1. Exponential and Logistic Functions
Objective: Students will be able to solve and graph exponential functions and apply to real world situations

2. Exponential Functions
Continuous for all real numbers
How do you know if a function is exponential or not?

3. Example
Are the following exponential functions

4. Basic Properties
What does the exponent tell us Repeated multiplication What if exponent is zero Answer will always be 1 What if exponent is negative Turns into a fraction What if it is a fraction Root is denom and power is exponent

5. Exponential Function From Table
Look for constant multiplication between answers from consecutive x values Function evaluated at zero will tell you the x value The constant multiplication value will be the base

6. Example G(x)=4=a Constant multiplication is 3 so 3 is the base
h(0)=8=a
Constant multiplication is ¼ so ¼ is the base

7. Growth and Decay

8. Exploration
Work on the exploration on page 279 in the book, it works on graphs of exponential patterns, key is what do you notice 1a The a value of 1, or point (0,1) 1b D all reals, Range 0 and all positive reals, Continuous, No Symmetry, No extremes, Bounded below by y=0, also an asymptote, Lim as x infinite is infinite, lim as x  -infinite is 0 2a Same as above the a value 2b same as above but decreasing, and limits the other way

9. Transforming exponential Functions
What happens when a changes a>1 vertically stretches - skinny a<1 vertically shrinks - fat What happens when you add or subtract a number to the exponent - Translates graph left and right What happens when you add or subtract a number to the function - moves the graph up or down What if a is negative - Reflex graph across x-axis What if exponent is negative - Reflect graph across y-axis

10. Natural Base e
Basic natural growth function Looked at in unit 1.3

11. Exponential Functions and the Base e

12. Exploration 2
Work through exploration 2 on page Graph 2 k=0.7 3 k=0.693

13. Transformation of Natural Log
What happens when you multiply the x by a constant - horizontal stretch or shrink What happens when you put a negative in front of x - reflect across y What happens with a number in front of e - vertically stretch or shrink What happens with a negative in front of e - reflect across x

14. Logistic Function

15. Graphing Logistic
Find y-intercept and horizontal aysmptote Y-int function evaluated at zero Asymptote – numerator is one and zero because denom is larger than numerator

16. Examples

17. Example

18. Example

19. Homework
Pg , 11, 13,14,15,25-30 a only, ,51 Honors 67