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Lesson 7.1 Exponential Models = = Aim: 1. What are the characteristics of an exponential function? 2. How can we graph an exponential function? 3. How can we apply exponential functions to growth and decay problems?

Today we will learn how to graph an exponential functions. You will also learn how to apply exponential equations to solve growth and decay problems.

An exponential function is in the for f(x)=b^x, where b > 0, and b is not equal to 0.
 * To begin, we will graph an exponential function.**

Here are some questions I want you to answer as you watch the videos:

1. What are the stipulations on b? 2. What happens to the graph when b=1? 3. What is an asymptote? 4. What is the y-intercept of any basic exponential function? 5. How do the graphs compare when b>1 and 0,b,1? 6. What is another way to write the equation for the function f(x)=3^(-x)?

Let's watch a video on how to graph an exponential function.

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Graph these two functions in your notebooks. You may use your calculator to find the values. Be sure to check your graph with the graphing calculator.

a. y=4^x b. g(x) = (1/3)^x


 * For this part we will apply the exponential function to problems involving growth and decay**

The equation we use for exponential growth is A(t) = a(1 + r)^t: A(t) = ending amount a = inital amount r = rate of growth t = number of time periods - usually in years

The equation we use for exponential decay is A(t) = a(1 - r)^t A(t) = ending amount a = initial amoung r = rate of growth t = number of years - usually in years

What is the difference between the two formulas? How will you remember which one to use for growth and which one you will use for decay?

Here is a video on the use with **exponential growth**. There may be some differences in the formula - that is ok.

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All you need to do is determine which formula you need to use, then substitute to solve.

Try these problems for exponential growth or exponential decay
 * 1) You invested $1000 in a savings account at the end of 6th grade. The account pays 5% annual interest. How much money will be in the account after six years? (Ans: $1340.10)
 * 2) Suppose you invest $500 in a savings account that pays 3.5% annual interest. How much will be in the account after 5 years? (Ans: $593.84)
 * 3) Suppose you invest $1000 in a savings account that pays 5% annual interest. If you make no additional deposits or withdrawals, how many years will it take for the account to grow to at least $1500? //Hint: Use table feature on your// //graphing calculator.// (Ans: After 9 years)
 * 4) The population of the Siberian lynx in 2003 was 150. Its population in 2004 is 120. Write an exponential equation to represent this. Use your formula to predict the population of the Siberian lynx in the year 2014. ( Ans: y = 150(0.8)^x. The population in 2014 will be approximately 13.)