Let’s all be ready . . . We’ll have an ACT problem and then Discuss the linear programming assignment

ACT of The Day 1 of 2 (2 min) When x = 3 and y = 5, by how much does the value of 3x2 – 2y exceed the value of 2x2 – 3y ? 4 14 16 20 50

ACT of The Day 2 of 2 (2 min) Sales for a business were 3 million dollars more the second year than the first, and sales for the third year were double the sales for the second year. If sales for the third year were 38 million dollars, what were sales, in millions of dollars, for the first year? 16 17.5 20.5 22 35

Flow Map - ACT Warm Up (10 min) - Discuss Linear Programming Assignment (20 min) - Assign calculator #s (3 min) - Introduce Exponential Functions (50 min) - Start HW (time permitting)

Exponential Functions The Big Brother of Linear Functions

Review Linear Functions Joe invests $1000 into a financial program that is guaranteed to grow in value by $200 every month for the next ten years. After ten years the investment is cashed out and the value is paid to Joe. Write a function that represents the value of his investment, J, after t months. How much will his investment be worth after the first year? How long will it take for the value of his investment to double?

Introduce Exponential Functions Mary invests $1000 into a financial program that is guaranteed to grow in value by 10% every month for the next ten years. After ten years the investment is cashed out and the value is paid to Mary. Write a function that represents the value of his investment, M, after t months. How much will her investment be worth after the first year? How long will it take for the value of her investment to double?

Compare Linear and Expo Functions Use function M and J to answer the following questions. Which investment is worth more at the end of the decade? Whose investment is worth more at the end of the first month? How many months does it take for their “rank” to switch?

Make Sure Linear Programming Assignment is AWESOME Find a suitable approach and answer for question C on the previous slide

End of Monday Gooz Fra Baaaaaa

We’ll start with the ACT of the day Let’s all be ready . . . We’ll start with the ACT of the day

ACT of The Day 1 of 3 (2 min) A car averages 27 miles per gallon. If gas costs $4.04 per gallon, which of the following is closest to how much the gas would cost for this car to travel 2,727 typical miles? $44.44 $109.08 $118.80 $408.04 $444.40

ACT of The Day 2 of 3 (1 min) The length, in inches, of a box is 4 inches less than twice its width, in inches. Which of the following gives the width, W inches, in terms of the length, L inches, of the box? 𝑊=2𝐿−4 𝑊= 1 2 𝐿−4 𝑊=2𝐿+4 𝑊= 1 2 𝐿−2 𝑊= 1 2 𝐿+2

ACT of The Day 3 of 3 (3 min) The graph below shows three out of four vertices of a rectangle. Which of the following points would be the the 4th vertex? (3, -7) (4, -8) (5, -1) (8, -3) (9, -3)

Flow Map - ACT of The Day (6 min) - Discuss Flow Map (2 min) - Review ACT of The Day (10 min) - Finish initial Expo Modeling Problem (5 min) - Graphs of Exponential Functions (20 min) - More exponential modeling problems (40 min) - Start HW (Time Permitting)

Questions and Comments? Answers to ACT Questions and Comments?

Compare Linear and Expo Functions Use function M and J to answer the following questions. Which investment is worth more at the end of the decade? Whose investment is worth more at the end of the first month? How many months does it take for their “rank” to switch? Just Part C

Graphs of Exponentials The Simple, Quick Version

Graphing Exponential Functions 𝑓 𝑥 =𝐴 𝑏 𝑥 +𝑐 If c = 0 then A – initial value, also the Y-Intercept, must not be zero b – exponential factor, must be positive and not equal to one (b < 1 means decay, b > 1 means growth)

Examples Sketch the graph of each function. Make sure to plot at least two points Ex1 - 𝑓 𝑥 =3 2 𝑥 Ex2 - 𝑓 𝑥 =−6 2 3 𝑥 Ex3 - 𝑓 𝑥 =5 0.4 𝑥 YT1 - 𝑓 𝑥 =5 5 2 𝑥 YT2 - 𝑓 𝑥 =− 1.2 𝑥

Graphing Exponential Functions 𝑓 𝑥 =𝐴 𝑏 𝑥 +𝑐 If c ≠ 0 then A+c – initial value, also the Y-Intercept b – exponential factor, must be positive and not equal to one (b < 1 means decay, b > 1 means growth) c – is the limiting value

Examples Sketch the graph of each function. Make sure to plot at least two points. Ex1 - 𝑓 𝑥 =6 1 3 𝑥 −4 Ex2 - 𝑓 𝑥 =−1.2 2 𝑥 +9 Ex3 - 𝑓 𝑥 =6−3 0.2 𝑥 YT1 - 𝑓 𝑥 =2 1.1 𝑥 +1

Side Note: How awesome was that ACT problem today? Back To Modeling Aka “Word Problems” Aka The Good Stuff Side Note: How awesome was that ACT problem today?

