11 General Mathematics Module 4 Exponential Functions Department of Education ● Republic of the Philippines General Mathematics – Grade 11 Alternative Delivery Mode Quarter 1 – Module 4: Exponential Functions First Edition, 2019 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, poems, pictures, photos, brand names, trademarks, etc.) included in this book are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education – Division of Cagayan de Oro Schools Division Superintendent: Dr. Cherry Mae L. Limbaco, CESO V Development Team of the Module Author/s: Macapangcat U.Mama Jr. Reviewers: Ray O. Maghuyop, EPS-Math Dr. Shirley A. Merida, PSDS Illustrators: Management Team Chairperson: Cherry Mae L. Limbaco, PhD, CESO V Schools Division Superintendent Co-Chairpersons: Alicia E. Anghay, PhD, CESE Asst. Schools Division Superintendent Lorebina C. Carrasco, OIC-CID Chief Members Joel D. Potane, LRMS Manager Lanie O. Signo, Librarian II Gemma Pajayon, PDO II Printed in the Philippines by Department of Education – Bureau of Learning Resources (DepEd-BLR) Office Address: Fr. William F. Masterson Ave Upper Balulang Cagayan de Oro Telefax: (08822)855-0048 E-mail Address: cagayandeoro.city@deped.gov.ph 11 General Mathematics Module 4 Exponential Functions This instructional material was collaboratively developed and reviewed by educators from public schools. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at action@ cagayandeoro.city@deped.gov.ph We value your feedback and recommendations. Department of Education • Republic of the Philippines This page is intentionally blank Table of Contents Overview ................................................................................................... 1 Module Content ......................................................................................... 1 Objectives .................................................................................................. 1 General Instructions .................................................................................. 2 Pretest ....................................................................................................... 2 Lesson 1..................................................................................................... 4 Activity 1 .......................................................................................... 4 Activity 2 .......................................................................................... 9 Lesson 2.................................................................................................. 10 Activity 3 ........................................................................................ 13 Lesson 3...................................................................................................14 Activity 4 .........................................................................................20 Enrichment Activity .................................................................................. 21 Summary/Generalizations ....................................................................... 22 Posttest ....................................................................................................23 Glossary ................................................................................................... 26 References ...............................................................................................27 Overview Have you observed that news stories, gossips or the latest trends in social media spread rapidly in modern society? With broadcasts televisions and radios, millions of people hear about important events within hours! Isn’t it amazing? In many problems, key variables are related by linear models. There are many other important situations in which variables are related by nonlinear patterns. One example is given in the exploration, the spread of information. Other examples include spread of disease, change in population, temperature, bank savings, drugs in the bloodstream, and radioactivity. Module Content In this module, you are expected to learn on how to represent real-life situations using exponential functions, distinguish between exponential functions, exponential equations, exponential inequalities, graphs exponential functions and solves problems involving exponential functions in real-life situations. Can we do it? But before we start, let us have a short agreement on what you are going to do in order for you to learn on this module. What I Need to Know 1. 