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UNIT 5: Exponential Growth / Decay Formula: y ab a = original amount (y-intercept) x b = growth factor (1 ± r) y = final amount x = unit of measure (time, bounces, etc.) Exponential Growth Exponential Decay Things to know about… y ab • b cannot be negative b > 1: growth x 0 < b < 1: decay • DOMAIN of all exponential functions is: ALL REAL NUMBERS (no restrictions for x) • RANGE of exponential functions: a = positive Range: y > 0 a = negative Range: y <0 • Y – INTERCEPT = a y-values follow multiply or divide patterns Example 1:Identify a Multiply Pattern to write an equation 3, 6, 12, 24, … 162, 54, 18, … 32, 56, 98, 171.5,… 72, 36, 18, 9, … 4, 12, 36, 108,… 20, 30, 45, 67.5, … Example 2 Identifying Initial Value, Growth & Decay EQUATION y = 2(1.03)x y = 20(0.85)x y = 900 (1.27)x y = 0.75(1.2)x y = 750(1.15)x y = 250,000(0.65)x y = 11,275(1.1)x y = (1.54)X 1.85 Initial Value Growth or Decay? Percent of Change Simplifying Exponential Expressions LAWS OF EXPONENTS • Remember when you multiply terms with same base, ADD exponents 2 2 5 3 2 5 3 5 4 x3 5 7 x8 11 x 5 5 • When you raise a power to a power, MULTIPLY exponents (3 x2 3 ) 3 3 x6 4 5 5 x 7 8 4 5 40 x 56 Practice: Simplify each Expression 1. 4 3. ab (7 4 2a 5 x3 2 x ) 2. 4. a 6 a 3 (a ) 3 6 Example 3: Algebraic Solving Exponential Equations Basic Steps:1] 2] 3] a) 2 4 n 3 2 3 n 9 FACTOR into common bases CANCEL common bases SOLVE equation / inequality 11 x 2 x 7 2 b) 5 21 5 x 5 9 x 2 c) 3 3 Example 3: Not Common Bases a) 3 2 n 5 27 b) 4 3 n 2 8 16 c) 9 4 3 x 64 27 2x Example 4: Algebraic Solving Exponential Inequalities 2 x 11 6x 3 n 2 n 9 5 x8 2 x 4 a) 7 7 b) 4 4 7 7 c) 5 5 Example 4: Not Common Bases a) 5 n 2 125 b) 32 5 x2 16 5x b) 1 3 4 x 1 9 Solving Exponential Equations can be done with the calculator like rational equations Calculator Active: [Y=] Y1 = Left Side of Equation Y2 = Right Side of Equation Check your [WINDOW] is large enough Find INTERSECTION: [2nd] [Trace] [5] When writing exponents [^] be careful of where you place your parentheses. Example 5 x Word Problems y ab 1) The population of rabbits is doubling every 3 months. There was initially had 15 rabbits. Write an equation to represent this growth and tell how many rabbits are in the population after 2 years. 2) The value of a car is depreciating at a rate of 7% each year. The car was purchased for $36,000 in 2005. What is the value of the car in 2013? 3) The number of cell phone sales is expected to increase by 25% every year. If 10,000 cell phones were sold in 1998, then how many cell phones were sold in 2005? 4) The number of mice is growing exponentially each month. Initially there were only 20 mice. After 2 months, the population of mice was 180 mice. What is the rate that the mice are growing at? How many mice would you expect at the end of the year (12 months)? 5) A bacteria colony is growing exponentially each day. There was initially had 100 bacteria and after 3 days it had 800. Write an equation to represent this growth, and tell how many bacteria after 10 days. 6) A towns population is growing exponentially. In 2000, the population was 10,000. By 2006 it had risen to 29,860. Let x = 0 represent 2000. Write an equation to represent the growth, and predict the population in 2010. 7) The increase in gas price is modeled by the equation y = 2.17(1.023)x, where x is years since 2004. When would we expect gas prices to be $3.25 8) The decrease in a computer’s value is modeled by the equation y = 4000(.87)x, where x is years since 2007. When would we expect the computer to be worth half it’s value? 9) y = 10,000(1. 08)x, where x is months, models the profit earned by a company. When will the company be able to triple its profit? 10) y = 40,000(.75)x, where x is years, models the value of a car. When will the car be worth only $10,000?