1. 3.1 Exponential and Logistic Functions

2. Exponential functions Let a and b real number constants. An exponential function in x is a function that can be written in the form: f(x) = a  b x Where a is nonzero, b is positive and b ≠ 1. a (the constant) is the initial value of f (the value at x = 0) b is the base

3. Identifying exponential functions Computing Exact Value y = x 8 y = 5 x y = x √x f(x) = 3  5 x f(x) = -2  3 x f(x) = 2 x x = 0 x = 1/3 x = -3/2

4. Determine a formula for the exponential function: g(x) = h(x) =

5. How an exponential functions changes For any exponential function f(x) = a  b x and any real number x, f(x +1) = b  f(x) If a > 0 and b > 1, the function f is increasing and is an exponential growth function. The base b is its growth factor If a > 0 and b < 1, the function f is decreasing and is an exponential decay function. The base b is its decay factor.

6. Exponential Functions f(x) = b x Domain: Range: Continuity: Symmetry: Bounded? Extrema: Horizontal Asymptote: Vertical Asymptote: End Behavior for function #1 End Behavior for function #2 f(x) = b x for b > 1 f(x) = b x for 0 < b < 1

7. Transforming exponential functions g(x) = 2 x-1 h(x) = 2 -x k(x) = 3  2 x

8. f(x) = e x Domain: Range: Continuity: Symmetry: Bounded? Extrema: Horizontal Asymptote: Vertical Asymptote: End Behavior: Definition:

9. Exponential Functions and the Base e Any exponential function f(x) = a  b x can be rewritten as f(x) = a  e kx for an appropriately chosen real number constant k. If a > 0 and k > 0, then f(x) = a  e kx is an exponential growth function. If a > 0 and k < 0, then f(x) = a  e kx is an exponential decay function.

10. Exponential function transformations g(x) = e 2x h(x) = e -x k(x) = 3e x

11. Logistic Growth Functions End Behavior: limit to growth If b < 1: growth If b > 1: decay Horizontal Asymptotes: y= 0 and y = c or

12. The Logistic Function Domain: Range: Continuity: Symmetry: Bounded? Extrema: Horizontal Asymptote: Vertical Asymptote: End Behavior

13. Graph the function and find the y-intercept and horizontal asymptotes. f(x) = 16 1 + 3 e -2x Y-intercept: Horizontal Asymptotes:

14. The population of North Carolina can be modeled by: P(t) = 12.79 1 + 2.402e -0.0309x When was the population of North Carolina 10 million? P: Population in Millions t: Number of years since 1900

15. Population Models: Assuming the growth is exponential, when will the population of Austin surpass 800,000 people?

16. Don’t forget your Homework ! Pg. 286-288 (2,6,8, 11, 12, 14,20, 24,32,34,42,56) Define Terms (Entire Sheet due Friday)