Intro to Exponential Functions Lesson 3.1 F.BF.5. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Monday, March 9, 2015 Monday Bellwork Solve for y (TRY IT without using a calculator) 1. y = 30 = 2. y = 2-1 = 3. y = 23 = 4. y = 4-2 = 5. 1 = 2y
Linear vs Exponential Functions Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Change at a changing Exponential rate Functions Change at a constant percent rate Contrast Suppose you have a choice of two different jobs at graduation
Start at $30,000 with a 6% per year increase Start at $40,000 with $1200 per year raise Which should you choose? One is linear growth One is exponential growth Which Job? How do we get each next value for Option A? When is Option A better? When is Option B better? Rate of increase a constant $1200 Rate of increase changing Percent of increase is a constant Ratio of successive years is 1.06 Year
Option A Option B 1 $30,000 $40,000 2 $31,800 $41,200 3 $33,708
$100.00 $105.00 $110.25 $115.76 $121.55 $127.63 $134.01 $140.71 $147.75 $155.13 $162.89 Compounded Interest Table of results from calculator Set y= screen y1(x)=100*1.05^x Choose Table (Diamond Y)
Graph of results Exponential Modeling Population growth often modeled by exponential function Half life of radioactive materials modeled by exponential function Growth Factor Recall formula new balance = old balance + 0.05 * old balance Another way of writing the formula new balance = 1.05 * old balance Why equivalent? Growth factor: 1 + interest rate as a fraction
Decreasing Exponentials Consider a medication Patient takes 100 mg Once it is taken, body filters medication out over period of time Suppose it removes 15% of what is present in the blood stream every hour Fill Fillininthe the rest restofofthe the table table At end of hour Amount remaining
exponentialisisaadecreasing decreasing function function Amount Remaining Amount Remaining Mg remaining Mg remaining Graph 100.00 100.00 80.00 80.00 60.00 60.00 40.00
40.00 20.00 20.00 0.00 0.00 0 0 1 1 2 2 3 4
5 3 4 5 At End of Hour At End of Hour 6 6 7 7 8 8 Solving Exponential Equations
Graphically For our medication example when does the amount of medication amount to less than 5 mg Graph the function for 0 < t < 25 Use the graph to determine when t M (t ) 100 0.85 5.0 General Formula All exponential functions have the general format: f (t ) A B Where A = initial value B = growth factor
t = number of time periods t Typical Exponential Graphs When B > 1 f (t ) A B t When B < 1 View Viewresults resultsof of B>1, B>1,B<1 B<1with with spreadsheet spreadsheet
I will graph a basic function f(x) = 2x Pre Calc Cookbook: 1. Make a table of at least 5 (x,y) values 2. Determine intercept(s) 3. Draw it on a graph!! WE graph a basic function f(x) = 2-x Pre Calc Cookbook: 1. Make a table of at least 5 (x,y) values 2. Determine intercept(s) 3. Draw it on a graph!! YOU graph a basic function f(x) = 2x + 3
Pre Calc Cookbook: 1. Make a table of at least 5 (x,y) values 2. Determine intercept(s) 3. Draw it on a graph!! Exponential Functions HW 1 Tuesday, March 10, 2015 Tuesday Bellwork Think pair share the HW from last night! THEN: Graph the function y = 3-x Definition of the Exponential Function The exponential function f with base b is defined by f (x) = bxx or y = bxx Where b is a positive constant, then x is any real number. Here are some examples of exponential functions. f (x) = 2x
g(x) = 10x h(x) = 3x+1 Base is 2. Base is 10. Base is 3. Characteristics of Exponential Functions 1. The domain of f (x) = bxx consists of all real numbers. The range of f (x) = bxx consists of all positive real numbers. 2. The graphs of all exponential functions pass through the point (0, 1) because f (0) = b00 = 1. 3. If b > 1, f (x) = bxx has a graph that goes up to the right and is an increasing function. 4. If 0 < b < 1, f (x) = bxx has a graph that goes down to the right and is a decreasing function. 5. f (x) = bxx is a one-to-one function and has an inverse that is a function.
6. The graph of f (x) = bxx approaches but does not cross the x-axis. The xaxis is a horizontal asymptote. f (x) = bx 01 Transformations Involving Exponential Functions Transformation Equation Description Horizontal translation g(x) = bx+c Shifts the graph of f (x) = bx to the left c units if c > 0.
