1. Algebra 2 Properties of Exponential Functions Lesson 7-2 Part 2

2. Goals Goal To graph exponential functions that have base e. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary Natural Base Exponential Function Continuously Compounded Interest

4. Essential Question Big Idea: Modeling What is a natural base exponential function?

5. Examine the graph of f(n) = (1 + ) n, as n gets very large. The function has a horizontal asymptote. As n becomes infinitely large, the value of the function approaches approximately 2.7182818…. This number is called e. Like , the constant e is an irrational number. n 1 The Natural Number e

6. Represented by e to honor Euler who discovered the number. An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to 2.72.

7. The number e is called the natural base. The function f (x) = e x is called the natural exponential function. Exponential functions with e as a base have the same properties as the functions you have studied. The graph of f(x) = e x is like other graphs of exponential functions, such as f(x) = 3 x. The domain of f(x) = e x is all real numbers. The range is {y|y > 0}. f (x) = e x f (x) = 2 x f (x) = 3 x (0, 1) (1, 2) 1 2 3 4 (1, e) (1, 3) The Natural Base e

8. f(x) = e x–2 + 1. For the given function. a)State the transformations to the parent function. b)Find the y – intercept. c)Find the horizontal asymptote. d)Domain and range. Example: a)Translated 2 units right, translated 1 unit up b)(0, 1+1/e 2 ) ≈ (0, 1.135) c) y = 1 d)Domain: all real numbers, Range: y > 1

9. f(x) = e x – 3. a)Translated down 3 units b)(0, -2) c) y = -3 d)Domain: all real numbers, Range: y > -3 For the given function. a)State the transformations to the parent function. b)Find the y – intercept. c)Find the horizontal asymptote. d)Domain and range. Your Turn:

10. Domain: Range: y-intercept: H.A.: Domain: Range: y-intercept: H.A.: Domain: Range: y-intercept: H.A.: Transformations Vertical stretch 3. Vertical shift up 2. Reflect over x-axis. Vertical shift down 1. Horizontal shift left 2. Vertical shift up 2 growth/decay?growth decay growth Transformations: All real numbers y > 2 (0, 5) y = 2 All real numbers growth/decay? All real numbers y < -1 y = -1 (0, -2) y > 2 (0, 9.39) y = 2 growth/decay?

11. The formula for continuously compounded interest is A = Pe rt, where A is the total amount, P is the principal, r is the annual interest rate, and t is the time in years. Compound Interest

12. What is the total amount for an investment of $500 invested at 5.25% for 40 years and compounded continuously? The total amount is $4083.08. A = Pe rt Substitute 500 for P, 0.0525 for r, and 40 for t. A = 500e 0.0525(40) Use the e x key on a calculator. A ≈ 4083.08 Example:

13. What is the total amount for an investment of $100 invested at 3.5% for 8 years and compounded continuously? The total amount is $132.31. A = Pe rt Substitute 100 for P, 0.035 for r, and 8 for t. A = 100e 0.035(8) Use the e x key on a calculator. A ≈ 132.31 Your Turn:

14. What is the total amount for an investment of $1000 invested at 7.25% for 15 years and compounded continuously? ≈ $2966.85 Your Turn:

15. Essential Question Big Idea: Modeling What is a natural base exponential function? A natural base exponential function is an exponential function with the irrational base e ≈ 2.72. Use a natural base exponential function to describe continuous growth or decay.

16. Assignment Section 7-2 Part 2, Pg 477 – 479; #1 – 4 all, 6 – 12 even, 16, 22, 24.