2. 3. 4. 5. 6. 7. 8. At the end of this module, you are expected to do the following: represent real-life situations using exponential functions M11GM-Ie-3 distinguish between exponential function, exponential equation, and exponential inequality M11GM-Ie-4 solve exponential equations and inequalities M11GM-Ie-f-1 represent an exponential function through its: (a) table of values, (b) graph, and (c) equation M11GM-If-2 find the domain and range of an exponential function M11GM-If-3 determine the intercepts, zeroes, and asymptotes of an exponential function M11GM-If-4 graph exponential functions M11GM-Ig-1 solve problems involving exponential functions, equations, and inequalities M11GM-Ig-2 General Directions To achieve the objectives of this module, can you try to follow the steps below? Great! First, bring with you a graphing paper or install a graphing calculator (Geogebra) on your cellular phone. Second, take ample of time reading the lessons carefully. Third, read and follow instructions honestly. Fourth, answer the pre-assessment on the first part of the module. Then, do all the given test and exercises thoroughly. And lastly, take note of the definitions of the terms that is being highlighted in through the discussion. What I Know GENERAL DIRECTIONS: Read the items carefully. Write your answer on the space provided before each item. 1. Which of the following functions represent an exponential function? x A. f(x) = 2x2 B. (𝑥) = 3𝑥 4 C. (𝑥) = 4𝑙𝑛𝑥 D. 𝑓(𝑥) = 3 2. Which of the following is an exponential equation? A. f(𝑥) = 𝑥2 B. x2 + y2 = 9 C. 𝑓(𝑥) = 4𝑙𝑛𝑥 D. 128 = 3𝑥 3. It is a function of the form f(𝑥) = b𝑥 where b>0 and not equal to 1. A. rational B. linear C. piece-wise D. exponential 4. Which among the choices below represents an exponential inequality? A. f(x) = x 2 B. x 2 + y2 ≥ 9 C. f(x) = 4lnx 5. What value of x can make the equation 5x+1 = 125 true? A. 1 B. 2 C. 3 D. 125≤ 5 x−3 D. 4 For items 6 and 7, refer to the situation below. The half-life of a radioactive substance is 72 hours with an initial amount of 55 grams. 6. Give an exponential function that models the situation above. A. f(x)= (55) C. f(x) = B. f(x)= (55) D. f(x) = (155) 7. How much of the substance remains after 5 days? A. 17.3 g B. 17.4 g C. 17.5 g x 8. What value of x can the expression 2 be equal to x 2 ? A. 1 B. 2 C. 3 D. 71.3 g D. 4 9. What happen to the exponential function as the value of x decreases without bound? A. B. C. D. The function increases without bound. The function approaches to infinity. The function approaches to the line x=0. The function approaches to the line y=0. x 10. What are the possible values of x in which the relation 3 <9 A. all values of x greater than 4 B. all values of x greater than 6 C. all values of x less than 4 D. all values of x less than 6 x-2 is true? Answer key on page 25 Your Pre-Test ends here! You may now start learning more about Representations of Functions and Relations! 3 Lesson Exponential Functions In Action 1 Imagine that in a certain school, the school head delivers a message of class suspension due to typhoon. His first goal is to send the message to two of his constituents and his constituents send the message to another two, and so on. How much time is needed so that everyone in the school know the announcement? What’s In Did you know that exponential function is a phenomenon that exists whenever a quantity grows or diminishes at a rate proportional to its present value? Such examples can be observed in businesses such as the compound interest, loans and mortgages. It can also be used to describe population growth, radioactive decay, growth of an epidemic and in many other fields of study. On this lesson, you are going to explore the characteristics and kinds of exponential expressions such as equations, functions and inequalities. What’s New Activity 1: Paper Folding to the Moon Directions. Your task in this activity is to perform the given steps or procedures below. Fill in the given table of the number of folds and its corresponding number of partitions. Write your answers on your journal notebook and submit next session. Enjoy! Materials: 1 whole sheet of paper, pen, worksheet Procedures: 1. Get a sheet of paper. Fold the paper once crosswise. 