter
Linear Model
Concepts/Skills to Maintain: Refer to TE
In order for students to be successful, the following skills and concepts need to be maintained:
 Know how to solve equations, using the distributive property, combining like terms and equations with
variables on both sides.
 Understand and be able to explain what a function is.
 Determine if a table, graph or set of ordered pairs is a function.
 Distinguish between linear and non-linear functions.
 Write linear equations and use them to model real-world situations.
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Opening:
Show video “Model exponential growth situations with 2 variables” from website
http://learnzillion.com/lessons/291-model-exponential-growth-situations-with-2variables
Use Exponential Function Concept Map Graphic Organizer for f ( x )  b
Distribute the exponential function concept map and have students complete the table and draw the
graph. Encourage them to answer as many questions as they can. Monitor to see that students are
able to complete the table and graph and check on their ability to answer the questions. After about
10 minutes, use the questions to begin a discussion of exponential functions.
x

Have students volunteer to share their answers on how linear and exponential functions
compare as far as x-intercepts, y-intercepts, slopes
Work Session:
Activity 1: Families of Exponential Functions
 Distribute the concept map on “Families of Exponential Functions f ( x)  b x  k ” and have
pairs use different colors to graph the four functions on one graph grid. Students should
complete the concept map by answering the questions and writing the three equations of the
functions indicated. Circulate among students to check their work.
o Explain that different letters are used to represent coefficients and constants in
an exponential function.

Have students volunteer to share how the graphs are similar and how they are different.
Activity 2: Exponential Functions in the form f ( x)  a b
Discuss increasing and decreasing and give handout with steps in calculating rate of
change on an interval of an exponential function
x
Activity 3: Graphic Organizer on Horizontal Transformations
x
Activity 4: Definitions of Properties of Exponential Functions f ( x)  a b (Guided)
 Note definitions:
Increasing (Positive slope) – line goes up as you move to the right
Decreasing (Negative slope) – line goes down as you move to the right
Positive - Where f(x) is positive depending on x values
Negative - Where f(x) is negative depending on x values
Parameter (use EOCT study guide definition, pg. 120)- the coefficient of the
variable and constant term in the function that affects the behavior of the function
 Guided practice with 3 graphs
 Independent practice with 2 graphs
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Activity 5: Graphic Organizer: Different representations of exponential graphs
Graphic Organizer:
Functions can be a table, equation, graph or verbal description.
Compare the following functions that are represented differently. What do they have in common?
What is different? Discuss the intercepts, slopes, shifts, rates of change, domain, range, etc.
Other activities included :
Exponential Growth/Decay Notes and Key (use where needed)
Worksheet A: Graphing Calculator Activity to Explore Exponentials
Closing:
Give three exponential equations. Students choose one equation, sketch a graph, and describe.
f ( x)  2 x  5
f ( x)  3 2 x
f ( x)  3(2) x
Closing at end of lesson: Ticket Out the Door—complete chart of exponential characteristics
(attached)
Corresponding Task(s) (if not in work session – there may be several tasks that fit) –
****All Tasks can be found at www.georgiastandards.org****
Highlight the Mathematical Practices that this lesson incorporates:
Make sense
of
problems
and
persevere
in solving
them
Reason
abstractly
and
quantitatively
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Construct
viable
arguments
and critique
the
reasoning
of others
Model with
mathematics
Use
appropriate
tools
strategically
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Attend to
precision
Look for
and make
sure of
structure
Look for
and
express
regularity
in repeated
reasoning
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Exponential
Functions
f (x)  b
x
Let's look at
f ( x)  2 x
(Parent Function)
Complete the table of
values.
b>0, b  1
x
f (x)
-4
Plot the points and sketch the graph below.
-3
-2
-1
0
1
2
3
4
1) Why do you think this is called an exponential function?
2) How does this compare to a linear function?
3) Can x ever have a value of 0?
4) Can f(x) ever have a value of 0?
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Vertical Transformations
Families of
Exponential Functions
f ( x)  b x  k
b>0, b  1
Complete the table for the following and
draw each in a different color on the
graph to the right.
A.
f ( x)  2 x  3
B.
f ( x)  2 x  4
x
-4
-2
x
-4
-2
f(x)
0
2
0
2
4
4
f(x)
1) How are the graphs above alike?
Asymptote
Asymptote
y-int=
y-int=
C.
f ( x)  2 x  7
D.
f ( x)  2 x  6
x
-4
-2
x
-4
-2
0
f(x)
0
2
4
f(x)
3) Write the equation of a function in this
family with a y–intercept of –2.
_________________
4) Write the equation of a function in this
family with a y–intercept of +5.
_________________
2
4
Asymptote
Asymptote
y-int=
y-int=
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2) How are they different?
5) Write the equation of a function in this
family with a y–intercept of –10.
_________________
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Vertical Stretching or Shrinking, Reflection across y-axis.
x
x
Exponential Functions in the Form f ( x)  ab or f ( x)  kb
with b>0, b  1
Graph each of the following
functions in different colors on the
graph at the right. (Sketch parent
graph in pencil. See pg 4.)
A. f ( x)  (2) x
B. f ( x) 
1 x
(2)
4
C. f ( x)  4(2) x
1) How are the graphs alike?
2) How are the graphs different?
3) What does the coefficient do to the exponential function f ( x)  2 x ?
4) How would the graph of f ( x)  5(2) x compare to the graph of f ( x)  2 x ?
5) How would the graph of f ( x)  3(2) x compare to the graph of f ( x)  2 x ?
6) How would the graph of f ( x)  .2(2) x compare to the graph of f ( x)  2 x ?
Definition: A function is said to be increasing on the interval (a, b) if,
for any two numbers in the interval, the greater number has the greater
function value. As you trace the graph from a to b (from left to right)
the graph should go up.
