Exponential Functions

1. Exponential Functions Section 3.1

2. What are Exponential Functions? • Exponential functions are functions whose equations contain a variable in the exponent • Categorized as Transcendental (Non-Algebraic) • Examples: f(x) = 2x g(x)=3x+1 h(x) = ( )x-1

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 ais defined by f(x) = axor y = ax where a is a positive number other and 1 (a>0 and a ≠ 1) and x is any real number.

5. Exponential Functionf(x) = ax • 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)

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 • Compound Interest: A • A = total (final) amount owed/earned • P=principal (initial amount borrowed/deposited) • r= annual interest rate (%) must convert to decimal • t= number of years • n= number of compoundings per year (“ly”)

8. Applications • 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 yearscompounded monthly.

9. Applications • Example 2 You deposit $1 into an account paying 100% interest compounded: • Yearly b) semiannually c) quarterly d) monthly e) weekly f) daily g) hourly h) by the minute i) by the second j) “continuously”

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 • Compound Interest for continuous compounding : A • A = total (final) amount owed/earned • P=principal (initial amount borrowed/deposited) • r= annual interest rate (%) must convert to decimal • t= number of years

12. Applications • 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 • Example 4 (Medicine): The radioactive substance iodine-131 is used in measuring heart, liver, and thyroid activity. The quantity, Q (in grams), remaining t days after the element is purchased is given by the equation: Calculate the amount remaining after 24 days.

14. Applications • Example 5: A baby that weighs 6.375 pounds at birth may increase her weight by 11% per month. Use the function: where K is the initial birth weight, r is percent of increase (in decimal form), and t is time in months. How much would the baby weigh at 6 months?

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