Shifts the graph of f (x) = bx to the right c units if c < 0. Vertical stretching or shrinking g(x) = c bx Multiplying y-coordintates of f (x) = bx by c, Stretches the graph of f (x) = bx if c > 1. Shrinks the graph of f (x) = bx if 0 < c < 1. Reflecting g(x) = -bx g(x) = b-x Reflects the graph of f (x) = bx about the x-axis. Reflects the graph of f (x) = bx about the y-axis. Vertical translation
g(x) = -bx + c Shifts the graph of f (x) = bx upward c units if c > 0. Shifts the graph of f (x) = bx downward c units if c < 0. Example 1: Use the graph of f (x) = 3x to obtain the graph of g(x) = 3 x+1. Solution Examine the table below. Note that the function g(x) = 3x+1 has the general form g(x) = bx+c, where c = 1. Because c > 0, we graph g(x) = 3 x+1 by shifting the graph of f (x) = 3x one unit to the left. We construct a table showing some of the coordinates for f and g to build their graphs. g(x) = 3x+1 (-1, 1) -5 -4 -3 -2 -1 f (x) = 3x
(0, 1) 1 2 3 4 5 6 Problems Sketch a graph using transformation of the following: 1. f ( x) 2 x 3 2. f ( x) 2 x 1 3. f ( x) 4 x 1 1 Recall the order of shifting: horizontal, reflection (horz., vert.), vertical. Note: This is similar to
section I on HW. Sketch the graph of the function WITHOUT the graphing calculator. Then give the domain and range in interval notations for each graph. Also give the intercept(s). The domain of f (x) = bx consists of all real numbers. The range of f (x) = bx consists of all positive real numbers. Y-intercept, let x=0 and solve for y! X-intercept, let y=0 and solve for x! (does not always exist) y2 Example 1: x 2
Note: This is similar to section II on HW. Sketch the graph of the function WITHOUT the graphing calculator. Then give the domain and range in interval notations for each graph. Also give the intercept(s). ( ,) Domain: all real numbers Range: all positive real numbers (0, ) Intercept: y2 0 2 1 1 y2 2 2
4 2 1 0, 4 EX 2: Determining the transformation between two functions! f(x)= 2x and g(x)= 2x-3 Shifts the graph of f (x) = 2x to the right 3 units because -3 < 0 Note: This is similar to section III on HW. EX 3: Determining the transformation between two functions!
f(x)= - 3x and g(x)= 6 - 3x Shifts the graph of f (x) = -3x upward 6 units because 6 > 0. Note: This is similar to section III on HW. Independent Practice Complete sections 1-3 Wednesday, March 11, 2015 Bellwork Pair-Share Sections 1-3 on your homework form last night We will review this briefly One-To-One Property For a>0 and a 1,
if am = an , then m = n Note: This is similar to section IV on HW. Example 1: 2a a 3 3 Pre Calc Cookbook: 1.Change fractions to numerator only 2.Find common bases (if not already) 3.Set exponents equal to each other 4.Solve for your
missing variable 2a a 2a a a a 3a0 a0 Example 2: x2 1 e Pre Calc Cookbook: 1.Change fractions to numerator only 2.Find common bases (if not already) 3.Set exponents equal to each other
4.Solve for your missing variable 2x e 2 x 1 2x x2 2x1 0 (x 1)(x 1) 0 x 1 0 x1 Example 3: Pre Calc Cookbook: 1.Change fractions to numerator only 2.Find common bases
(if not already) 3.Set exponents equal to each other 4.Solve for your missing variable 1 x3 4 32 1 36 4 5 1 (2 ) x3
( 2 2 x3 ) 2 5 2 2 x6 5 2x 6 11 2x 5.5 x The Natural Base e An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to 2.72. More accurately, e 2.71828... The number e is called the natural base. The function f (x) = ex is called the
natural exponential function. f (x) = 3x f (x) = ex 4 (1, 3) f (x) = 2x 3 (1, e) 2 (1, 2) (0, 1) -1 1 Formulas for Compound Interest
After t years, the balance, A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas: 1. For n compoundings per year: rt r A P 1 n 2. For continuous compounding: A = Pert. Example: Choosing Between Investments You want to invest $8000 for 6 years, and you have a choice between two accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment? Solution We use the compound interest model with P = 8000, r = 7% = 0.07, n = 12 (monthly compounding, means 12 compoundings per year), and t = 6. r
A P 1 n ( ) nt 0.07 8000 1 12 ( ) 12*6 12,160.84 The balance in this account after 6 years is $12,160.84.
Note: This is similar to section V on HW. more Example: Choosing Between Investments You want to invest $8000 for 6 years, and you have a choice between two accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment? Solution For the second investment option, we use the model for continuous compounding with P = 8000, r = 6.85% = 0.0685, and t = 6. A Pe rt 8000e 0.0685(6) 12, 066.60 The balance in this account after 6 years is $12,066.60, slightly less than the previous amount. Thus, the better investment is the 7% monthly compounding option. Example Use A= Pert to solve the following problem: Find the accumulated value of an investment of $2000 for 8 years at an interest rate of 7% if
the money is compounded continuously Solution: A= Pert A = 2000e(.07)(8) A = 2000 e(.56) A = 2000 * 1.75 A = $3500 Independent Practice Complete sections 4 & 5
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