Name Date HW Packet Lesson 3 Introduction to Exponential Functions HW Problem 1 In this problem, we look at the characteristics of Linear and Exponential Functions. Complete the table below. Function If Linear, is it Increasing or Decreasing or Constant If Exponential, is it Exponential Growth or Exponential Decay If Linear, state the slope If Exponential, state the base Identify the y-intercept as an ordered pair. Y = 2x + 4 Linear Increasing Slope = 2 y-intercept = (0,4) f(x) = 3(2) x g(x) = -1.5x - 2 p(t) = 100(1.2) x f(c) = 1.8c + 32 g(x) = 1000(0.75) x HW Problem 2 In this problem, we look at the characteristics of exponential functions in more depth. For the following three equations, identify each of the characteristics. Initial Value (a)? f(x) = 125(1.25) x g(x) = 125(0.75) x h(x) = -125(1.25) x Base (b)? Domain? Range? X-intercept? Y-intercept? Horizontal Asymptote? Increasing or Decreasing?
HW Problem 3 In this problem, we look at how the Rate of Change can show what type of function we are dealing with. You need to analyze each set of data and figure out if it is Linear Increasing, Exponential Increasing or Exponential Decreasing. Make sure you state the behavior of the function (i.e.) a) The function has a constant rate of change of b) The function is changing exponentialy by a factor of Write the equation that goes with each data set. y.04.2 1 5 25 125 625 y -1.375 -.5.375 1.25 2.125 3 3.875 y -3-5.5-8 -10.5-13 -15.5-18 y 98.224 99.108 100 100.9 101.81 102.72 103.65 y.111.333 1 3 9 27 81 2
Problem 4 WRITING EXPONENTIAL EQUATIONS/FUNCTIONS The rabbit population in several counties is shown in the following table. Rabbit Population Year Coconino Yavapai Harestew 2006 15000 8000 25000 2007 18000 12800 18750 2008 21600 20480 14063 2009 25920 32768 10547 Assume this growth is exponential. Let t = 0 represent the year 2006 and let a represent the initial population in 2006. Let b represent the ratio in population between the years 2006 and 2007. a) Write the equation of the exponential mathematical model for each situation. Round any decimals to two places. Be sure your final result uses proper function notation. Use C(t) for Coconino, Y(t) for Yavapai and H(t) for Harestew. b) Using these models, forecast the rabbit population in 2012 (to the nearest rabbit) for each county. c) Also using these models, predict which will happen first, 1) The Rabbit Population in Coconino Country reaches 60,000 2) The Rabbit Population in Yavapai Country reaches 340,000 3) The Rabbit Population in Harestew goes below 5000. Explain your reasoning. 3
HW Problem 5 Assume you can invest $1000 at 5% Simple Interest or 4% Compound Interest (Annual). The equation for Simple Interest is modeled by: A = P + Prt. Compound Interest is modeled by A = P(1+r) t. Given that, your two equations are: S(t) = 1000 + 50t C(t) = 1000(1.04) t a) How much does each investment return after 1 year? b) How much does each investment return after 10 years? c) How much does each investment return after 20 years? d) When would the two investments return the same amount? How much would they return (Use your calculator and take advantage of the Calc Intersect function) e) Which investment would you go with? Why? 4
HW Problem 6 Solve each of the following equations Algebraically and using your calculator. Example: You invest $50,000 at 6% interest compounded yearly. It is modeled by the formula f(x) = 50000(1.06) x. Determine how much money you would have after 20, 30 and 40 years. Algebraically: Using your calculator: Enter 50000(1.06)^20 Hit Y= Calc: 160357 Set Y 1 = 50000(1.06)^x Enter 50000(1.06)^30 Hit 2 nd /Window Calc: 287175 Set TblStart = 20 Enter 50000(1.06)^40 Set Tbl = 10 Calc: 514286 Hit 2 nd /Graph Record the results a) f(x) = 50(1.25) x Find f(50), f(55), f(60), f(65), f(70), f(75) b) f(x) = 1000(0.65) x Find f(0), f(20), f(40), f(60), f(80), f(100) 5
HW Problem 7 Solve each of the following equations using your calculator. Example: You invest $50,000 at 6% interest compounded yearly. It is modeled by the formula f(x) = 50000(1.06) x. Determine how long it will take to make $75,000 Using your calculator: Hit Y= Set Y 1 = 50000(1.06)^x Set Y 2 = 75000 Adjust the Window so the intersection of the two lines show up. Hit 2 nd /Trace and use the Calc/Intersect Function Results: You will have $75,000 in 6.9 years. a) f(x) = 50(1.25) x Find x for each of the following: f(x) = 100 f(x) = 200 f(x) = 300 b) f(x) = 1000(0.65) x Find x for each of the following: f(x) = 750 f(x) = 500 f(x) = 250 6