Definition: A function is said to be decreasing on the interval (a, b) if,
for any two numbers in the interval, the greater number has the smaller
function value. As you trace the graph from a to b (from left to right) the
graph should go down.
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Definitions for Properties of an Exponential Function:
f ( x)  ab
1) Domain:
2) Range:
x
with b>0, b  1
3) Maximum:
6. End Behavior
5) Asymptote
4) Minimum:
6) Increasing:
8) What is the x-intercept?
11) Where is the function
negative?
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7) Decreasing:
9) What is the y-intercept?
10) Where is the function
positive?
12) Parameters:
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Horizontal Transformation in Exponential Functions:
f(x) = bx + k where k represents a horizontal movement left or right. When moving
horizontally, you always move opposite of k.
Graph the following (create a table for points) – use different color pencils for each!
f(x) = 2x
f(x) = 2x + 2
f(x) 2x-2
Did you notice that when k is +2 that you moved left and when k is -2, you moved to the right?
REMEMBER to always take the opposite of k and move in that direction (negative k = move to the
right, positive k = move to the left)
So when k is attached to the x in the exponent, you are moving the graph left and right that many
units.
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When k is in the exponent but being multiplied by x, you are making a horizontal shrink
or stretch!
f(x) = bkx
-
when k is greater than 1 it is a horizontal stretch and when k is less than one
(greater than 0) it is a horizontal shrink.
Graph the following (create table for points) – use different colors for each
exponential function.
f(x) = 2x
f(x) = 23x
f(x) =
2
1
x
3
*if k is negative here f(x) = b-kx the graph will be reflected over the y
axis!
Graph f(x) = 2-3x on the above graph!
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Properties of Exponential Functions Practice (Guided)
Look at the graphs below and identify each of the following:
1)
a. Domain: ___________________________
b. Range: ____________________________
c. x-intercept: ________________________
d. y-intercept: ________________________
e. Increasing: _________________________
f. Decreasing: ________________________
g. Positive: ___________________________
h. Negative: __________________________
i. Minimum or Maximum: ______________
j. Rate of change: _____________________
k. Asymptote:_________________________
l. End Behavior:_______________________
m.Vertical Transformation:_______________
n. Horizontal Transformation:_____________
o. Parameters:_________________________
2)
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
Domain: ___________________________
Range: ____________________________
x-intercept: ________________________
y-intercept: ________________________
Increasing: _________________________
Decreasing: ________________________
Positive: ___________________________
Negative: __________________________
Minimum or Maximum: ______________
Rate of change: _____________________
Asymptote:_________________________
l. End Behavior:_______________________
m.Vertical Transformation:_____________
n. Horizontal Transformation:_____________
o. Parameters:_________________________
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1
2
3. Graph f ( x)   (2) x and answer a-o as in problems 1 and 2 above.
y
x
a. Domain: ___________________________
b. Range: ____________________________
c. x-intercept: ________________________
d. y-intercept: ________________________
e. Increasing: _________________________
f. Decreasing: ________________________
g. Positive: ___________________________
h. Negative: __________________________
i. Minimum or Maximum: ______________
j. Rate of change: _____________________
k. Asymptote:_________________________
l. End Behavior:_______________________
m. Vertical Transformation:______________
n. Horizontal Transformation:_____________
o. Parameters:__________________________
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Independent Practice
Describe the characteristics of each function:
1.
a. Domain: ___________________________
b. Range: ____________________________
c. x-intercept: ________________________
d. y-intercept: ________________________
e. Increasing: _________________________
f. Decreasing: ________________________
g. Positive: ___________________________
h. Negative: __________________________
i. Minimum or Maximum: ______________
j. Rate of change: _____________________
k. Asymptote:_________________________
l. End Behavior:_______________________
m. Vertical Transformation:_______________
n. Horizontal Transformation:______________
o. Parameters:___________________________
2.
a. Domain: ___________________________
b. Range: ____________________________
c. x-intercept: ________________________
d. y-intercept: ________________________
e. Increasing: _________________________
f. Decreasing: ________________________
g. Positive: ___________________________
h. Negative: __________________________
i. Minimum or Maximum: ______________
j. Rate of change: _____________________
k. Asymptote:_________________________
l. End Behavior:_______________________
m. Vertical Transformation:_______________
n. Horizontal Transformation:_____________
o. Parameters:__________________________
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Activity 5: Graphic Organizer: Different representations of exponential graphs
Graphic Organizer:
Functions can be a table, equation, graph or verbal description.
Compare the following functions that are represented differently. What do they have in common?
What is different? Discuss the intercepts, slopes, shifts, rates of change, domain, range, etc.
1. y = 3 * 2x
2. y = 3x + 1
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and
and
X
Y
0
1
1
3
2
9
3
27
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Ticket Out the Door: Complete missing parts of the chart.
Transformation
_______________
Equation
f(x) = bx + k
_______________
Vertical Stretching
or Shrinking
Reflecting
____________
f(x) = - bx
f(x) = b- x
_________________
________________
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f(x) = bx +k
Description
- Shifts the graph f(x) = bx to the
left k units if k>0
- Shifts the graphs f(x) = bx to the
right k units if c<0
- Stretches the graph of f(x) = bx if
k>1
- Shrinks the graph of f(x) = bx if
0<k<1
-________________________
-________________________
- Shifts the graph of f(x) = bx
upward k units if k>0
- Shifts the graph f(x) = bx
downward k units if k<0
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Exponential Growth/Decay Notes
Exponential Equations:
Exponential Growth:
Examples:
Graph:
*the graphs have asymptotes:
Exponential Decay:
Examples:
Graph:
Finding Multipliers:
Percentage Increase
Percentage Decrease
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