Algebra 2 Notes Section 7.1: Graph Exponential Growth Functions Objective(s): To graph and use exponential growth functions. