1. Exponential Functions Section 1

2. Exponential Function f(x) = a x, a > 0, a ≠ 1 The base is a constant and the exponent is a variable, unlike a power function.

3. For an exponential function f(x) = a x, a > 0, a ≠ 1, if x is any real number, then

4. Page 502 21.

5. Properties of the Exponential Function when a > 1 Domain: All reals Range: Positive reals No x-intercepts y-intercept: 1 x-axis is a horizontal asymptote as x  -∞ Increasing function One-to-one Graph contains the points (0, 1), (1, a), and (-1, 1/a) Graph is smooth and continuous, with no corners or gaps

6. Example: Graph the following equations on your calculator: y 1 = 2 x y 2 = 3 x y 3 = 5 x For what values of x is 2 x < 3 x < 5 x ? For what values of x is 2 x > 3 x > 5 x ? For what values of x is 2 x = 3 x = 5 x ?

7. Properties of the Exponential Function when 0 < a < 1 Domain: All reals Range: Positive reals No x-intercepts y-intercept: 1 x-axis is a horizontal asymptote as x  ∞ Decreasing function One-to-one Graph contains the points (0, 1), (1, a), and (- 1, 1/a) Graph is smooth and continuous, with no corners or gaps

8. Example: Graph the following equations on your calculator: y 1 = (1/2) x y 2 = (1/3) x y 3 = (1/5) x For what values of x is (1/2) x < (1/3) x < (1/5) x ? For what values of x is (1/2) x > (1/3) x > (1/5) x ? For what values of x is (1/2) x = (1/3) x = (1/5) x ?

9. Graphing Using Transformations Page 503 37.

10. Pages 502-504 (22-44 even, 75, 77, 79, 85)

11. The number e is defined as the number that the expression approaches as n  ∞

12. Page 503 #45

13. Solving Exponential Equations If a u = a v, then u = v

14. Page 503 53.55.

15. Page 503 61.63.

16. Page 503 67.69.

17. Pages 503-504 (46-70 even)