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

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

Example Are the following exponential functions

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

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

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

Growth and Decay

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

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

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

Exponential Functions and the Base e

Exploration 2 Work through exploration 2 on page 282 1 Graph 2 k=0.7 3 k=0.693

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

Logistic Function

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

Examples

Example

Example

Homework Pg 286 1-6, 11, 13,14,15,25-30 a only, 31-34 41,51 Honors 67