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Chapter 2: Exponential and Inverse Functions

Entry Task
Activity and Journal Reflection
Homework and Goals*
An employer tasks you to complete a job and agrees to pay you $0.01 on day one, $0.02 on day two, and $0.04 on day three, doubling the total amount on a given day for an entire month.  Complete the table using the facts.






Exponential relationships:
  • From the ET data, calculate the best fit regression in your calculator. Which model works best?
  • Use the TRACE function to ensure the model passes through all of the points.

Pressure vs. Altitude model relationship.

JR: Without a calculator, reduce each exponential function, if possible write as a rational number:
  • 23
  • (23)2
  • 21/2
  • 2-1
  • 2324
  1. Graph (by hand, NO computer printout) Richter Magnitude vs. Energy for earthquakes.
  2. Then choose what you believe is the best model for the data.
  3. Compute the regression formula for the model you chose.
  4. Explain why you believe your model is best.
  5. Use the equation to:
    1. Compute the energy from a 8.0 magnitude earthquake.
    2. Compute the Richter Magnitude for the Mt. Saint Helen's Eruption on May 18, 1980.
Test Corrections: Wednesday (Tomorrow) 3/11 (AM/PM)

Today's Goal
: Students will graph and interpret functions that are best characterized by exponential relationships.
Select your preferred mathematics class for next year HERE.

What x-value will give a y-value of 3?
  • y = 5x + 2
  • y = 5x2
  • y = 5*2x
Differentiate between a power function and exponential function.

Graph (by hand!) Richter Magnitude vs. LOG(Energy).  Use the graph to find the following (on the graph):
  • The y-intercept.
  • The slope of Richter Magnitude vs. LOG(Energy).
Calculate: 10^( y-intercept) and 10^(slope).  Use these answers in the following equation:
y = (first answer)*(second answer)^x
What does this equation represent?  Why?

JR: What to power function and exponential functions have in common? How to power functions and exponential functions operate differently?
Use the Laws of Exponents to explain the following properties of the exponential function f(x) = bx using complete sentences.
  • f(m) * f(n) = f(m + n)
  • f(m)/f(n) = f(m - n)
  • [f(m)]n = f(m*n)

Test Corrections: TODAY (AM/PM)

Today's Goal
: Understand the existence of a relationship between exponential functions and linear functions. (This relationship is the inverse)
Independently complete the ET on a spare sheet of paper, show all your work, turn this into Dr. Edgerton:

Assume, 2 = 100.30103
Without using a calculator, find the value of x that will make the equation true.
  • 10x = 2
  • 10x = 10
  • 10x = 5
  • 10x = 20
  • 10x = π

Formal introduction of Logarithms:

JR: What are at least two things you would like to better understand about logarithms? Please be specific.
Watch Khan Academy Video about Logarithms

Pg 113 # 1, 2, 3, 5

Today's Goal: Students will expand and formalize understanding of a logarithm of base 10.
(6 Periods)
What is an inverse function? What purpose does it serve? Provide an example.
Not Tree-Logs Activity - Understanding Logarithms

JR: Explain what a logarithm is and why they are useful.
Complete the lesson and HOMEWORK page for today's activity. You will turn in today's homework for credit.

Today's Goal: Students gain familiarity with the logarithmic scale and the relationship between logarithms and exponential functions.

Associated Standard: CCSS F.BF.5 - (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
(6 Periods)
From last nights activity, Emily made a claim about adding logarithms to multiply the numbers on the power scale. Using the fact that 10log(x) = x, quantify the claims.

  1. log(ab) = log(a) + log(b)
  2. log(a/b) = log(a) - log(b)
  3. log(ab) = alog(b)
No More Wood Logs Activity- Change of Base

Expand on the idea, show a proof of a different base and build understanding of the relationship between a base other than 10 and how to symbol manipulate.

JR: Perform the following:
  1. log(100)
  2. log(1000)
  • Add the answers of the above.
  • Compute 10^(the answer)
  • What significance does the answer you just computed have?
  • Write a paragraph on the relationship (as demonstrated above) about how to multiply large numbers using logarithms.  Demonstrate using numbers that are not multiples of ten.
Problems 1-4 in homework packet

(Extra Credit: On the log2 created in class, what does the point halfway between 0 and 1 represent. Explain.)

Today's Goal: Students will be able to explain what "change of base" means and how they are calculated for logarithms other than base 10.

