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; LE5Time allotted for this Lesson: 4 to 5 days

Materials Needed:

Colored pencilsGraph 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 TETier 1: already knows Tier 2: needs review Tier 3: New Vocabulary

Tier 1CoefficientSlopex-intercepty-intercept

Tier 2Average Rate of ChangeConstant Rate of ChangeDomainExponential FunctionExponential ModelLinear FunctionLinear Model

Tier 3Asymptote ContinuousEnd BehaviorInterval NotationVertical TransformationHorizontal TransformationParameter

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.

Opening:

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9th Grade Unit 3

Show video “Model exponential growth situations with 2 variables” from website http://learnzillion.com/lessons/291-model-exponential-growth-situations-with-2-variables

Use Exponential Function Concept Map Graphic Organizer for 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.

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 ” 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 Discuss increasing and decreasing and give handout with steps in calculating rate of change on an interval of an exponential function

Activity 3: Graphic Organizer on Horizontal Transformations

Activity 4: Definitions of Properties of Exponential Functions (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 valuesParameter (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 graphsGraphic 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.

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

Construct viable

arguments and critique

the reasoning of others

Model with mathematics

Use appropriate

tools strategically

Attend to precision

Look for and make

sure of structure

Look for and

express regularity

in repeated reasoning

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9th Grade Unit 3

(Parent Function)

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1) Why do you think this is called an exponential function?

2) How does this compare to a linear function?

Plot the points and sketch the graph below.

Complete the table of values.

x f (x)-4

-3

-2

- 1

0

1

2

3

4

9th Grade Unit 3

Vertical Transformations

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Complete the table for the following and draw each in a different color on the graph to the right.

1) How are the graphs above alike?

2) How are they different?

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.

_________________

5) Write the equation of a function in this family with a y–intercept of –10.

_________________

A.

x f(x) -4-2024

Asymptote

y-int=

B.

x f(x) -4-2024

Asymptote

y-int=

C.

x f(x) -4-2024

Asymptote

y-int=

D.

x f(x) -4-2024

Asymptote

y-int=

9th Grade Unit 3

Vertical Stretching or Shrinking, Reflection across y-axis.Exponential Functions in the Formwith b>0, b 1

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.Unit 3 Lesson3 Student Edition 6

Graph each of the following functions in different colors on the graph at the right. (Sketch parent graph in pencil. See pg 4.)

1) How are the graphs alike?

2) How are the graphs different?

3) What does the coefficient do to the exponential function

4) How would the graph of compare to the graph of

5) How would the graph of compare to the graph of

6) How would the graph of compare to the graph of

9th Grade Unit 3

Definitions for Properties of an Exponential Function:

with b>0, b 1

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10) Where is the function positive?

11) Where is the function negative?

12) Parameters:

8) What is the x-intercept? 9) What is the y-intercept?

1) Domain:

5) Asymptote 3) Maximum:

4) Minimum:

6) Increasing: 7) Decreasing:

6. End Behavior

2) Range:

9th Grade Unit 3

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.When k is in the exponent but being multiplied by x, you are making a horizontal shrink or stretch!Unit 3 Lesson3 Student Edition 8

9th Grade Unit 3

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) =

*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)

2)

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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:_________________________

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:_________________________

y

x

9th Grade Unit 3

3. Graph and answer a-o as in problems 1 and 2 above.

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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:__________________________

9th Grade Unit 3

Independent Practice

Describe the characteristics of each function:

1.

2.

Activity 5: Graphic Organizer: Different representations of exponential graphs

Graphic Organizer:Functions can be a table, equation, graph or verbal description.

Unit 3 Lesson3 Student Edition 12

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:___________________________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:__________________________

9th Grade Unit 3

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 and

2. y = 3x + 1 and

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X Y

0 1

1 3

2 9

3 27

9th Grade Unit 3

Ticket Out the Door: Complete missing parts of the chart.

Transformation

Equation Description

_______________

_______________

f(x) = bx + k - 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

Vertical Stretching or Shrinking ____________

- Stretches the graph of f(x) = bx if k>1- Shrinks the graph of f(x) = bx if 0<k<1

Reflecting f(x) = - bx

f(x) = b- x

-________________________

-_________________________________________

________________

f(x) = bx

+k- 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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9th Grade Unit 3

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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Name:_____________________________________ Date:____________

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Name:________________________________ Date:_____________

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