9th Grade Unit 3
9thGrade 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
Coefficient
Slope
x-intercept
y-intercept / Tier 2
Average Rate of Change
Constant Rate of Change
Domain
Exponential Function
Exponential Model
Linear Function
Linear Model / Tier 3
Asymptote
Continuous
End Behavior
Interval Notation
Vertical Transformation
Horizontal Transformation
Parameter
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:
Show video “Model exponential growth situations with 2 variables”from website
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.
- 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.
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:
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
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.
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 ****
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(Parent Function)
Complete the table of values.x / f (x)
-4
-3
-2
- 1
0
1
2
3
4
Unit 3 Lesson3Student Edition1
9th Grade Unit 3
Vertical Transformations
A.x / f(x)
-4
-2
0
2
4
Asymptote
y-int=
B.
x / f(x)
-4
-2
0
2
4
Asymptote
y-int=
C.
x / f(x)
-4
-2
0
2
4
Asymptote
y-int=
D.
x / f(x)
-4
-2
0
2
4
Asymptote
y-int=
Vertical Stretching or Shrinking, Reflection across y-axis.
Exponential Functions in the Form
with b>0,b1
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.
Definitions for Properties of anExponential Function:
with b>0, b1
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) = 2xf(x) = 2x + 2f(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!
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) = 2xf(x) = 23xf(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!
Properties of Exponential Functions Practice (Guided)
Look at the graphs below and identify each of the following:
1)
2)
3. Graph and answer a-o as in problems 1 and 2 above.
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.
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.
- y = 3 * 2x and
X / Y
0 / 1
1 / 3
2 / 9
3 / 27
- y = 3x + 1 and
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
Exponential Growth/Decay Notes
Exponential Equations:
Exponential Growth:
Examples:Graph:
*the graphs have asymptotes:
Exponential Decay:
Examples:Graph:
Finding Multipliers:
Percentage Increase
Percentage Decrease
Name:______Date:______
Name:______Date:______
Name:______Date:______
Unit 3 Lesson3Student Edition1