Warm Up Find a partner at your table. One of you is Partner A, the other is Partner B. Each of you will answer the first problem. Stop and compare answers. They should be the SAME answer! If not, work together to figure out what mistake was made. Move on to the next problem. Work at the same pace, do not get ahead of your partner.

Exponential Equations 𝒇(𝒙)=𝒂 𝒃 𝒙 a= starting value b= multiplier x= time 𝑨 𝒕 = 𝑨 𝟎 (𝟏+𝒓) 𝒕 Exponential growth and decay- given a rate 𝑨 𝟎 = the initial amount, r = the rate as a decimal, t = time Exponential growth and decay- given an outcome and the time to achieve it 𝑨 𝒕 = 𝑨 𝟎 (𝒃) 𝒕 𝒌 or 𝑨 𝟎 𝒃 𝟏 𝒌 𝒕 𝑨 𝟎 = the initial amount, b= multiplier t = time k = the time needed to multiply 𝑨 𝟎 by b

𝑨 𝒕 = 𝑨 𝟎 (𝒃) 𝒕 𝒌 or 𝑨 𝟎 𝒃 𝟏 𝒌 𝒕 𝑨 𝒕 = 𝑨 𝟎 ∙ 𝟐 𝒕 𝟏𝟐 P 𝒕 = 𝑨 𝟎 ∙ 𝟒 𝒕 𝟕 A bank advertises that if you open a savings account, you can double your money in 12 years. Express A(t), the amount of money after t years. 𝑨 𝒕 = 𝑨 𝟎 ∙ 𝟐 𝒕 𝟏𝟐 A certain type of plant will quadruple its population in 7 years. P 𝒕 = 𝑨 𝟎 ∙ 𝟒 𝒕 𝟕 A certain bacteria will decrease by a third every 6 hours. P 𝒕 = 𝑨 𝟎 ∙ 𝟐 𝟑 𝒕 𝟔

𝑨 𝒕 = 𝑨 𝟎 (𝒃) 𝒕 𝒌 or 𝑨 𝟎 𝒃 𝟏 𝒌 𝒕 $𝟏𝟎𝟎𝟎 𝟏𝟎 𝒚𝒆𝒂𝒓𝒔 𝟗𝟎 𝟗𝟎𝟎𝟎 Suppose you invest some money that grows to the amount 𝑨 𝒕 =𝟏𝟎𝟎𝟎∙ 𝟐 𝒕 𝟏𝟎 in t years. How much did you invest? How long does it take to double your money? $𝟏𝟎𝟎𝟎 𝟏𝟎 𝒚𝒆𝒂𝒓𝒔 Suppose that t hours from now the population of a bacteria colony is given by P 𝒕 =𝟗𝟎∙ 𝟏𝟎𝟎 𝒕 𝟖 What is the population when t = 0? What will the population be in 8 hours? 𝟗𝟎 𝟗𝟎𝟎𝟎

Exponential Equations Half-Life 𝑨 𝟎 = the initial amount, t = time k = the time needed to multiply 𝑨 𝟎 by 𝟏 𝟐 (the half-life)

The Rule of 72 If a quantity is growing at m% per year (or month), then the doubling time is approximately ( 72 𝑚 ) years (or months.) About how long will it take for a savings account of $1000 to grow to $2000 if it earns a 9% annual rate of interest? 72 9 =8 𝑦𝑒𝑎𝑟𝑠

Exponential Equation from Two Points 𝒙 𝒚 𝒙 𝒚 0,3 𝑎𝑛𝑑 (2,12) 𝑦=𝑎 𝑏 𝑥 3=𝑎 𝑏 0 12=𝑎 𝑏 2 𝒇 𝒙 =𝟑 ∙𝟐 𝒙 3=𝑎 12=3 𝑏 2 4= 𝑏 2 𝒇 −𝟐 =𝟑 ∙𝟐 −𝟐 𝒇 −𝟐 = 𝟑 𝟒 b=2

2. 𝑓 1 =25, 𝑓 6 =9 Find an exponential function with the given values: 1. 𝑓 0 =5, 𝑓 3 =50 2. 𝑓 1 =25, 𝑓 6 =9 0,5 𝑎𝑛𝑑 (3, 50) 50=𝑎 𝑏 3 5=𝑎 𝑏 0 𝒇 𝒙 =𝒂 𝒃 𝒙 5=𝑎 50=5 𝑏 3 𝒇 𝒙 =𝟓 𝟑 𝟏𝟎 𝒙 10= 𝑏 3 b= 3 10 1,25 𝑎𝑛𝑑 (6, 9) 9=𝑎 𝑏 6 25=𝑎 𝑏 1 𝒇 𝒙 = 𝟐𝟓 𝒃 .𝟖𝟏𝟓 𝒙 9= 25 𝑏 𝑏 6 25=𝑎𝑏 25 𝑏 =𝑎 9=25 𝑏 5 𝒇 𝒙 =𝟑𝟎.𝟕 .𝟖𝟏𝟓 𝒙 b=0.815

Homework Page 183 #1-17 Odds Page 178 #25,27,38,46,51*