2. Record the number of partitions using Table 1 as a guide. 3. Fold the paper again, this time lengthwise. Repeat procedure 2 until the 7th fold. Refer to the Table 1. 4 Table 1 Number of Folds and Its Corresponding Parts Number Number of “How many times would you have to fold a piece of paper for it to reach the Moon? How much paper do you need? of Folds Partitions As it is observed if you fold a paper in half, the number of partitions is doubled. Let’s say that a 500 page ream of bond paper is 5 cm high which means that each bond paper is 0.01 cm and the mean distance of the moon is 384,400 km from the Earth! Converting that figure into centimeters and number of pages, we have 3.844 x 10^12 pages away. Amazing right!? Guide questions: 1. How many partitions will you have by folding the paper 8 times? 10 times? X times? 2. How many folds you must perform in order to have 512 partitions? 3. What pattern can you observe in Table 1? Answer key on page 24 Definition An exponential expression is an expression of the form a b x−c + d, where b > 0 and b ≠ 1. An exponential function with base b is the function of the form f(x) = bx or y= bx where b > 0 or b ≠ 1. Example: f(x) = 2x (base is 2) f(x) = 3x (base 3) An exponential equation is an equation involving exponential expressions. Example: 9 = 3x 3x+1 = 27 252x−2 = 125 An exponential inequality is an inequality involving exponential expressions. Example: 9 ≥ 3x 3x+1 ≤ 27 Inequality Symbols: greater than (>) greater than or equal (≥) less than (<) less than or equal (≤) 5 What is it? Example 1. Determine whether the given expression is an exponential function, exponential equation or exponential inequality. 1. f(x) = 2x + 1 2. f(x) = 2x3 exponent) 3. 32 = 2x−2 4. y = e x (Exponential function with a base 2) (Not an exponential function since the variable is not in the (Exponential equation) (Exponential function with a base e also known as natural exponential function) (Exponential inequality) 5. 64 ≤ 2x+3 KINDS OF EXPONENTIAL FUNCTIONS/MODELS Some of the commonly known real-life applications of exponential functions are the computations on population growth, exponential decay and compound interest. A. EXPONENTIAL GROWTH AND DECAY The rule of exponential growth and decay can be modeled by f(x) = (ab)x where a is the initial amount, b is the growth factor, and x is the number of intervals (minutes, hours, days, years, and so on. The half-life of a radioactive substance is the time it takes for half of the substance to decay. Example 2 (Exponential Growth). In the beginning, God created the first man and woman on Earth. Suppose their number doubles every 2 years. Give an exponential function that models the situation. How many people on Earth will be after 30 years? Solution : Initially, At t = 0 At t = 2 At t = 4 At t = 6 At t = 8 Number of people on Earth = 1 Number of people on Earth = 21= 2 Number of people on Earth = 22= 4 Number of people on Earth = 23= 8 Number of people on Earth = 24= 16 Have you figured the pattern already? What pattern can you derive from this? Graph of f(x)=2(x/2) Figure 1. Amount of population (y) per period of time (x) 6 Given that the initial amount a is 2 and the growth factor is 2 since it uses the term double (3 if triples and so on.) and x is ranging from 1,2,3,4, and so on until it reaches to 15. Thus, The exponential model for this situation is f(x)=2t/2 . After 30 years, the number of people on Earth is given by (𝑥) = (2)30/2 = (2)15 = 32,768. Example 3 (Exponential Decay). The half-life of a radioactive substance is 1200 years. If the initial amount of substance is 300 grams, give an exponential model for the amount remaining after t years. What amount of substance remains after 2400 years? Solution: Let t = time in