Associated Standard: CCSS F.LE.4 - For exponential models, express as a logarithm the solution to abcx = d where a, c, and d are real numbers and the base b 2, 10 or e; evaluate the logarithm using technology
Use your computer to do brief search of how human kidney's operate. As best you can, explain the process.
It's ALL in There Activity - Modeling Exponential Functions

Drug Filtering (NCTM) activity (modified to include several drug doses).
EQ: How long does it take a drug to completely exit the body? (Hint: It doesn't leave the body... :O)

JR: Describe a unique situation where you would use logarithms to solve for a real world problem. (Your situation should be different from all other students).
All problems in homework packet assigned 3/17

Today's Goal: Students will model an application-based scenario and use logarithms with non-standard bases to solve for unknowns.

Associated Standard: CCSS F.BF.5 - (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents
Get ready for the quiz! 
  • Move to a seat where you have ample room,
  • Obtain all the materials you need before class starts,
  • Seat at most two at the square "cafe tables"
  • Place the paper "blinders" between each pair of people. Put yourself in a positive mental state.
Quiz 2.1 (Exponents and Logarithmic Scale) You may use a 3 x 5 note card* along with any measurement devices you brought, but NO CALCULATORS  Materials will NOT be loaned during class.

Topics on Quiz: Use logarithms to find the inverse of basic exponential functions; explain change of base in a new context; build exponential functions from a word situation.

JR: What advice would you give to yourself at the beginning of this logarithms unit?
Pg 130 #15, 16a and Pg. 132 # 2.

Today's Goal: Assessing student's knowledge of logarithms and exponent relationships.
Change your calculator's graphing window to the following settings
0 ≤ x ≤ 10
-1 ≤ y ≤ 5
  • y = log2(x),
  • y = log3(x),
  • y = log5(x),
  • y = log10(x).
Explain the similarities and differences in the graphs of the functions.
Review concepts some students missed:
  • Piecewise Functions (particularly that from the Drug Filtering Activity)
  • Calculate 4-1 and 5-1
  • Practice with logarithm laws through some practice problems
Logarithmic Functions; Lesson 2.4.  Activity 2.6 Omit 3a. (Pgs. 132-136).

JR: Graph these pairs of functions separately on your calculator. What relationship do their graphs have?
  • y = ex  and y = ln(x)
  • y = 2x + 1 and y = 1/2x - 1/2
  • y = x3 and y = x1/3
Pg 137 # 3, 5, 11

Today's Goal: See the inverse relationship graphically between exponential functions and logarithmic functions.
Explain the equivalence of these two statements: logba = x and bx=a Explore the nature of Euler's number.

Convert between different forms of exponential functions (y = abx ⇒ y = aemx ⇒ y = a10nx).

Use common ratios to model data with exponential functions.

Pg. 128 #6, 7, 11.

Pg. 148 Table 2.33
model with an exponential equation.
convert to  y = abx.
Plot some of the data and the second exponential equation to check its fit.
JR: Explain how you would create an exponential model if you are only given two data points.
Given the two points (0, 4) and (3, 5.5):
  1. Use this data to create an exponential model in the general form (y = abx).
  2. Convert this equation as an exponential function with a base of e.
  3. What is the domain and range for these functions? Does every output have a unique input?
  4. Repeat steps 1, 2 and 3 above for the points (12, 3) and (14, 2).
Today's Goal: Investment modeling with exponential functions. Compounded continuously vs compounded regularly. Explore Euler's number as a natural phenomena of doubling with infinitely many compoundings. Explore how e is used through logarithms and mathematics.
Plot the following values in your calculator and describe a function in the form y = aekx where a, k and x are constants.
Input (x)
Output (y)

Activity 2.7 (Pg. 144 - 148, Omit 1 and 2)

JR: Explain the purpose of creating "semi-log" and "log-log" graphs of data.  What are the results of each?
Pg 149 # 2, 4, 8, (Optional #12)

Note: Re-expression is a way of rewriting an equation by manipulating a function with logarithms. Recall the semi-log "re-expression" of the Richter scale on 3/11.

Today's Goal: Students will learn techniques of exponential and logarithmic modeling. This includes using data to generate an exponential function and inverting to become logarithmic function.

Work at your four-tops and begin Activity 2.8 on page 155.
Composition and Inverse Functions; Lesson 2.6.  Activity 2.8 (Pgs. 155-166) #1-2.
  • Omit 2d & 2e.
  • Check-in with Mr. G after completing 2g.
  • Begin homework notes.

JR: Let f(x) = 2x + 1 and g(x) = 3x - 7. Write the function f(g(x)) in simplest terms.
Complete Today's activity AND
One full page of notes on Pg. 157-163

Today's Goal: Composition and Inverse Functions
Determine whether each of the following sets of ordered pairs defines a function. If not, explain. If yes, is it one-to-one?
  • {(1, 3), (2, 4), (3, 5), (4, 3)}
  • {(3, 5), (4, 6), (6, 8), (8, 10)}
  • {(1, 1), (1, 3), (3, 5), (4, 6)}
  • {(35, 48), (48, 35), (20, 20), (15, 15)}

Composition Function Activity

Pg 165 #4, 5, 7, 9, 10

JR: Explain the result should you expect by taking the composition of two functions that are inverses of one another? Provide some examples.
Test Review Questions:
Pg.171 - 177 # 3, 4 (sketch graphs), 6, 7, 8, 10, (optional 15).

