Name: ______________________________ Period: ___________ Exponential Functions—Basics and Applications According to http://www.purplemath.com/modules/expofcns.htm : “Exponential functions look somewhat similar to functions you have seen before, in that they involve exponents, but there is a big difference, in that the variable is now the power, rather than the base. Previously, you have dealt with such functions as f(x) = x2, where the variable x was the base and the number 2 was the power. In the case of exponentials, however, you will be dealing with functions such as g(x) = 2x, where the base is the fixed number and the power is the variable.” Recall the graph of 𝑦 = 𝑥 2 : What do you remember/notice about the graph of quadratics? _____________________________ _____________________________ _____________________________ _____________________________ Compare this to the graph of 𝑦 = 2𝑥 What do you remember/notice about the graph of quadratics (solid line) compared with the exponential (dashed line)? ______________________________ ______________________________ ______________________________ Name: ______________________________ Period: ___________ Evaluate the following exponentials at the given values. Be careful of order of operations! 𝑦 = 3𝑡 ; 𝑤ℎ𝑒𝑛 𝑡 = 0 𝐴𝑁𝐷 𝑤ℎ𝑒𝑛 𝑡 = 4 𝑦 = 2.5𝑥 ; 𝑤ℎ𝑒𝑛 𝑥 = 3 𝑦 = 100(2.5)𝑥 ; 𝑤ℎ𝑒𝑛 𝑥 = −1 𝑎𝑛𝑑 𝑤ℎ𝑒𝑛 𝑥 = 3 Name: ______________________________ Period: ___________ Exponentials have many applications to real world situations. Can you think of times when you have heard about exponential growth and/or exponential decay? ________________________________________________________________________________ _________________________________________________________________________________________ Let’s try a few examples of application problems using exponentials: Ten grams of Carbon 14 is stored in a container. The amount C (in grams) of Carbon 14 present after t years can be modeled by C = 10(0.99987) t. How much is present after 1000 years? The population of Winnemucca, Nevada, can be modeled by P=6191(1.04) t T is the number of years since 1990. What was the population in 1990? When was the population about 7833? Hint: use the graphing calculator. Try this window: X min =0 X max =10 Y min = 7500 Y max = 8000 Name: ______________________________ Period: ___________ Comparing linear growth to exponential growth: Suppose you receive $150 for your birthday. You are considering two options for investing your money. Option 1: You will put your money in a back account paying interest according to the equation 𝑦 = 𝐵(1.05)𝑡 where y is the amount saved, B is the initial balance, and t is the time in months. You will not add to the account, but you will not make any withdrawals. Option 2: You will hide the $150 in your mattress. Each month you will add $7.50. You will not make any withdrawals. Write the two equations that represent these options here: When would the balance be the same for both options?