years. We use the fact that the mass is halved every 1200 years. Thus we, have: Initially, at t = 0 Amount of Substance = 300 g at t = 1200 years Amount of Substance = 150 g at t = 2400 years Amount of Substance = 75 g Graph of f(x)=300(1/2) x Figure 1. Amount of substance (y) per period of time (x) Given: a = 300 g , b = ½ (from the definition of half-life), x = Substituting the given to the exponential model, we have: f(x) = (ab)x So the exponential model for this situation is 𝑓 𝑥 3 ( ) x To find the amount of substance remains after 2400 years, let f(x) = (300) (1/2) 𝑓 𝑥 3 ( ) (replace x by t/1200) 𝑓 𝑥 3 ( ) (replace t by 2400) f 𝑥 3 ( ) (solve for f(x)) Thus, the amount of substance remains after 2400 years is 75 grams. 7 B. COMPOUND INTEREST If a principal P is invested at an annual rate r, compounded annually, then the amount after t years is given by t A = P(1 + r) . EXAMPLE 3. Determine the amount of money that will be accumulated if a principal of ₱10,000 is invested at an annual rate of 5% compounded yearly for 10 years. Solution Given: Principal (P ) = ₱10,000 𝑟𝑎𝑡𝑒 (𝑟) = 5% 𝑜𝑟 0.05 𝑡im𝑒 (𝑡) = 10 𝑦𝑒𝑎𝑟𝑠 Find: Amount after 10 years To solve for A, substitute the values given in the formula. Then, 𝐴 = (1 + 𝑟) = 10,000 (1 + 0.05)10 = ₱16,288.95. Hence, there will be ₱16,288.95 after 10 years. C. NATURAL EXPONENTIAL FUNCTION The natural exponential function is the function (𝑥) = 𝑒𝑥 . EXAMPLE 4. The population of the Philippines can be approximated by the function (𝑥) = 20000000𝑒0.0251𝑥 (0 ≤ 𝑥 ≤ 40) where x is the number of years since 1955. Use this model to approximate the Philippines population during the years of 1955, 1965, 1975 and 2005. Round off answers to the nearest thousand. Solution. x=time in years P(x) 1955 (x=0) 1965 (x=10) 1975 (x=20) 2005 (x=50) 20,000,000 25,706,000 33,040,000 70,156,000 Congratulations! You have finished the lesson. Now let me know how much you learn! Please turn to the next page and answer the activity. 8 What’s More ACTIVITY 2 GENERAL DIRECTIONS: Answer as directed. Write your answers on your Activity Notebook. A. Determine whether the given expression is an exponential function, exponential equation or exponential inequality. Choose your answer among the choices below. (2 points each) A. Exponential Function C. Exponential Equation B. None of these D. Exponential Inequality 1. 𝒇(𝒙) = 𝒙𝟐 + 𝟔𝒙 – 𝟗 _______________________________ 6. 𝟔𝟒 = 𝟐𝒙+𝟐 _______________________________ 2. 𝒇(𝒙) = 𝒆𝟐𝒙 _______________________________ 7. 𝟖𝟏 = 𝟑𝟐𝒙−𝟏 _______________________________ 3. 𝒇(𝒙) = 𝟒𝒍𝒏𝒙 _______________________________ 8. 𝒚 ≥ 𝟑𝒙 _______________________________ 4. 𝒚 = 𝟐𝒙𝟑 _______________________________ 9. 𝟐𝟐𝟓 ≤ 𝟓𝟑𝒙−𝟐 _______________________________ 5. 𝒚 = 𝟑𝒙 _______________________________ 10. 𝒚 = 𝟑𝒙−𝟒 _______________________________ B. Solve the following problems. Show all your solutions. (5 points each) 1. A population in a barangay starts with 1,000 individuals and triples every 10 years. Give an exponential model to represents the population. What is the population of the barangay after 20 years? 2. A bank offers a 2% annual interest rate, compounded annually for a certain amount. Give an exponential model if ₱20,000.00 is invested under this condition. How much money will there be in the account after 5 years? 3. The half-life of a radioactive substance is 1500 years. If the initial amount of the substance is 500 grams, give an exponential model for the amount remaining after t years. What amount of substance remains after 3000 years? Answer key on page 25 9 Lesson 2 Solving Exponential Equations And Inequalities To start our discussion, we will be using some properties of exponential functions to solve its equations and inequalities. Take a look at this! Property 1 (One-to-One Property) If 𝒙𝟏 ≠ 𝒙𝟐, then 𝑏 𝑥 ≠ 𝑏 𝑥 . Similarly, if 𝒙𝟏 = 𝒙𝟐, then 𝑏 𝑥 = 𝑏 𝑥 . What is it EXAMPLE 1 A. Solve the equation 2𝑥 = 8. Solution By looking at the equation, you must think of the value of x in such a way that when you raise 2 by that number, the answer is 8. What do you think is the number? That’s right! That number should be 3 since 23 is 2 x 2 x 2 which is 8. Thus, x is equal to 3. B. Solve