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## 3.1 Exponential Functions

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**3.1 Exponential Functions**By Thersia Weber and Rachel Eron**Exponential Format**Only an exponential function if the x is in the exponent. f(x) = abx a = initial amount b = rate of growth/base x = variable b > 1: exponential growth b < 1: exponential decay**Identifying Exponential Functions**f(x) = 4x7 Exponential Function? Initial Value? Base? Growth or Decay? f(x) = 3x Exponential Function? Initial Value? Base? Growth or Decay? Yes X is in exponent 1 A isn’t listed, automatically 1 3 B=3 Growth B=3 > 1 No X isn’t in exponent - - -**Table of Values**a = x = 0 then plug it into y=abx y=6bx then plug in another value on the table 3/2 = 6b1 solve for b b= 3 . 1 2 6 b= 3 = 1 12 4 plug b into equation y= 6(1/4)x Power x f(x) -2 96 -1 24 0 6 1 3/2 2 3/8**Table of Values Examples**x f(x) -2 108 -1 36 0 12 1 4 2 4/3 x f(x) y = abx y = 4.8bx 5.6 = 4.8b10 1.167 = b10 ln 1.167 = 10 ln b ln b = .0154 b = e.0154 y = 4.8 e.0154x y = abx y = 12bx 4 = 12b1 4/12 = b 4/12 = 1/3 = b y = 12 (1/3)x y = 12 (1/3)x y = 4.8 e.0154x**Review of Graph Transformations**Graph transformations previously learned also apply to exponential functions http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=104# Horizontal Translations y = abx-c graph shifts to right y = abx+c graph shifts to left Vertical Translations y = abx +c graph shifts up y = abx-c graph shifts down Axis Flips y = ab-x graph flips over y axis y = -abx graph flips over x axis Use URL to explore transformations of Exponential functions on graph**Plugging into Equations**Exponential equations can be found in many real life situations. To solve real life equations, look at it as y = abx and find initial amount and other variables. Plug in the numbers given and find the information asked for in problem. The number B of bacteria in a petri dish culture of t hours is given by: B = 100e0.693t Find initial number of bacteria present. How many bacteria are present after 6 hours? B = 100e0.693t B = 100e0.693x6 B = 6394 Initial: 100 # Present after 6 hours: 6394**Allowance**• Suppose your parents are going to pay you a weekly allowance for doing your household chores this coming year. Also, imagine that they are going to give you a choice of two payment schemes. With Choice A, your parents will pay you $10 every week. Therefore, you will have earned $10 after the first week, $20 after the second week, $30 after the third week, and so on. With Choice B, just to be nice, they are going to pay you $0.01 before you even start working. After you complete chores for week one, they will double your money, thus giving you a total of $0.02. After you complete chores for week two, they will once again double your money, thus giving you a total of $0.04. They will continue to double your money at the end of every week. • Write an equation for each payment scheme and use www.graphsketch.com to make graphs. Explain which payment scheme you will choose and why using the graphs. • Each project will be turned in and graded according to rubric, worth 20 points.**Allowance Answers**• Each payment scheme will give you a fair amount of allowance, but judging by the graphs, the second payment plan would be a better choice because you will earn more eventually, faster. 2-3 groups to present explanation. If present, 5 extra credit points.**References**• Precalculus Textbook by Franklin Demana, Bert Waits, Bregory Foley, and Daniel Kennedy • www.graphsketch.com (Graph, slide 10) • http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=104# (Graphing Activity, slide 7) • http://www.cartoonstock.com/directory/e/exponential_growth.asp (Introduction Comic) • http://education.ti.com/calculators/downloads/US/Activities/Detail?id=7727 (Supplemental Activity, Allowance)**Quiz**• Now take a 10 question assessment on what you’ve learned. • Assessment • Assessment Answers