Properties of Exponentials Functions. What you’ll learn To explore the properties of functions of the form To graph exponential functions that have base e. Vocabulary Natural base exponential function, Continuously compounded interest

You can apply the four types of transformations Take a note: You can apply the four types of transformations -stretches, compressions, reflections, and translations- to exponential functions. The factor a in can stretch or compress and possible reflects the graph of the parent function For example: x -2 -1 1 2

How does the graph of compare to the graph of the parent function? x Problem 1: Graphing How does the graph of compare to the graph of the parent function? x -2 -1 1 2

Your turn How does the graph of compare to the graph of the parent function? Answer: Reflects across the x-axis; compress by a factor of 0.5

Problem 2 Translating the Parent Function How does the graph of each function compare to the graph of the parent function? Graph then translate 4 units to the right Make the table of values x -1 ½=0.5 1 2 4 3 8

x Graph then translate 10 units up. (x,y) Make a table of values -1 1 40 (-1,40) (-1,50) 20 (0,20) (0,30) 1 10 (1,10) (1,20) 2 5 (2,5) (2,15) 3 2.5 (3,2.5) (3,12.5)

Your turn How does the graph of each function compare to the graph of the parent function? Translate 2 units to the left; the y-intercepts becomes 16 Answers: b) Stretch the graph of by a factor of 5 And translate the graph of 5 units.

Problem 3: Using an exponential Model The best temperature to brew coffee is between 1950F and 2050F. Coffee is cool enough to drink at 1850F. The table 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? Use an exponential model. The room temperature is 68 degrees Step 1: plot the date to determine if an exponential model is realistic. STAT then 1 then ENTER plot the data in L1 and L2. Time Temp 203 5 177 10 153 15 137 20 121 25 111 30 104 Step 2: the graphing calculator model assumes the asymptote is y=0. Since room temperature is about 680F, Subtract 68 from each temperature value. Calculate the third list by letting L3=L2-68 Step 3: Use the ExpReg L1,L3 (from STAT then to the right to CALC then 0 then ENTER) function on the transformed data to find en exponential model. Step 4: Translate vertically by 68 units to model the original data. Use the model to find how long it takes to the coffee to cool to 1850F.Use the y function and plug it in.

Your turn Use the exponential model. How long does it take for the coffee to reach a temperature of 100 degree F. Answer: About 31.9 min(go to the table and look for it) b) In Problem 3, would the model of the exponential Data be useful if you did not translate the data by 68 units? Explain Answer No; a hot coffee cannot cool below room temperature. So, to use exponential data, it is important to translate the data by 68 units

Take a note Read your note about “e”

Amount in account at time t Problem 4: Evaluating How can you use a graphing calculator to evaluate ? In the lesson before we learned the interest that was Compound annually. The formula for continuously compounded interest uses the number e. Interest rate annual Amount in account at time t Time in years Principal

Problem 4: Continuously Compounded Interest Suppose you won a contest at the start of 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? Your turn: About how much will be in the account After 4 years of high school? Answer $4475

Classwork odd Homework even TB pgs 447-449 exercises 7-42