the equation 4x+1 = 16. Solution Write both sides with 4 as the base. +1 4𝑥 = 16 (Copy the given equation) +1 4𝑥 = 42 (Express the equation having the same base) 𝑥+1 =2 (Use of One-to-one property) 𝑥 = 2−1 (Addition Property of Equality) 𝑥=1 (Solved value of x) Thus, the value of x is 1 to make the equation true. C. Solve the equation 125 125 5 x-1 3(x-1) = 25 =5 x+3 2(x+3) 3(𝑥 − 1) = 2(𝑥 + 3) x-1 = 25 x+3 . (Copy the given equation) (Express the equation having the same base) (Use of One-to-one property) 10 3𝑥 − 3 = 2𝑥 + 6 (Distributive Property) 3𝑥 − 2𝑥 = 6 + 3 (Combine all the variables in one side and constants on the other) 𝑥=9 (Solved value of x) Solution: Note that both 125 and 25 can be expressed with the base 5. 2 D. Solve the equation 9 𝑥 = 3𝑥+3. Solution: Both sides of the equation can be expressed with the base 3. 2 9 𝑥 = 3𝑥+3 (Copy the given equation) 2 32(𝑥 ) = 3(𝑥+3) (Express the equation having the same base) 2(𝑥2) = (𝑥 + 3) (Use of One-to-one property) 2𝑥2 = 𝑥 + 3 (Distributive Property) 2𝑥2 − 𝑥 − 3 = 0 (Equate to 0.) (2𝑥 − 3)(𝑥 + 1) = 0 (Factor the equation to get the value of x) (2𝑥 − 3) = 0 or (𝑥 + 1) = 0 (Addition Property of Equality)* 𝑥= 3 2 or 𝑥 = −1 (Solved values of x) *Addition Property of Equality- adding both sides of the equation with the same number to make the equation true. EXAMPLE 2 A. Solve the inequality 3x < 9x-2. 3x < 9x-2 (Copy the given inequality) 3𝑥 < 32(𝑥−2) (Express the equation having the same base) 𝑥 < 2(𝑥 − 2) (Since the base 3>1, then the direction of inequality is retained) 𝑥 < 2𝑥 − 4 (Distributive Property) 4 < 2𝑥 − 𝑥 (Combine similar terms) 4<𝑥 (Solved value of x) Thus, the solution set to the inequality is {𝑥 ∈ ℝ ∥ 𝑥 > 4} Solution: Both 3 and 9 can be written using 3 as the base. 11 Property 2 (Exponential Inequality) If 𝑏 > 1, then the exponential function 𝑦 = 𝑏𝑥 is increasing for all x. This means that 𝑏𝑥 < 𝑏𝑦 if and only if 𝑥 < 𝑦. If 0 < 𝑏 < 1, then the exponential function 𝑦 = 𝑏𝑥 is decreasing for all x. This means that 𝑏𝑥 > 𝑏𝑦 if and only if 𝑥 < 𝑦. x+5 B. Solve the inequality ( ) ≥( 3x ) Solution: 2 = ( ) , then we write both sides of the inequality with the base Since ( )x+5 ≥ ( ) ( )x+5 ≥ ( ) 3x 2(3x) 𝑥 + 5 ≤ 2(3𝑥) 𝑥 + 5 ≤ 6𝑥 (Copy the given inequality) (Express the equation having the same base) (Since the base < 1, then the direction of inequality is reversed) (Distributive Property) 5 ≤ 6𝑥 − 𝑥 (Combine similar terms) 5 ≤ 5𝑥 (Divide both sides by 5) 1≤𝑥 (Solved value of x) Thus, the solution set to the inequality is {𝑥 ∈ ℝ ∥ 𝑥 ≥ 1} EXAMPLE 3. Solving Problems Involving Exponential Equations and Inequalities A. The half-life of Zn-71 is 2.45 minutes. At 𝑡 = 0, there were 𝑦0 grams of Zn-71, but only 1 of this amount remains after some time. How much time has 256 passed? Solution: Using exponential models that you have learned previously, we can determine that after t minutes, the amount of Z n-71 is 𝑦𝑜 12 1 2 𝑡 4 Initial amount of substance time of half-life t/2.45 y0 t/2.45 t/2.45 4 = y0 = = Copy from the given equation Cancel y0 since it is common to both sides of the equation 8 Use the one-to-one property =8 Distributive property t = 8(2.45) t = 19.6 Solve for t Thus, 19.6 minutes have passed since t = 0 We solve the equation y0 t/2.45 = y0. What’s More ACTIVITY 3 DIRECTIONS. Solve the following exponential equations and inequalities. 1. 169x= 13x 4. 2. ( ) 5. 4 4 < 4 4 3. ( ) Answer key on page 25 13 Lesson 3 Graphing Exponential Functions What’s In Can you still recall on plotting of points in the Cartesian plane? Let us start! Look at the Cartesian plane below! QII QI QIII QIV How many parts are there in the x-y plane? Is it four? No, there are seven parts in the coordinate plane! Can you name them? These are the following: a. b. c. d. e. f. g. Quadrant I (x,y) Quadrant II (-x,y) Quadrant III (-x,-y) Quadrant IV (x, -y) X-intercept (x,0) or (-x,0) Y-intercept (0,y) or (0, -y) Origin (0,0) The x-coordinate is called the abscissa and the y-coordinate is the ordinate. What do you think is the use of the Cartesian plane? (graphing) You are right! We can illustrate the graph of a function using the x- and y-plane. So, let us start with our topic on exponential functions. 