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2 nd period: Come in and sit down in your usual seat with your tracking sheet and homework out. As soon as the bell rings Ill 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?

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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?

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Weve used linear and polynomial models to help us model real life situations.

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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…

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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)

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An exponential function with base b has the form f(x) = ab x 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) = 2 x G(x) = 10 -x

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Lets compare the two: Exponential:Polynomial: f(x) = 4 x f(x) = 5x 3 With polynomial functions, the variable was in the BASE. With exponential functions, the variable is in the exponent.

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

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

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A line that a graph approaches but never actually touches. A couple of examples:

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Domain: Range: Intercepts: Asymptotes: End behavior: Increasing or decreasing? x -20123 F(x)

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Domain: Range: Intercepts: Asymptotes: End behavior: Increasing or decreasing? x -20123 F(x)

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Exponential Growth Exponential Decay

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So far, weve 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: Its named for a Swiss mathematician named Leonhard Euler. (pronounced OILER)

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x -20123 F(x) Domain: Range: Intercepts: Asymptotes: End behavior: Increasing or decreasing?

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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.

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None!

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1. Given the function f(x) = 3e x : 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 -20123 F(x)

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8-3: The Number ‘e’ (Day 1) Objective (Ca. Standard 12): Students know the laws of fractional exponents, understand exponential function, and use these.

8-3: The Number ‘e’ (Day 1) Objective (Ca. Standard 12): Students know the laws of fractional exponents, understand exponential function, and use these.

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