Exponential Modeling - 1 A small herd of sheep is initially populated with 100 sheep. If the Sheppard tends to his flock with the utmost care we can expect a growth of 12% every breeding season. Use this to answer the following questions: Write the function P(s) that represents the size of the population at the end of every breeding season. How many seasons will it take for the population to triple in size? By what percent will the population grow after 5 seasons?

Exponential Modeling - 2 Amoxicillin is a commonly used antibiotic that the average body metabolizes at a rate of 2.8% per hour. What is the body’s daily metabolic rate for amoxicillin?

Exponential Modeling - 3 Bob and his son hang 10,000 Christmas lights. Due to over linkage and excessive use of duct tape the lights burn out at rate of 1/3 every two days. It snows the day after the lights are hung and then it takes three days for the snow to melt. How many lights will Bob and his son have to replace? (Hint: The lights have been going for four days)

Assignment Review notes and examples for exponential graphing and exponential modeling There is a set of graphing practice problems on weebly: m3h_graphing-practice-2-21.pdf Answer key included

End of Tuesday Gooz Fra Baaaaaa

Notebooks out and open That ACT goodness is coming at you Let’s all be ready . . . Notebooks out and open That ACT goodness is coming at you

ACT of The Day 1 of 2 (2 min) A neighborhood recreation program serves a total of 280 children who are either 11 years old or 12 years old. The sum of the children’s ages is 3,238 years. How many 11-year-old children does the recreation program serve? 55 122 132 158 208

ACT of The Day 2 of 2 (1 min) Which of the following is an equation of the circle with its center at (0,0) that passes through (3, 4) in the standard (x, y) coordinate plane? 𝑥−𝑦=−1 𝑥+𝑦=7 𝑦 2 − 𝑥 2 =7 𝑥 2 + 𝑦 2 =49 𝑥 2 + 𝑦 2 =25

Flow Map - ACT of The Day (3 min) - Discuss Flow Map (1 min) - Review ACT of The Day (5 min) - Review expo modeling (10 min) - Review expo graphing (3 min) - Quiz on Expo Graphing & Modeling (15 min, 6 items) - Lesson on solving exponential equations by hand (20 min) - Introduction to the logarithmic operator (15 min) - Start HW (Time permitting)

Questions and Comments? Answers to ACT Questions and Comments?

Exponential Modeling - 3 Bob and his son hang 10,000 Christmas lights. Due to over linkage and excessive use of duct tape the lights burn out at rate of 1/3 every two days. It snows the day after the lights are hung and then it takes three days for the snow to melt. How many lights will Bob and his son have to replace? (Hint: The lights have been going for four days)

Exponential Modeling - 4 Under the correct environmental conditions a yeast culture can increase in mass by 17% every week. How many days will it take for a culture to go from 2 kilograms to 8 kilograms?

Exponential Graphing Sketch the graph of each function. Make sure to “clearly” plot two points. Ex1: 𝑔 𝑥 =−3 4 3 𝑥 Ex2: 𝑓 𝑥 = 6 2 3 𝑥 −5

Quiz Time Open notes (paper only), Calculator Permitted, No Other Resources You have ~15 min

Quiz Time Open notes (paper only), Calculator Permitted, No Other Resources You have ~15 min Moved to tomorrow

A Quick Review of The Properties of Exponents Buckle Up

Properties & Facts About Exponents - The basic idea of exponents is repeated multiplication (as with positive integer exponents, e.g., 5 3 =5∙5∙5=125 ) - Given the concept of inverses negative exponents translate to “repeated division” (really repeated multiplication of the reciprocal value, e.g., 2 3 −2 = 3 2 ∙ 3 2 = 9 4 or 4 −2 = 1 4 ∙ 1 4 = 1 16 ) - As a corollary to the aforementioned definitions and the properties of exponents: “Any number (except zero) raised to the zero power is 1” - “Fractional” exponents translate into radicals (e.g. 8 2 3 = 3 8 2 = 3 8 2 =4 or 6 = 6 1 2 )

Solving Exponential Equations Both with and without the use of the logarithmic operator

