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4G Ol on 17:42 (VPN) ( 7 Explain the lesson and explain your answers must have concrete evidence Exponential Function Definition: An exponential function can be written as f(x) = b*, where b > 0, b # 1, and x is any real number In the equation f (x) = b*, b is a constant called base and x is an independent variable called exponent Examples: f(x) = 3x g(x) = 10* h(x) = 2x+1 51 real numbers. The domain of an exponential function is the set of real numbers and the range is the set of all positive Can you cite instances where we can apply exponential function in our life? 1. Population growth 2. Growth and decay 3. Increase of prices of commodities 4. The value of brand-new cars Examples 1. The number of pupils at Exponent Primary School has increased by 40 each of the past five years. If the population was 500 five years ago, what is the present population? SOLUTION: The number of pupils increased by the same amount each year, so this represents a linear function. Because the pupils' population increased by 40 per year, in five years it will grew by 5 x 40 =200 pupils, from 500 it will become 700. If P represents the number of pupils in t years, the function would be P 500+401 2. The initial population at Brgy 127 is 10,000 people. Brgy 127 grows at a constant rate of 10% per year. What would be the population of this barangay after 4 years? SOLUTION: If P represents the total population and t represents time, then P-initial population times (rate), thus P = 10000 +(1.10)" = 14641 population of Brgy. 127 after 4 years. 1.10 - the growth factor - 100% +10% = 110% = 1.10 3. Determine the growth factor of the quantity that increases by the given percent. a. 50% 100% +50% - 150% = 1.50 b. 75% 100% +75% = 175% = 1.75 C. 10% 100% +10% = 110% = 1.10 d. 12.5% 100% + 12.5% = 112.5% = 1.125 Emerson deposits P50000 in a savings account. The account pays 6% annual interest. If he makes no more deposits and no withdrawals, calculate his new balance after 10 years. Solution: 6% the interest rate per year Isuponl Inclusnoga I gaivloe Growth factor would be: 100% +6% = 106% = 1.06 Equation would be y = P50000 (1.06)", where y represents the new balance y = P89542.38 New balance after 10 years The rule for exponential growth can be modeled by y=ab , where a is the starting number, b is the growth factor, and x is the number of intervals 5. A bacteria grows at a rate of 25% each day. There are 500 bacteria today. How many bacteria will be a. Tomorrow? SOLUTION: y=ab" , a=500, b= 1.25, x=1 y = 500(1.25) Waupand to thogort y = 625 bacteria therefore, there will be 625 bacteria tomorrow b. One week from now? SOLUTION: y= ab* 01 6: 09001 bagroxa a= 5004?" 3'3" 17.42 3-500 ' - 53> 2 pi 5 Division Property e. 1 3x+s l *'6 (93 5 (as) (3)3):4-5 S (_15_ X4) 3-2036) 53.56:\" QUHS) 5 -5(K-6} 45x40 5 -Sx+3tl reunite 9 as 32 and 243 as 35 negative law of exponcnts , Distributive properly '6)' +5x330+10 Put similar terms on the same side x S 40 in 240 simplify (multiplying -number to both sides reverse the inequality symbol} REVISED KNOWLEDGE: Actual answer to the process questions! focus questions. 1. Why are exponential equations and exponential inequalities iinponant'? . Exponential functions can he used to model growth and decay. Exponential functions are ever-increasing so saying that an exponential function models population growth exactly means that human population will grow without bound. 2. How can exponential Functions help solve reel-life problems? ! Exponential functions are used to represent realworld applications. such as bacteria] grou-'thr'decay, population groml't'decline and compound interest. FINAL KNOWLEDGE: Generalization! Synthesis! Summary l\" .. _. _ . .. i l Exponential mctions have many applications in real life. They are used to solve problems that l involve population growth and decay, interest of an investment or a loan, and appreciation or I i depreciation ofmarltct value of a particular product. They can be even utilized to model a particular [ : phenomenon at some constant conditions. tyl Tell whether each ot'thc following statements is true or false. Write your answer before the number. I. In exponential function dened by J['(x) = 3\