WARM UP 2nd period: Come in and sit down in your usual seat with

your tracking sheet and homework out. As soon as the bell rings I’ll give you your test back and you will have 10 minutes to finish it. If you do not finish in that time, you must come back after school to finish it.

If you are finished, see if you can figure out the problem below:

There are three switches downstairs. Each corresponds to one of the three light bulbs in the attic. You can turn the switches on and off and leave them in any position. How would you identify which switch corresponds to which light bulb, if you are only allowed one trip upstairs?

WARM UP Come in and have your tracking sheet and

homework out. Answer the following questions:

1. Solve for x: (a – bx)4 = c

2. Find (g(f(x)) if f(x) = 2x + 3 and g(x) = 9x + 4

3. Find f(g(8)) for the above functions.

4. How do you know if a function is increasing or decreasing when given its graph?

UNIT 5! EXPONENTIAL AND LOGARITHMIC FUNCTIONS

SO FAR….

We’ve used linear and polynomial models to help us model real life situations.

NOW…

We are going to use exponential and logarithmic models to do the same thing.

These show up WAY more often in real life than polynomial functions do…

A FEW PLACES YOU MIGHT SEE EXPONENTIAL FUNCTIONS:

At the bank (compounded interest)

In science (scientists figure out how old fossils are by determining their half life)

In medicine (doctors and sociologists use exponential functions to study how fast diseases spread)

EXPONENTIAL FUNCTION (THE FORMAL DEFINITION)

An exponential function with base b has the form f(x) = abx where x is any real number and a and b are real number constants such that a does NOT equal 0, b is positive, and b does NOT equal 1.

Some examples: F(x) = 2x

G(x) = 10-x

HOW ARE EXPONENTIAL FUNCTIONS DIFFERENT FROM POLYNOMIAL FUNCTIONS?

Let’s compare the two:

Exponential: Polynomial:f(x) = 4x f(x) = 5x3

With polynomial functions, the variable was in the BASE. With exponential functions, the variable is in the exponent.

BEFORE WE GET INTO THE FUNCTIONS THEMSELVES, LET’S REVIEW SOME OF THE RULES OF EXPONENTS

HOW FAST CAN YOU GO?

Your goal: simplify as many of the expressions involving exponents below as you can within the 2 minute time limit.

Ready… Set…

SWITCH WITH YOUR NEIGHBOR AND CHECK THEIR ANSWERS BELOW:

NOW THAT WE’RE COMFORTABLE WORKING WITH EXPONENTS, LET’S TALK ABOUT GRAPHS

What’s important: Domain Range X and Y intercepts Asymptotes (what are these?) End Behavior

SIDE NOTE ON ASYMPTOTES

A line that a graph approaches but never actually touches.

A couple of examples:

WHAT IS THE ASYMPTOTE OF THIS GRAPH?

F(X) = 3X

Domain: Range: Intercepts: Asymptotes: End behavior: Increasing or

decreasing?

x -2 -1 0 1 2 3

F(x)

F(X) = 2-X

Domain: Range: Intercepts: Asymptotes: End behavior: Increasing or

decreasing?

x -2 -1 0 1 2 3

F(x)

THESE TWO GRAPHS SHOWED US THE TWO BASIC TYPES OF EXPONENTIAL FUNCTIONS

Exponential Growth Exponential Decay

NATURAL LOG So far, we’ve looked at bases that are regular

numbers: 2, 3, 10, etc. But actually, for most real-world applications,

the base of an exponential function is actually e, the natural log.

This is an irrational number that shows up all the time in the real world.

Fun fact: It’s named for a Swiss mathematician named Leonhard Euler. (pronounced OILER)

CAN YOU FILL IN THE TABLE OF VALUES AND GRAPH THIS ONE YOURSELF?

x -2 -1 0 1 2 3

F(x) Domain: Range: Intercepts: Asymptotes: End behavior: Increasing or

decreasing?

PRACTICE

There are 10 exponential functions around the room. Complete them in any order you wish, but use them to fill in the back of your guided notes.

You only need to do ____ out of 10.

HOMEWORK

None!

EXIT TICKET

1. Given the function f(x) = 3ex: a. Fill in the following table of values:

b. Sketch the graph of the function.

c. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing.

x -2 0 1 2 3

F(x)