Solving Exponential Equations – You Try Solve each equation. Show your steps. Ex1: 2 𝑥 −7=1 Ex2: 2− 1 3 x =−25 Ex3: 4 𝑥 =32 Ex4: 18=2+ 1 2 𝑥 I’ll take questions when I’m ready

Solving Exponential Equations – You Try Solve each equation. Show your steps. YT1: 10−3 9 𝑥 =1 YT2: 1 4 = 32 𝑥

Solving Exponential Equations – Examples Solve each equation. Write answers as exact values. Ex1: 13= 3 𝑥 −7 Ex2: 1000= 2345 1 3 𝑥 +200 Ex3: 2 1 5 𝑥−1 =8 YT: −17=9− 2 𝑥+2

The Logarithmic Operator Defined as the inverse operation to “exponentiating” i.e., If 𝐵 𝑥 =𝐴 then log 𝐵 𝐴 =𝑥 And vice versa

The Logarithmic Operator There are three variants of logs that you should be familiar with: Log Base 10: 10 𝑥 =𝑦⟺𝑥= log 𝑦 Log Base e: 𝑒 𝑥 =𝑦⟺𝑥= ln 𝑦 (aka The Natural Log) Log base b: 𝑏 𝑥 =𝑦⟺𝑥= log 𝑏 𝑦

Napier’s Constant - The number 𝑒≈2.718 is known as either Euler or Napier’s Constant (for Leonhard Euler & John Napier) - It was discovered by Jacob Bernoulli while trying to stack that cheddar - A formal, and simple, definition of e is: 𝑒= 𝑛=0 ∞ 1 𝑛! = 1 1 + 1 1 + 1 1∙2 + 1 1∙2∙3 + 1 1∙2∙3∙4 +… =1+1+ 1 2 + 1 6 + 1 24 +… Used for its “friendly” involvement with many concepts in calculus

You need to know a simple property of logs For Now You need to know a simple property of logs Which is . . . ln 𝑎 𝑥 =𝑥∙ ln 𝑎

Solving Exponential Equations – Examples Solve each equation. Write answers as exact values. Ex1: 13= 3 𝑥 −7 Ex2: 1000= 2345 1 3 𝑥 +200 Ex3: 2 1 5 𝑥−1 =8 YT: −17=9− 2 𝑥+2

Assignment On the weebly File: m3h_solving-expo-equations.pdf Make sure to show work for all problems There are some sample expo problems posted on the weebly

End of Wednesday Gooz Fra Baaaaaa

One Warm Up Problem Quiz will begin immediately after Let’s all be ready . . . One Warm Up Problem Quiz will begin immediately after

Expo Warm Up The function 𝑓 𝑥 =1.25 0.98 𝑥 represents the amount of radioactive material, in metric tons, that remains poisonous after x months of storage. How many months and days (e.g., 3 months and 19 days) does it take for half of the initial amount of material to be non-poisonous? (i.e., what is the material’s half-life?)

Quiz Time Open notes (paper only), Calculator Permitted, No Other Resources You have ~15 min

Now for those delightful ACT problems

ACT of The Day 1 of 3 (1 min) If 3.1358 10 𝑥 =0.0000031358 what is the value of x? A. -7 B. -6 C. -5 D. 6 E. 7

ACT of The Day 2 of 3 (1 min) A triangle is composed of angles represented as: 3x + 10, −2x + 40, and x + 40. What is the value of x? F. 45 G. 50 H. 55 I. 60 J. 65

ACT of The Day 3 of 3 (1 min) If the number 79.8756 is rounded to the nearest hundredth, what will be the sum of the tens and hundredths place of the resulting number? K. 12 L. 13 M. 14 N. 15 O. 16

Flow Map - Quiz (15 min) - ACT of The Day (3 min) - Discuss Flow Map (2 min) - Review ACT of The Day (5 min) - Check/review HW (20 min) - Lesson on “logs” (40 min) - Start HW (Time permitting)

Questions and Comments? Answers to ACT Questions and Comments?