14 One way to graph exponential functions is with the use of the table of values to show the points. Consider the examples below. What’s More How do you graph an exponential function? 1. Sketching the graph of (𝑥) = 2𝑥. Steps: 1. Construct a table of values of ordered pairs for the given function. The table of values for 𝑓(𝑥) = 2𝑥 is as follows: x -3 -2 -1 0 1 2 3 4 y 1/8 1/4 1/2 1 2 4 8 16 2. Plot the points found in the table and connect using a smooth curve. a. Plotting of points from the table of vales in Step 1. 15 b. connecting the points through the smooth curve. What can you observed in the graph above? That’s right! The graph is increasing. Observe also that for all values of x, it gives a positive y- values. And also, as the values of x decreases without bound, the function approaches to the horizontal axis but never actually touch reach the line. Hence, the line y=0 is called the horizontal asymptote. Do you know what an asymptote is? Try to read again the underlined sentence above. Did you get it now? Asymptotes are line where the graph approaches but never touches. Well, it sounds like your crush, right? But anyway, are you having fun so far? Let us continue! How would you call a point that is located exactly on the y-axis? The point that is on the y-axis is (0, 1). This point is called the y-intercept. Sketch the graph of (𝑥) =(1/2)x Can you fill in the corresponding values of y? Just follow the steps above. 1. Construct a table of values of ordered pairs for the given function. The table of x values for f(x) = 1 is as follows: 2 x y -3 -2 -1 16 0 1 2 3 4 2. Plot the points found in the table and connect using a smooth curve in the Cartesian plane below. What can be observed in the graph above? Fill in the blanks. The graph is . For all values of x, it gives a As x increases without bound, the value of the function approaches to Hence, the line y=0 is called the . y-values. . How would you call a point that is located exactly on the x-axis? Does exponential function has this point? The point where the graph crosses the x-axis is called the x-intercept. And exponential function has no x-intercept. Why do you think so? Note that, exponential functions do not have x-intercept since its graph has horizontal asymptote. This implies that the graph does not intersect the x-axis. 17 REMARKS: In general, the graph of the function depends on the value of the base (e.i., b>1 or 0<b<1) 0<b<1 b>1 3. Transformation of 𝒇(𝒙) = 𝒃𝒙 We’re almost there! This time let us have a comparison of our graphs to determine its transformation. What do you know about transformation? Sketch the graph of f(𝑥) = 2𝑥, 𝑔(𝑥) = 3𝑥 and (𝑥) = 4𝑥 in one plane. Observe the graph above and together, let us analyze its behavior. All the graphs are increasing since b>1. The y-intercept of f(𝑥) = 2𝑥 is . 𝑥 The y-intercept of f (𝑥) = 3 is . The y-intercept of f (𝑥) = 4𝑥 is 1. 18 The line y=0 is the horizontal asymptote. The functions f, g and h have no zero. This means that there is no x-values that makes the function 0. The base determines the steepness of the graph. Observe that in every 1 unit change in x, f(𝑥) = 2𝑥 increases by 2 times, 𝑔(𝑥) = 3𝑥 and (𝑥) = 4𝑥 increases by 4 times. Definition. Let b be a positive number not equal to 1. The transformation of an exponential function with the base the base b is the form (𝑥) = 𝑎 𝑏𝑥 𝑐 + 𝑑. It is defined as the process where the graph of the function changes position without changing its shape or size. Transformations Involving Exponential Functions Type of Transformation Equation Horizontal Translation 𝒇(𝒙) = 𝒃𝒙 + 𝒄 Vertical Translation Reflection Description 𝒇(𝒙) = 𝒃𝒙 + 𝒄 𝒇(𝒙) = −𝒃𝒙 𝒇(𝒙) = 𝒃−𝒙 Vertical Stretching or Shrinking 𝒇(𝒙) = 𝒄𝒃𝒙 Shifts the graph of (𝒙) = 𝒃𝒙 c units to the left, if c>0 and to the right, if c<0. Shifts the graph (𝒙) = 𝒃𝒙 c units upward, if c>0 and downward, if c<0. Reflects the graph of 𝒇(𝒙) = 𝒃𝒙 about the x-axis. Reflects the graph of 𝒇(𝒙) = 𝒃𝒙 about the y-axis. Multiplying y-coordinates of (𝒙) = 𝒃𝒙 by c. Stretches the graph of 𝒇(𝒙)=𝒃𝒙 if c>1 and shrinks if c<1. 19 ACTIVITY 4 DIRECTIONS. Consider the given exponential function. Sketch the graph of x+2 (𝑥) = 2 . Determine its domain, range, x-intercept and asymptote. 