4.1 exponential functions 2
- 1. Exponential Functions Section 4.1 JMerrill, 2005 Revised 2008
- 2. Definition of Exponential Functions The exponential function f with a base b is defined by f(x) = bx where b is a positive constant other than 1 (b > 0, and b ≠ 1) and x is any real number. So, f(x) = 2x , looks like:
- 3. Graphing Exponential Functions Four exponential functions have been graphed. Compare the graphs of functions where b > 1 to those where b < 1 2x y = 7x y = 1 2 x y = 1 7 x y =
- 4. Graphing Exponential Functions So, when b > 1, f(x) has a graph that goes up to the right and is an increasing function. When 0 < b < 1, f(x) has a graph that goes down to the right and is a decreasing function.
- 5. Characteristics The domain of f(x) = bx consists of all real numbers (-∞, ∞). The range of f(x) = bx consists of all positive real numbers (0, ∞). The graphs of all exponential functions pass through the point (0,1). This is because f(o) = b0 = 1 (b≠o). The graph of f(x) = bx approaches but does not cross the x-axis. The x-axis is a horizontal asymptote. f(x) = bx is one-to-one and has an inverse that is a function.
- 6. Transformations Vertical translation f(x) = bx + c Shifts the graph up if c > 0 Shifts the graph down if c < 0 2x y = 2 3x y = + 2 4x y = −
- 7. Transformations Horizontal translation: g(x)=bx+c Shifts the graph to the left if c > 0 Shifts the graph to the right if c < 0 2x y = ( 3) 2 x y + = ( 4) 2 x y − =
- 8. Transformations Reflecting g(x) = -bx reflects the graph about the x-axis. g(x) = b-x reflects the graph about the y-axis. 2x y = 2x y = − 2 x y − =
- 9. Transformations Vertical stretching or shrinking, f(x)=cbx : Stretches the graph if c > 1 Shrinks the graph if 0 < c < 1 2x y = 4(2 )x y = 1 (2 ) 4 x y =
- 10. Transformations Horizontal stretching or shrinking, f(x)=bcx : Shinks the graph if c > 1 Stretches the graph if 0 < c < 1 2x y = 4(2 )x y = 1 (2 ) 4 x y =
- 11. You Do Graph the function f(x) = 2(x-3) +2 Where is the horizontal asymptote? y = 2
- 12. You Do, Part Deux Graph the function f(x) = 4(x+5) - 3 Where is the horizontal asymptote? y = - 3
- 13. The Number e The number e is known as Euler’s number. Leonard Euler (1700’s) discovered it’s importance. The number e has physical meaning. It occurs naturally in any situation where a quantity increases at a rate proportional to its value, such as a bank account producing interest, or a population increasing as its members reproduce.
- 14. The Number e - Definition An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. It models a variety of situations in which a quantity grows or decays continuously: money, drugs in the body, probabilities, population studies, atmospheric pressure, optics, and even spreading rumors! The number e is defined as the value that approaches as n gets larger and larger. 1 1 n n + ÷
- 15. The Number e - Definition n 1 2 2 2.25 5 2.48832 10 2.59374246 100 2.704813829 1000 2.716923932 10,000 2.718145927 100,000 2.718268237 1,000,000 2.718280469 1,000,000,000 2.718281827 1 1 n n + ÷ 0 1 1 n A n + ÷ 1 , 1 n As n e n →∞ + → ÷ The table shows the values of as n gets increasingly large. n → ∞As , the approximate value of e (to 9 decimal places) is ≈ 2.718281827
- 16. The Number e - Definition For our purposes, we will use e ≈ 2.718. e is 2nd function on the division key on your calculator. y = e 1 1 n y n = + ÷
- 17. The Number e - Definition Since 2 < e < 3, the graph of y = ex is between the graphs of y = 2x and y = 3x ex is the 2nd function on the ln key on your calculator y =e y = 2x y = 3x y = ex
- 18. Natural Base The irrational number e, is called the natural base. The function f(x) = ex is called the natural exponential function.
- 19. Compound Interest The formula for compound interest: ( ) 1 = + ÷ nt r A t P n Where n is the number of times per year interest is being compounded and r is the annual rate.
- 20. Compound Interest - Example Which plan yields the most interest? Plan A: A $1.00 investment with a 7.5% annual rate compounded monthly for 4 years Plan B: A $1.00 investment with a 7.2% annual rate compounded daily for 4 years A: B: 12(4) 0.075 1 1 1.3486 12 + ≈ ÷ 365(4) 0.072 1 1 1.3337 365 + ≈ ÷ $1.35 $1.34
- 21. Interest Compounded Continuously If interest is compounded “all the time” (MUST use the word continuously), we use the formula where P is the initial principle (initial amount) ( ) = rt A t Pe
- 22. ( ) = rt A t Pe If you invest $1.00 at a 7% annual rate that is compounded continuously, how much will you have in 4 years? You will have a whopping $1.32 in 4 years! (.07)(4) 1* 1.3231e ≈
- 23. You Do You decide to invest $8000 for 6 years and have a choice between 2 accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment?
- 24. You Do Answer 1st Plan: 2nd Plan: 0.0685(6) (6) 8000 $12,066.60P e= ≈ 12(6) 0.07 (6) 8000 1 $12,160.84 12 A = + ≈ ÷