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Precalculus Lesson 3.1

Precalculus Lesson 3.1. Check: NONE. Warm-up. Find the Domain and identify any asymptotes. Answers. Objective(3.1). Evaluate and graph exponential functions. Use the natural base e and compound interest formulas. The graph of f ( x ) = a x , a > 1.

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Precalculus Lesson 3.1

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  1. PrecalculusLesson 3.1 Check: NONE

  2. Warm-up Find the Domain and identify any asymptotes.

  3. Answers

  4. Objective(3.1) • Evaluate and graph exponential functions. • Use the natural base e and compound interest formulas.

  5. The graph off(x) = ax, a > 1 Graph of Exponential Function (a > 1) y 4 Range: (0, ) (0, 1) x 4 Horizontal Asymptote y = 0 Domain: (–, )

  6. The graph off(x) = ax, 0 < a < 1 Graph of Exponential Function (0 < a < 1) y 4 Range: (0, ) Horizontal Asymptote y = 0 (0, 1) x 4 Domain: (–, )

  7. Exponential Function: An equation in the formf(x) = Cax. If 0 < a < 1 , the graph represents exponential decay If a > 1, the graph represents exponential growth Examples: f(x) = (1/2)x f(x) = 2x Exponential Decay Exponential Growth These graphs “shift” according to changes in their equation...

  8. Take a look at how the following graphs compare to the original graph of f(x) = (1/2)x : Vertical Shift: The graphs of f(x) = Cax + k are shifted vertically by k units. f(x) = (1/2)x f(x) = (1/2)x + 1 f(x) = (1/2)x – 3

  9. Take a look at how the following graphs compare to the original graph of f(x) = (2)x : (3,1) (0,1) (-2,-2) Notice that f(0) = 1 Notice that this graph is shifted 3 units to the right. Notice that this graph is shifted 2 units to the left and 3 units down. f(x) = (2)x f(x) = (2)x – 3 f(x) = (2)x + 2 – 3 Horizontal Shift: The graphs of f(x) = Cax – h are shifted horizontally by h units.

  10. Take a look at how the following graphs compare to the original graph of f(x) = (2)x : (0,1) (0,-1) (-2,-4) Notice that f(0) = 1 This graph is a reflection of f(x) = (2)x . The graph is reflected over the x-axis. Shift the graph of f(x) = (2)x ,2 units to the left. Reflect the graph over the x-axis. Then, shift the graph 3 units down f(x) = (2)x f(x) = –(2)x f(x) = –(2)x + 2 – 3

  11. Example: Sketch the graph off(x) = 2x. Example: Graph f(x) = 2x y 4 2 x –2 2

  12. Example: Sketch the graph ofg(x) = 2x – 1. State the domain and range. Example: Translation of Graph y f(x) = 2x The graph of this function is a vertical translation of the graph of f(x) = 2x downone unit . 4 2 Domain: (–, ) x y = –1 Range: (–1, )

  13. Example: Sketch the graph ofg(x) = 2-x. State the domain and range. Example: Reflection of Graph Also written asg(x)=(1/2)x y f(x) = 2x The graph of this function is a reflection the graph of f(x) = 2x in the y-axis. 4 Domain: (–, ) x –2 2 Range: (0, ) Complete front of notes 3.1

  14. The graph off(x) = ex Graph of Natural Exponential Function f(x) = ex y 6 4 2 x –2 2

  15. The number e The irrational number e, where e 2.718281828… is used in applications involving growth and decay. Using techniques of calculus, it can be shown that Complete back of notes 3.1

  16. Continuous Compounding Formula 6.6 The Natural Base, e Formula A = Pert A: amount of the investment with interest P: principal (initial investment) r: interest rate t: time in years

  17. Example 6.6 The Natural Base, e Continuous Compounding Formula An investment of $7400 at 12% interest is compounded continuously. How much will the investment be worth in 15 years? A = Pert = 7400e0.12(15)  44,767.39

  18. Formula • “n” Compoundings per year n: # of compoundings per year Quarterly: semi-annually: Daily:

  19. “N” Compounding Formula An investment of $7400 at 12% interest is compounded quarterly. How much will the investment be worth in 10 years?

  20. Assignments Classwork: Notes hand-out 3.1 Practice with formulas p. 225 # 61, 65, 71 Homework(3.1) p. 224 #10, 13-16, 22, 24, 28, 38, 42, 60-66 even,72 Quiz 3.1-3.2 Friday

  21. Closure Describe the transformation. f(x) = 4x

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