The Logarithmic Operator Defined as the inverse operation to “exponentiating” i.e., If 𝐵 𝑥 =𝐴 then log 𝐵 𝐴 =𝑥 And vice versa

The Logarithmic Operator There are three variants of logs that you should be familiar with: Log Base 10: 10 𝑥 =𝑦⟺𝑥= log 𝑦 Log Base e: 𝑒 𝑥 =𝑦⟺𝑥= ln 𝑦 (aka The Natural Log) Log base b: 𝑏 𝑥 =𝑦⟺𝑥= log 𝑏 𝑦

Napier’s Constant - The number 𝑒≈2.718 is known as either Euler or Napier’s Constant (for Leonhard Euler & John Napier) - It was discovered by Jacob Bernoulli while trying to stack that cheddar - A formal, and simple, definition of e is: 𝑒= 𝑛=1 ∞ 1 𝑛! = 1 1 + 1 1∙2 + 1 1∙2∙3 +…=1+ 1 2 + 1 6 +… Used for its “friendly” involvement with many concepts in calculus

You need to know a simple property of logs For Now You need to know a simple property of logs Which is . . . ln 𝑎 𝑥 =𝑥∙ ln 𝑎

Apologies for the sloppy answer key Yesterday’s HW Apologies for the sloppy answer key

From Yesterday Solve the equation. −200=−70 1 9 𝑥−3 +10

Clarification on The Log Operator The idea of “the log of x” isn’t much different from saying “the square root of x”. The logarithmic operator is the inverse of an exponential function. (i.e., expos are to logs as quadratics are to square roots)

The Properties of Logs Sum/Product: log 𝐴𝐵 = log 𝐴 + log 𝐵 Diff/Quotient: log 𝐴 𝐵 = log 𝐴 − log 𝐵 Coef/Power: log 𝐴 𝐵 =𝐵 log 𝐴 Change of Base: log 𝑠 (𝑇) = log (𝑇) log (𝑠)

Expanding Log Expressions Expand each expression. Ex1: log 3 𝑥 2 𝑦 Ex2: log 100 𝑎 3 𝑏 2 Ex3: log 5 𝑝 𝑟 3 4

Expanding Log Expressions Expand each expression. YT1: log 2 𝑥 2 8𝑦 YT2: log 𝑎 3 𝑏 2 2

Condensing Log Expressions Condense each log expression. (i.e., re-write as a single log) Ex1: log 4 − log 𝑥 +2 log 𝑦 Ex2: 3log 𝑎 +2 log 𝑏−1 − log 𝑐+3 Ex3: log 3 3𝑥+1 +2 log 3 𝑥−2

Condensing Log Expressions Condense each log expression. (i.e., re-write as a single log) YT1: log 𝑡−1 −3log⁡𝑝 +2 log 𝑟 YT2: log 5 2𝑥 +3 log 5 3 𝑥 2 YT3: 4− log 3 10

Assignment Have “B for D” Tonight

End of Thursday Gooz Fra Baaaaaa

Let’s all be ready . . . Notebooks out and open Laptops open and have google ready That ACT goodness is coming at you

ACT of The Day 1 of 2 (5 min) Line L passes through the point (1, -2) and is perpendicular to line M. The standard form equation for line M is 2x – 6y = 12. Which of the following is not a point that lies on line L? A. (-3, 10) B. (2, -5) C. (6, -17) D. (9, -28) E. (12, -35)

ACT of The Day 2 of 2 (3 min) In the figure to the right, A, C, D, and E are collinear; B, C, and F are collinear; and the angles at B, D, and F are right angles, as marked. Which of the following statements is NOT justifiable from the given information? F. 𝐴𝐵 ∥ 𝐸𝐹 (parallel) G. 𝐵𝐶 ≅ 𝐶𝐹 (congruent) H. 𝐸𝐹 ⊥ 𝐵𝐹 (perpendicular) I. ∠BCA≅∠𝐷𝐶𝐹 (congruent) J. ∆𝐵𝐴𝐶~∆𝐷𝐶𝐹 (similar)

Flow Map - ACT of The Day (6 min) - Discuss Flow Map (1 min) - Review ACT of The Day (9 min) - Hand back and review quiz (10 min) - Progress Reports (4 min) - SchoolNet Practice Assignment (51 min) - Surprise (9 min)

Questions and Comments? Answers to ACT Questions and Comments?

Quiz Results Boxed/circled # is your grade #s 1 – 4 each worth 5 points #5 worth 10 points #6 worth +3 points Name and Attendance worth 20 points Grade = # of Points/50 Questions/comments?

Progress Reports Review for accuracy Speak to me today and send follow up Email with any discrepancies

Make Up Time Make sure to tie off all loose ends

SchoolNET – Expo Assignment Counts as a quiz grade (maybe double) Feel free to use your notes, neighbors, google, or the paid professional educator (i.e., Mr Jones) Must be completed by 11:59pm Monday (2/27) CODE: lufthansa

Done Early? Go to the weebly Under the ACT section Open “Math Set 03” Open “Practice Test 5”

Surprise… It’s Game Day