1. Exponential Functions Section 3.1

2. What are Exponential Functions?

3. Why study exponential functions? Many real-life situations can be described using exponential functions, including – Population Growth – Growth of epidemics – Radioactive decay – Compound Interest

4. Definition of Exponential Function The exponential function f with base a is defined by f(x) = a x or y = a x where a is a positive number other and 1 (a>0 and a ≠ 1) and x is any real number.

5. Exponential Function f(x) = a x Domain: (-∞, ∞) Range (0, ∞) y-intercept: (0, 1) NO zero (has a horizontal asymptote at y=0) Increasing (-∞, ∞) No relative minimum or maximum Neither even nor odd Continuous Has an inverse (logarithm) Xy=f(x) -2 0 1 2

6. f(x) = a (bx-c) + d Transformations learned in Chapter 1 still apply Parent is exponential function with base a Vertical translation –”d” Horizontal translation –”bx-c=0” Reflection on x-axis – “sign of a” Reflection on y-axis-”sign of b” Vertical Stretch or Shrink – “numeric value of a” EXAMPLES

7. Applications

8. Example 1: You take out a loan of $30,000 to buy a new car. The bank loans you the money at 7.5% annual interest for 5 years compounded monthly.

9. Applications Example 2 You deposit $1 into an account paying 100% interest compounded: a)Yearly b) semiannually c) quarterly d) monthly e) weekly f) daily g) hourly h) by the minute i) by the second j) “continuously” PrnA 1112annually 1122.25semiannually 1142.44140625quarterly 11122.61303529monthly 11522.69259695weekly 113652.71456748daily 1187602.71812669hourly 115256002.71827924minute 11315360002.71828178second

10. “e” ---Natural number An irrational number (lots of decimal places) Denoted by “e” in honor of Leonard Euler As n→∞, the approximate value of “e” to nine decimal places is e ≈ 2.718281827…….

11. Applications

12. Example 3: You invest $5000 for 10 years at an interest rate of 6.5%. If continuous compounding occurs, how much money will you have in 10 years?

13. Applications

15. Assignment Page 396 #25-31 odd, 35-45 odd, 53-55 odd, 65-67 odd, 73