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Exploring and Graphing Exponential Functions in Mathematical Studies

This section delves into evaluating and graphing exponential functions, a crucial topic in Mathematics for International Students. It covers essential concepts such as the properties of exponential growth, transformation of graphs, and real-world applications. Students will learn how to interpret exponential functions and their significance in various contexts. By mastering these skills, learners will gain a deeper understanding of mathematical concepts and enhance their analytical abilities.

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Exploring and Graphing Exponential Functions in Mathematical Studies

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  1. Material Taken From:Mathematicsfor the international student Mathematical Studies SLMal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark BruceHaese and Haese Publications, 2004
  2. Section 16AB – Evaluating and Graphing Exponential Functions Objectives To graph an exponential function and describe its features. To use exponential functions to model growth and decay.
  3. y= 2x y = 3x y = 10x y = (1.3)x Using Autograph or Calculator Exploration 1 On the same set of axes, graph the following: y = y = y = 0.875x Make observations about: the overall shape of the graph the y-intercept the equation of the horizontal asymptote
  4. y = 2x y = 2x + 1 y = 2x– 2 Using Autograph or Calculator Exploration 2 On the same set of axes, graph the following: Make observations about: the overall shape of the graph the y-intercept the equation of the horizontal asymptote
  5. y = 2x y = 32x y = ½  2x Using Autograph or Calculator Exploration 3 On the same set of axes, graph the following: y = -2x y = -32x y = -½  2x Make observations about: the overall shape of the graph the y-intercept the equation of the horizontal asymptote
  6. y = 2x y = 22x y = 24x Using Autograph or Calculator Exploration 4 On the same set of axes, graph the following: y = 2x Make observations about: the overall shape of the graph the y-intercept the equation of the horizontal asymptote
  7. Summary lambda y = (k)aλx+c, where k, a, λ, c ∊ Q a and λ control = c controls = If k is negative, then the steepness the vertical translation y = c is the equation of the horizontal asymptote the graph is reflected in the x-axis
  8. Summary y = (k)aλx+c, where k, a, λ, c ∊ Q Found on Page 522 of the text.
  9. To graph an exponential function find: the horizontal asymptote the y-intercept two other points (e.g. x = 2 and x = -2)
  10. Example 1 Sketch the graph of y = 2-x - 3
  11. IB Example 2 The graph below shows the curve y = k(2x) + c, where k and c are constants. Find the values of c and k.
  12. IB Example 3 The following diagram shows the graph of y= 3–x + 2. The curve passes through the points (0, a) and (1, b). (a) Find the value of (i) a; (ii) b. (b) Write down the equation of the asymptote to this curve.
  13. Homework 16A, Pg 519 #1ace, 2ace, 3ace, 4ace, 5ace 16B, Pg 520-522 #1abcd, 3ab, 5abcd
  14. Section 16CD – Exponential Growth and Decay Real World Examples: Bacteria Viruses
  15. Electricity – Ohm’s Law Population Air Pressure and Altitude
  16. Temperature and Cooling Radioactive Decay
  17. Depreciation Compound Interest
  18. Consider the rat population growing during the plague: week 0  100 rats each week the population doubles Population after t weeks = 100  2t
  19. What if the rat population only grew by 50% each week? week 0  100 rats Each week the population is 50% more than it was the previous week 150 100 + 50 100  1.5 225 150 + 75 100  1.5  1.5 337.5 225 + 112.5 100  1.5  1.5  1.5 337.5 + 168.75 100  1.5  1.5  1.5  1.5 506.25 Population after t weeks = 100  (1.5)t
  20. What if the rat population only grew by 50% each week? Week 0  100 rats Each week the population is 50% more than it was the previous week Population after t weeks = 100  (1.5)t = 100  (1 + 0.5)t initial amount rate of growth
  21. FA Cup Tournament The competition begins with 256 teams in a single elimination. 128 256  0.5 64 256  0.5  0.5 32 256  0.5  0.5  0.5 256  0.5  0.5  0.5  0.5 16 Teams after x rounds = 256  (0.5)x
  22. FA Cup Tournament The competition begins with 256 teams in a single elimination. Teams after x rounds = 256  (0.5)x = 256  (1 – 0.5)t initial amount rate of decay Find the number of rounds needed until there is one winner.
  23. Summary: Exponential Growth y = a(1 + r)x a is the initial amount r is the growth rate x is the number of time intervals y is the final amount Exponential Decay y = a(1 – r)x r is the decay rate
  24. Summary: Exponential Growth y = a(1 + r)x Exponential Decay y = a(1 – r)x These should seem familiar… geometric sequence compound interest
  25. Example 4: The population size of rabbits on a farm is given, approximately by R = 50(1.07)n where n is the number of weeks after the rabbit farm was established.a) What was the original rabbit population? b) How many rabbits were present after 15 weeks? c) How many rabbits were present after 30 weeks? d) How long would it take for the population to reach 500?
  26. IB Example 5: The value of a car decreases each year. This value can be calculated using the function v= 32 000r t, t 0, 0 < r < 1, where v is the value of the car in USD, t is the number of years after it was first bought and r is a constant. a) (i) Write down the value of the car when it was first bought. (ii) One year later the value of the car was 27 200 USD. Find the value of r. b) Find how many years it will take for the value of the car to be less than 8000 USD.
  27. IB Example 6: The graph below shows the temperature of a liquid as it is cooling. a) Write down the temperature after 5 minutes. b) After how many minutes is the temperature 50°C? The equation of the graph for all positive x can be written in the form y = 100(5–0.02x). c) Calculate the temperature after 80 minutes. d) Write down the equation of the asymptote to the curve. temperature (C°) time (minutes)
  28. Homework 16C,Pg 525 #1-4 16D,Pg 527 #1-3
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