20 Enrichment Activity Read the following situation and write a reflection based on the questions below. Radioactive Substances In July 2002, National Geographic ran an article about the problems that America faces with its ever-growing amount of nuclear waste. Currently the United States has over 77,000 tons of waste. Environmentalists talk about how the radioactive material will be dangerous for thousands of years because of its long halflife. In fact, it will take 240,000 years for plutonium 239 to become safe! When scientists talk about half-life, they are referring to how long it will take for half of a sample to decay. In the case of nuclear waste, it refers to how long it takes for half of the radioactive material to turn into lead. Waste Material In the Philippines, about 35,580 tons of garbage daily and on average each person produces 0.5 and 0.3 kg of garbage in urban and rural areas, respectively. Imagine this rate after 10 years. Moreover as the population increases, the amount of garbage produce also increase. Bacteria The most common example is the growth of bacteria colonies. Bacteria multiply at an alarming rate. If we assume that bacteria can double every hour and if we start with just a single bacteria, then after one day there will be over 16 million bacteria! Your Task In your own community, conduct a survey on the possible application of exponential functions. Create a situation similar to the above situations and predict what will happen after a certain time. What value can be obtained from this situation? What can you do to improve/prevent a situation like this? Reflect on the following. 1. What is the importance of exponential growth and decay in the life of human beings? 2. How do you know whether the exponential function is growth or decay? 21 Let us summarize… An exponential equation is an equation involving exponential expressions. An exponential inequality is an inequality involving exponential expressions. The transformation of an exponential function with the base the base b is the form 𝑓(𝑥) = 𝑎 ∙ 𝑏𝑥−𝑐 + 𝑑. Asymptotes are line where the graph approaches but never touches. The natural exponential function is the function (𝑥) = 𝑒𝑥. Transformations Involving Exponential Functions Type of Transformation Equation Description Horizontal Translation 𝒇(𝒙) = 𝒃𝒙+𝒄 Vertical Translation 𝒇(𝒙) = 𝒃𝒙 + 𝒄 Reflection 𝒇(𝒙) = −𝒃𝒙 Reflects the graph of 𝒇(𝒙) = 𝒃𝒙 about the x-axis. 𝒇(𝒙) = 𝒃−𝒙 Reflects the graph of 𝒇(𝒙) = 𝒃𝒙 about the y-axis. 𝒇(𝒙) = 𝒄𝒃𝒙 Multiplying y-coordinates of 𝒇(𝒙) = 𝒃𝒙 by c. Vertical Stretching or Shrinking Shifts the graph of (𝒙) = 𝒃𝒙 c units to the left, if c>0 and to the right, if c<0. Shifts the graph (𝒙) = 𝒃𝒙 c units upward, if c>0 and downward, if c<0. Stretches the graph of (𝒙) = 𝒃𝒙 if c>1 and shrinks if c<1. 22 Posttest GENERAL DIRECTIONS: Read the items carefully. Write your answer on the space provided before each item. 1. Which of the following functions represent an exponential function? A. (𝑥) = 4𝑥2 D. 𝑓(𝑥) = 2𝑥+1 C. 𝑓(𝑥) = 𝑙𝑛 𝑥 2. It is a function of the form (𝑥) = 𝑏𝑥 where b>0 and not equal to 1. A. Rational C. Piece-wise D. Exponential Function B. Linear 3. Which among the choices below represents an exponential inequality? A. (𝑥) = 𝑥 2 3x+3 B. 𝑥2 + 𝑦2 ≥ 9 C. 𝑓(𝑥)= 4𝑙𝑛𝑥 D. 125 ≤ 25 4. What value of x can make the equation 5𝑥+1 = 125 true? A. 2 B. 3 C. 4 D. 5 For items 5 and 6, refer to the situation below. The half-life of a radioactive substance is 24 hours with an initial amount of 100 grams. 