Unit 5 Eponential/Logarithmic Functions Eponential Functions Unit 5.1) William Bill) Finch Mathematics Department Denton High School Lesson Goals When ou have completed this lesson ou will: Recognize and evaluate eponential functions of base a and base e. Appl transformations to eponential functions and graph eponential functions of an base. Appl eponential functions to real-life situations. Eponential 2 / 25 Eponential Functions A base-b eponential function has the form f ) = ab where a 0, b > 0 and b 1, and is an real number. Eponential Growth: f ) = ab, b > 1 Domain, ) Range 0, ) -Intercept 0, a) -Intercept none Increasing, ) Hor Asmptote -ais Continuous, ) End Behavior lim f ) = 0 lim f ) = 0, a) Eponential 3 / 25 Eponential 4 / 25
Eponential Deca: f ) = ab, 0 < b < 1 Domain, ) Range 0, ) -Intercept 0, a) -Intercept none Decreasing, ) Hor Asmptote -ais Continuous, ) End Behavior lim f ) = lim f ) = 0 0, a) Eample 1 Sketch the graph of f ) = 2. Then describe the following: 1. Domain 2. Range 3. Intercept 4. Asmptote 5. End behavior 6. Interval incr/decr Eponential 5 / 25 Eponential 6 / 25 Translations Reflections Horizontal Translation c Units Vertical Translation c Units Reflect wrt -ais Reflect wrt -ais f ) = b g) = b +c h) = b c f ) = b g) = b + c h) = b c f ) = b g) = b f ) = b g) = b Eponential 7 / 25 Eponential 8 / 25
Dilations Vertical Stretch/Shrink f ) = b g) = ab, a > 1 h) = ab, 0 < a < 1 Horizontal Stretch/Shrink f ) = b g) = b c, 0 < c < 1 h) = b c, c > 1 Eample 2 Use the graph of f to describe the transformation that produces the graph of g. 1. f ) = 4, g) = 4 +2 2. f ) = 3, g) = 2 3 ) 1 3. f ) = 2, g) = 2 Eponential 9 / 25 Eponential 10 / 25 The Natural Base e Graph of f ) = e Named for the Swiss mathematician Leonhard Euler pronounced Oiler ). Also called the natural base. Irrational number e 2.71828.... Graph 1 = 1 + 1 ) and 2 = e on the same set of aes. What happens the graph of 1 as increases? Note that e can be found two places on the calculator keboard: 1) the second function of the division ke, and 2) the second function of the LN ke. Domain, ) Range 0, ) -Intercept 0, 1) -Intercept none Increasing, ) Hor Asmptote -ais Continuous, ) End Behavior lim f ) = 0 lim f ) = 8 6 4 2 0, 1) 3 2 1 1 2 3 Eponential 11 / 25 Eponential 12 / 25
Eample 3 Use a calculator to evaluate f ) = e for the indicated values of. 1. f 3) Eample 4 Use the graph of f ) = e to describe the transformation that results in the graph of each function. 1. g) = e 4 2. f 2) 2. h) = e + 3 3. j) = 1 2 e Eponential 13 / 25 Eponential 14 / 25 Compound Interest Suppose an initial amount of mone called the principal) P is invested at an annual interest rate r and compounded once a ear. This means that at the end of the first ear the earned interest is added to the principal creating a new balance A1: A0 = P A1 = A0 + A0r A1 = A01 + r) A1 = P1 + r) Compound Interest Notice the pattern of multipling the principal b 1 + r each ear. Year 0 A0 = P Balance After Compounding 1 A1 = P1 + r) 2 A2 = A11 + r) = P1 + r)1 + r) = P1 + r) 2 3 A3 = A21 + r) = P1 + r) 2 1 + r) = P1 + r) 3. t. At = P1 + r) t Eponential 15 / 25 Eponential 16 / 25
Compound Interest The pattern of annual compounding can be modified to accommodate compounding more often such as quarterl or monthl, for eample) to produce the following Formula for Compound Interest: At) = P 1 + r ) nt n where t is time in ears At) is the total amount after t ears P is the initial principal r is the interest rate n is the number of times per ear the interest is compounded Eponential 17 / 25 Compound Continuousl At) = P 1 + n) r nt Compound Interest) At) = P 1 + 1 ) nt n r n n/r r At) = P 1 + 1 ) r)t n = and n = r) r Eample 5 Suppose ou get a summer internship that allows ou to save $5000 to be invested in an interest-bearing account. Calculate how much mone ou will have if ou invest our mone for 10 ears at 7% and: 1. compound semiannuall 2. compound quarterl 3. compound monthl 4. compound dail Eponential 18 / 25 Compound Continuousl At) = P 1 + 1 ) rt = n and n = r) r [ ) ] rt At) = P 1 + 1 Power Prop Ep) [ ) ] rt At) = P 1 + 1 lim 1 + 1/) = e) At) = Pe rt Substitution) Eponential 19 / 25 Eponential 20 / 25
Compound Continuousl If ou increase the number of times ou compound without bound this means to infinit... and beond!) the Formula for Compound Interest becomes At) = Pe rt Eample 6 Suppose ou get a summer internship that allows ou to save $5000 to be invested in an interest-bearing account. Calculate how much mone ou will have if ou invest our mone for 10 ears at 7% if the interest could be calculated continuousl. where t = time in ears At) is the total amount after t ears P is the initial principal r is the interest rate e is the natural base Eponential 21 / 25 Eponential 22 / 25 Eample 7 A population is declining at a rate of 2.5% annuall. The current population is approimatel 11 million people. Assuming the population decline continues at this rate, predict the population after: 1. 30 ears annual deca. 2. 30 ears continuous deca. Eample 8 The table below shows the population growth of deer in a forest from 2000 to 2010. Deer Population Year Deer 2000 125 2010 264 Assume an eponential rate of growth: 1. Identif the rate of increase. 2. Write an eponential equation to model this situation. 3. Predict the number of deer in 2020. Eponential 23 / 25 Eponential 24 / 25
What You Learned You can now: Recognize and evaluate eponential functions of base a and base e. Appl transformations to eponential functions and graph eponential functions of an base. Appl eponential functions to real-life situations. Do problems Chap 3.1 #1, 3, 11-19 odd, 21, 25, 29, 31, 35, 37, 41, 43 Eponential 25 / 25