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Remember that there are two types of exponential functions: Remember that the criteria for each type is . . . b > 1 Growth Calculating Exponential Values 0 < b < 1 Decay Let's discuss first. Remember that there are two types of exponential functions: Remember that the criteria for growth is . . . b > 1 Calculating Exponential Values growth y = a•(b)x y = a•( )x The common form of the equation will be used for calculations: The base, b will be expressed differently to make it more obviousthat it is greater than 1 . . . where "r" is the percent increase.Also, these problems are time based so the variable "x" is replaced by "t". So the modified equation looks like: y = a•( ) 1+r 1+r Calculating Exponential Values t Let's see if you can identify the numerical values given the terms: The cost of a movie ticket is $8, which increases by 3% each year.Find the cost of the ticket after 7 years. Here is the equation for the exponential function: y = 8•(1+0.03)t or y = 8•(1.03)t Growth factor:(a single value) Overview of terminology for growth: Initial amount: $ y = a•( ) initial amount 1+r Calculating Exponential Values growth factor: a single value 1+r t Percent increase:(as a percent) What value will be put in for "t": percent increase time % Let's see if you can calculate the cost of the movie ticket 7 years later. The cost of a movie ticket is $8, which increases by 3% each year.Find the cost of the ticket after 7 years. Here is the equation for the exponential function: y = 8•(1+0.03)t y = 8•(1.03)7 y = $ y = a•( ) 1+r since we are calculating money roundyour answer to the nearest penny. Calculating Exponential Values t Growth factor:(one value) Round your answer to the nearest dollar. Rents in a particular area are increasing by 4% every year. Predict what therent of the apartment would be after 5 years, if its rent is $400 per month now. Initial amount: $ Subtituting into the equation: y = y = a•( ) 1+r Calculating Exponential Values t What value will be put in for "t": Percent increase:(as a percent) •( Answer: $ % ) The population of the United States was about 250 million in 2003,and is growing exponentially at a rate of about 0.7% (r = 0.007) peryear. What will be its population in the year 2013? Round your answer to the nearest million. y = a•( ) 1+r Calculating Exponential Values t Answer: A business man made a profit of $15153 in 1990. The profit increasedby 2% per year for the next 10 years. Identify an exponential growthmodel for the profit and find the annual profit in the year 2000. Round your answer to the nearest dollar. Subtituting into the equation: y = y = a•( ) 1+r Calculating Exponential Values t initial amount •( Answer: $ ) time An initial population of 456 starfish doubles (gets 100% bigger which meansr = 1) each year for 4 years. What i the population after 4 years? Round your answer to the nearest integer. y = a•( ) 1+r Calculating Exponential Values t Answer: A special application of exponential growth is "compounded Interest". Interest is often compounded (or calculated) more than once a year sothe above equation needs to be modified once again. We need to divide the rate, "r" by the number of times we compoundin a year and multiply time, "t" by that same amount. y = a•( ) y = a•( ) 1+r 1+ Calculating Exponential Values r n t n•t Let's see if you can identify the numerical values given the terms: Laura invests $600 for 2 years at 5% interest compounded quarterly.What is the value of the account in 2 years?Here is the equation for the exponential function: y = 600•(1+0.05/4)4•2 or y = 8•(1.0125)8 Overview of terminology for compound interest: Compounding times:(per year) Initial amount: $ y = a•( ) initial amount 1+ Calculating Exponential Values compounding times per year r n n•t Annual Rate:(as a percent) What value will be put in for "t": annual rate time % Let's see if you can calculate the value of the account 2 years later. Laura invests $600 for 2 years at 5% interest compounded quarterly.What is the value of the account in 2 years? Here is the equation for the exponential function: y = 600•(1+0.05/4)4•2 y = $ y = a•( ) 1+ Calculating Exponential Values r n since we are calculating money roundyour answer to the nearest penny. n•t Dan borrowed $700 for 3 years at 10% interest compounded annually(once per year, n = 1) from his dad. How much does Dan owe his dadafter 3 years? Round your answer to the nearest penny. Subtituting into the equation: y = y = a•( ) 1+ Calculating Exponential Values r n n•t •(1+ Answer: $ ) A bank pays 4% interest compounded monthly on a deposit of $5000.What will be the balance in the account after 5 years? Round your answer to the nearest dollar. y = a•( ) 1+ Calculating Exponential Values r n n•t Answer: $ |