EQ: How do exponential functions behave and why?

Would you rather have $1 million NOW or 1 penny doubled each day for 30 days? Day 30: $5,368, Amount each day = 00.01(2) t

x f(x) = a (b) x Variable is in the exponent Initial Amount (y-intercept) Growth/Decay Factor

Growth b>1Decay b< 1 (and not negative) y = 100(1.3) x y = 100(0.3) x

Growth/Decay Rate b = 1 means b = 100% (stays the same) b – 1 = Growth/Decay RATE Ex) y = 5(1.12) x 1.12 is growth factor 0.12 (or 12%) is growth rate

1. y = 12(0.92) x 2. y = 5(1.5) x Decay Decay factor =0.92 Rate of decay = -.08 Y-intercept = 12 Growth Growth factor = 1.5 Rate of growth = 0.5 Y-intercept = 5

x f(x) = a (b) x XY

f(x) = a (b) (x – h ) + k Stretch/Compress Left/Right Up/Down Transformations

1. y = 2(4) x – Stretched vertically by “2” Moved right 6 units Moved up 5 units

2. y = -½ (1.2) x+1 – 2 Reflected over the x-axis Compressed by ½ Moved left 1 unit Moved down 2 units

Compound Interest Formula P(t) = total amount at time “t” P o = total amount at time “zero” r = growth/decay rate (decimal) n = number of compounds t = time

Not compounded: n = 1 monthly:Compounded monthly: n = 12 quarterly:Compounded quarterly: n = 4 bi-monthly:Compounded bi-monthly: n = 24 Compounded daily: n = 365

Compounded Continuously The number “e” 2 nd, 2 nd,

1) The population of a city is modeled by the equation f(x) = 1,203,368(1.08) t What is its initial population? Is the population growing or decaying? What is the growth rate? What might the population be in 5 years?

2) Isabella invested $500 at a 6% annual interest, compounded quarterly. Write an exponential equation representing this situation. How much will be in the account after 10 years?

3) Jerald has $2,000 to invest in an account that pays 6.5% annual interest compounded continuously. a. Write an exponential equation representing this situation. b. How much will be in the account after 10 years?

4) A student wants to save $8000 for college in 5 years. How much should be put into an account that pays 5.2% annual interest compounded continuously? P(t)= ________ P 0 = ________ r = _________ n= ________ t = _________

5) The value of an industrial machine has a decay factor of 0.75 per year. The machine was originally worth 150,000. How much is it worth after 3 years? P(t)= ________ P 0 = ________ r = _________ n= ________ t = _________

6) A computer engineer is hired for a salary of $28,000. If she gets a 5% raise each year, how much money will she make in 10 years?

7) How much money will a trust that started with $30,000 and earns 10% annual interest have in it after 5 years?

8) A savings account that earns 4.62% annual interest, compounded quarterly contains $250,000 after 25 years. How much money was initially invested?

9) A man invested $2000 at a annual interest rate of 5.1% compounded monthly. Will the man be able to at least double his money in 10 years? Explain.

10) A population of rabbits grows at a rate of 3.5% per day. If the initial population is 2 rabbits, how many rabbits will there be after 10 days?

Exit Ticket Select a rate and an investment amount Write down your rate & initial amount Calculate how much money you will have in 10 years

y = 0.7(5) x y = 16(0.5) x Growth Growth factor = 5 Rate of growth= 4 Y-intercept = 0.7 Decay Decay factor = 0.5 Rate of decay =.95 Y-intercept = 16