lesson+7.2

Lesson 7.2: Properties of Exponential Functions

Aim: 1. What are the properties of an exponential graph? 2. How can we perform transformations on a parent function? 3. How can we determine the exponential regression from a set of data? 4. Knowing the exponential regression, how can we interpolate or extrapolate data? 5. What is the natural base exponential function and how can we use apply it to exponential growth or decay functions?

Before you begin watching a video of graphing transformations of exponential functions, copy down these questions that can be answered after watching the video
 * 1) What does the parent function, f(x) = b^x, look like?
 * 2) When /a/ > 0, what happens? When 00? How does the graph change when h<0? What type of transformations are these?
 * 4) How does the graph change when k>0? How does the graph change when k<0? What type of transformations are these?

Here is a video on how the transformations work.

Be sure to copy down the information on how to transform an exponential function from its parent function. media type="youtube" key="jC21l8nG7Aw" width="425" height="350"

Now, graph a few of your own. Remember, sketch the parent function first, then finish with the transformations. 1. y=2^(x-4) 2. y=3*2^x 3. y=20(1/2)^x + 10

Now we will use the graphing calculator to apply exponential regressions to problems. We have looked at other nonlinear regressions prior to this. Here is a video on how to use the graphing calculator to determine exponential regressions.

media type="youtube" key="IE-FdDsJcPw" width="425" height="350"

Here is a problem for you to solve: The best temperature to brew coffee is between 195 degree F and 205 degree F. Coffee is cool enough to drink at 185 degree F. The table below shows temperature readings from a sample cup of coffee. how long does it take for a cup of coffee to be cool enough to drink if the room temperature is 68 degrees F? Use you graphing calculator to determine the exponential regression.

Answer: y=134.5(0.956)^x + 68, it will take 3.1 minutes to reach 185 degrees F)
 * Time (min) || Temp (F) ||
 * 0 || 203 ||
 * 5 || 177 ||
 * 10 || 153 ||
 * 15 || 137 ||
 * 20 || 121 ||
 * 25 || 111 ||
 * 30 || 104 ||

Here is the last thing you need to do... there is a special number that is related to exponents and to logarithms. It is called is called the natural base. Its symbol is e. It is a naturally occurring number in logarithms like pi is in circles. We use this number in population problems and continuously compounded interest problems.

The formula: A(t) = P//e//^(rt); A is the ending amount, P is the beginning amount, r is the rate as a percent, and t is the time in years.

Notice this formula is just a little different than the one we had in the last lesson. How is interest compounded continuously different than the interest we looked at yesterday?

Here is a video that demonstrates how to determine interest compounded continuously:

media type="youtube" key="LaN7n4sefH8" width="425" height="350"

And...here is a problem for you to solve:

Suppose you won a scholarship at the start of the 5th grade that deposited $3000 in an account that pays 5% annual interest //compounded continuously//. How much will you have in the account when you enter high school 4 years later? Express the answer to the nearest dollar.

(Ans: $3664)