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Notes #3-___ Date:______ 8-8 Exponential Growth and Decay (437) W.1 Evaluate each exponential function for the given domain. x a) y 5 {2,3,4} b) g(x) 2 3x {-2,0,3} W.2 Find the next term of each sequence. a) 2.5, 2, 5, 10… b) -8, 4, -2, 1… W.3 Simplify each expression. 2a 3b 4 a 5 a) 4b 2 1 b) 3mn m 3 2 Exponential Growth: y a bx a > 0, starting amount b > 1, growth factor Ex. 1 In 1998, a certain town had a population of about 10,000 people. Since 1998, the population has increased about 2% a year. Write an equation to model the population increase and approximate the population in 2000. a = 10,000 b = 100% + 2% = 1.02 y 10,000 1.02 x x = 2000 – 1998 = 2 y 10,000 1.022 = 10,404 annually 1x/yr semi-annual 2x/yr quarterly 4x/yr monthly 12x/yr Ex.2 A savings certificate of $1000 pays 6.5% annual interest compounded yearly. What is the balance when the certificate matures after 5 years? Ex.3 Suppose the savings certificate in Example 2 paid interest compounded quarterly instead of annually. Write the equation representing the balance after 5 years. Exponential Decay: y a bx a > 0, starting amount 0 < b < 1, decay factor Ex.4 Technetium-99 has a half-life of 6 hours. Suppose a lab has 80 mg of technetium-99. How much technetium-99 is left after 16 hours? a = 80 b = 100% - 50% = .5 y 80 0.5x x 24 6 4 y 80 0.54 5mg Ex.5 From 1983 to 1997, the ratio of students per computer at the school has dropped by about 6.8% per year. If there were 103 students per computer in 1983, what was the number of students per computer in 1997? a = 103 b = 100% - 6.8% = .932 y 103 .932 x x = 1997 – 1983 = 2 y 103 .93214 ≈ 38 students Ex.6 You bought a computer for $1800. The value of the computer will be less each year because of depreciation. The computer depreciates at the rate of 29% per year. a) Write an exponential decay model for this situation. b) Estimate the value of the computer in two years. Ex.7 Classify as growth or decay. Identify the growth or decay factor and the percent of increase or decrease. a) y40 0.75t Summary: 6 b) y40 5 t Notes #3-___ Date:______ 10-3 Finding and Estimating Square Roots (525) W.1 Evaluate the expression without using a calculator. Write the result in decimal form. 12 a) (9 10 ) (6 10 ) 3 1.8 10 2 b) 2.4 10 7 W.2 The function y10 1.08 x models the cost of annual tuition (in thousands of dollars) at a local college x years after 1997. a) What is the annual percent increase? b) How much was tuition in 1997? c) How much will the tuition be the year you graduate from high school? (write the function) * If b2 = a, then b is a square root of a. * The convention is to use * symbol names the principal square root and names the negative square root. The symbol ± indicates both. The number inside is the radicand . * Numbers with rational square roots are perfect squares. The 2 is called the index. instead of 2 . The Ex.1 Simplify each expression. 9 a) 25 b) 25 d) *The 49 e) c) 1 16 f) 64 0 negative number results in an imaginary number. Rational number: can be expressed as a ratio of two integers *Some square roots are rational numbers and some are irrational numbers. (integers, fractions, ending/repeating decimals) Ex.2 Tell whether each is rational or irrational. 1 a) 144 b) c) 6.25 5 1 9 d) List the first 12 perfect square natural numbers. e) f) 7 What if we do not have a perfect square? We can estimate a square root by using the two nearest perfect squares. Ex.3 What two whole numbers is the root closest to? a) 2 ≈ 1.41 b) 3 ≈ 1.73 c) 14 ≈ d) 53 ≈ Ex.4 Using a calculator, approximate the square root to the nearest hundredth. a) 17.81 b) 28.34 Summary: Ex.5 The formula d x 2 (3x)2 gives the length of the diagonal of a rectangular field that has a length three times its width x. Find the length of the diagonal if x = 8 ft.