5. Give an exponential function that models the situation above. A. f(x)= (100) B. f(x)= (100) (24/t) C. f(x)= (t/24) (t/24) D. f(x)= (100) (t/72) 6. How much of the substance remains after 5 days? A. 3.125 g B. 7 g C. 12.5 g D. 50 g 7. John and Peter are solving (0.6)x-3 > (0.36)-x-1. Shown below are their solutions. Who get the correct answer? John Peter A. John (0.6)𝑥−3 > (0.62)-x-1 (0.6)𝑥−3 > (0.62)-x-1 (0.6)𝑥−3 > (0.6)2(-x-1) (0.6)𝑥−3 > (0.6)2(-x-1) (0.6)𝑥−3 > (0.6)−2x-2 (0.6)𝑥−3 > (0.6)−2x-2 x-3<-2x-2 x-3<-2x-2 3x>1 3x<1 x> X< B. Peter C. Both John and Peter 23 D. Neither John nor Peter 8. Determine the amount of substance remaining after 12 hours in situation number 5-6. A. 7.05 g B. 7.5 g C. 70.71 g D. 71.70 g 2x3 4 x2 9. Solve for x : 16 A. B. C. D. 4 10. Which of the following best describes the graph of an exponential function at the left? A. The function is decreasing, define for all values of x and as the function approaches 0, x increases without bound. B. The function is increasing, defined for all values of x, and as the function approaches 0, y increases without bound. C. The function is increasing, defined for all x values, attains only positive y-values and the line y=0 is the horizontal asymptote. D. The function is decreasing, defined for all x values and attains x and y values. Answer key on page 25 24 ACTIVITY 2 25 POST TEST 10.B 9. D 9. C 8. D 8. C 7. C 7. B 6. C 6. A 5. A 5. B 4. B 4. A 3. A 3. D 2. A 2. D 1. B 1. D PART A PRETEST 1. D 2. D 3. D 4. D 5. B 6. B 7. A 8. B 9. C 10.A 10.A 5. ACTIVITY 3 ACTIVITY 2 2. 22,081.62 1. [-3/2, ∞) 1. 9000 2. -3 PART B 1. 0 2. -2 3. (-∞, 1/3) ACTIVITY 1 1. 256; 1024 2. 9 3. 3. f(x)= 500(1/2)x/1500 ; 125 grams 2 raised to the number of folds KEY ANSWERS GLOSSARY OF TERMS Exponential Decay and Growth. The exponential growth and decay can be modeled by (𝑥) = (𝑎𝑏)𝑥 where a is the initial amount, b is the growth factor, and x is the number of intervals. The half-life of the substance is the time it takes for half of the substance to decay. Exponential Equation. An exponential equation is an equation involving exponential expressions. Exponential Expression. An exponential expression is an expression of the form 𝑎 ∙ 𝑏𝑥 − 𝑐 + 𝑑, where 𝑏 > 0 and 𝑏 ≠ 1. Exponential Function. An exponential function with base b is the function of the form f(𝑥) = 𝑏𝑥 or 𝑦 = 𝑏𝑥 where 𝑏 > 0 𝑜𝑟 𝑏 ≠ 1. Exponential Inequality. An exponential inequality is an inequality involving exponential expressions. Exponential Transformation. Let b be a positive number not equal to 1. The transformation of an exponential function with the base the base b is the form f(𝑥)=𝑎∙𝑏𝑥−𝑐+𝑑. It is defined as the process where the graph of the function changes position without changing its shape or size. 26 References A. Books / Manuals / Other Printed Materials Crisologo, L., Hao, L., Miro, E., Palomo, E., Ocampo, S., and Tresvalles, R. General Mathematics Teacher’s Guide. Department of Education- Bureau of Learning Resources, Ground Floor Bonifacio Bldg, DepEd Complex Meralco Avenue, Pasig City, Philippines 1600. Lexicon Press Inc. 2016. blr.lrpd@deped.gov.ph. B. Websites Eisegel. "Paper Folding To The Moon | Scienceblogs". 2009. Scienceblogs.Com. https://scienceblogs.com/startswithabang/2009/08/31/paper-folding-to-themoon. Nykamp DQ, “The exponential function.” Insight. http://mathinsight.org/exponential_function. From Math https://www.youtube.com/watch?v=xZn4f1eIl3g file:///C:/Users/Intel/Downloads/Applications_of_Exponential_Functions_Student.pdf http://teachtogether.chedk12.com/teaching_guides/view/14 https://www.math-exercises.com/equations-and-inequalities/exponential-equations-andinequalities https://www.math-exercises.com/equations-and-inequalities/exponential-equations-and-inequalities https://www.youtube.com/watch?v=dV1mUjnrGY4 https://courses.lumenlearning.com/waymakercollegealgebra/chapter/characteristics-of-graphs-ofexponential-functions/ https://www.google.com/search?q=find+the+domain+and+range+of+an+exponential+function.&spell=1& sa=X&ved=2ahUKEwjm98yvytXpAhWOUt4KHcmOBXIQBSgAegQIDRAm&biw=1366&bih=646#kpvalbx =_SSvPXpvsENXmwQPjuLvgBg54 https://www.youtube.com/watch?v=gVGlA4jdQso https://www.youtube.com/watch?v=CwclkRAJmm8 https://www.youtube.com/watch?v=A7tB_ycw_Z0 https://www.youtube.com/watch?v=6FeKywTfAw0 Mobile Application(s) International Geogebra Institute. Wolfauser 90, 4040 Linz, Austria. “GeoGebra Calculator”. Google Store, Version 5.0.366.0-3D (2017). http://www.geogebra.org/. Accessed on October 12, 2019. 27 For inquiries and feedback, please write or call: Department of Education – Bureau of Learning Resources (DepEd-BLR) DepEd Division of Cagayan de Oro City Fr. William F. Masterson Ave Upper Balulang Cagayan de Oro Telefax: ((08822)855-0048 E-mail Address: cagayandeoro.city@deped.gov.ph