Today's Goal: Continue Composition functions, emphasis on domain and range
Model a function that passes through the points (0, 32) and (100, 212) using a:
  • linear model
  • exponential model of the form y = abx and y = aekx
In-class Review

JR:Which skills do you need to review most for the test tomorrow?
Finish review problems from yesterday's HW.

Review session with Mr. G. immediately after class from 3:30pm - 4:30pm.

Today's Goal: Review for exam.
Get ready for the Test! 
  • Move to a seat where you have ample room,
  • Obtain all the materials you need before class starts. No materials will be loaned during class!
  • Seat at most two at the square "cafe tables"
  • Place the paper "blinders" between each pair of people. Put yourself in a positive mental state.
Test 2.1 (No Foolin')  You may use YOUR calculator and a 3 x 5 note card* (writing on both sides is permitted).  Measurement devices (e.g. ruler and protractor) are also allowed.  No materials will be loaned during class!  Be prepared to:
  • Write an exponential function that models a situation given two data points.
  • Answer question using your function.
  • Solve for unknown parts of logarithm functions and exponential functions using log and exponent laws.
  • Convert between the forms y = abx and y = aekx.
  • Use compositions to show that functions are inverses of one another (power, exponential, logarithmic and trigonometric functions are eligible).
JR: Explain in simple terms why a calculator is unnecessary to simplify log3(81).  Give several other examples (using other bases) that use the same idea.
Write the first 11 terms in the list as fractions {1/0!, 1/1!, 1/2!, 1/3!, ..., 1/10!}.

Then, add all of these numbers together in your calculator and express your result as a decimal (do not round.)

If you continued making this sequence and adding numbers together, what value do you approach?
Use your calculator to model the data on p. 152 #12 (year vs. population) exponentially.  Write the equivalent logarithmic expression and use it to solve for the year the population will reach 300,000,000.  Is your answer realistic?
Post-Spring Break Plans

Review previous night's homework:

Exponential functions as recursion:
  • Create a list of values for y = 20 (1.5)x where x is integers from 0 to 10.
  • How can you develop the 11th number by observing the pattern from only the 10th number?
  • Create an equation that represents this formula.
JR: In what ways will you stay current in mathematics over spring break?
Upon return, all Sophomores will miss class 1-4 class April 13, 14, 20 and 27 for the environmental challenge field-trips. During this time, instruction will continue and ALL pre-arranged absence procedures will be followed according to the class syllabus. Please review the class syllabus if you are unclear on these procedures. Week 1 of Quarter 4 is posted for your convenience.

Otherwise, no additional homework assigned :-)

Today's Goal: Sequences and the relationship with exponential functions.
*Unless otherwise noted, homework is due the next class day.

Standards to Cover


Distinguish between situations that can be modeled with linear functions and with exponential functions.

Common Core State Standards for Mathematics

Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Common Core State Standards for Mathematics


Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Common Core State Standards for Mathematics


Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Common Core State Standards for Mathematics


For exponential models, express as a logarithm the solution to abct =d where a ,c ,and d are numbers and the base b is 2,10,or e; evaluate the logarithm using technology.

Common Core State Standards for Mathematics


Interpret the parameters in a linear or exponential function in terms of a context.

Common Core State Standards for Mathematics

(P) F.BF.1.c

(+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

Common Core State Standards for Mathematics


(+) Verify by composition that one function is the inverse of another.

Common Core State Standards for Mathematics


(+) Read values of an inverse function from a graph or a table, given that the function has an inverse.

Common Core State Standards for Mathematics


(+) Produce an invertible function from a non-invertible function by restricting the domain.

Common Core State Standards for Mathematics

(P) F.BF.5

(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Common Core State Standards for Mathematics


Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Graph exponential and logarithmic functions, showing intercepts and end behavior.

Common Core State Standards for Ma

Common Core State Standards for Mathematics


Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

Common Core State Standards for Mathematics


Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Common Core State Standards for Mathematics

BIG Ideas, Things that I think are valuable

Students will model with exponential functions and use data to find the function y = abcx + d and the inverse within an application context.
Students will calculate and use inverse and composition functions for exponential, logarithmic, trigonometric and simple polynomial functions.
Students will describe the domain and range of a function for logarithmic, exponential, linear, polynomial, rational and trigonometric functions.

Things That Grow Exponentially