2. Exponential Functions Brought to you by Tutorial Services The Math Center

3. Exponential Functions Def: An exponential function with base a is a function of the form f(x) = a x, where a and x are real numbers such that a > 0 and a ≠ 1. i.e. The base is constant and the exponent is a variable.

4. Properties of Exponential Functions (1) f(x) is increasing for a > 1 and decreasing for 0 < a < 1. (2) The y-intercept is at (0,1). (3) The x-axis is a horizontal asymptote. (4) Has domain (-∞, ∞) and range (0, ∞). (5) f(x) is a one-to-one function.

5. Exponential Family of Functions (Graphing) y = a bx+c + d (1)By Table…(pick a number according to properties listed above, use a calculator if necessary) (2) By Transformation (translating, stretching, shrinking and reflecting)

6. Example: the graph of y = 2 x Notice the y-intercept is at (0,1) The x-axis is a horizontal Asymptote, i.e. the line which the graph of y = 2 x approaches as x→ - but never crosses. Graphing ∞

7. Graph of Graphing

8. Translate (i.e. shifted) 3 units to the left The domain is (-∞,∞). Graphing

9. Logarithmic Functions Def: For a > 0 and a ≠ 1, the logarithmic function with base a is denoted f(x) = log a (x), where y = log a (x) if and only if a y = x. i.e. instead of f -1 (x) we use log a (x) as the inverse of an exponential function

10. Two Popular Bases (1)Common log…base 10 (log) (2)Natural log…base e (ln) Properties of Logarithmic Functions 1)f(x) is increasing for a >1 and decreasing for 0<a< 1 2)The x-intercept is at (1,0) 3)The y-axis is a vertical asymptote 4)Has Domain (0,∞) and Range (-∞,∞) 5) f(x) is a one-to-one function

11. Logarithmic Family of Functions (Graphing) Graphing Logarithms 1.By Setting up table 2.By Transformation

12. The common Log is base 10, when the base a is not written, it is understood to be base 10. The x-intercept (i.e. zero or root) is (1,0) The vertical asymptote is y-axis Example y = log (x)

13. Graph of Logarithmic Functions

14. The exponential function e x and Natural Log (Ln) are inverses of each other, notice the reflection over the diagonal line y = x Note: e ≈ 2.71828… Note: Ln rises more rapidly than the common log v.s Exponential Functions vs Logarithmic Functions

15. Shifted one points to the right, (x-1) New vertical asymptote at x = 1 Zero at (2, 0) Domain is from (-∞,∞) Example y = log (x – 1)

16. Graph of a Logarithmic Transformation y = log (x - 1)

17. Exponential and Logarithmic Functions Links Exponential Functions Handout Algebra and Logarithmic Functions HandoutAlgebra and Logarithmic Functions Handout Exponential Functions Quiz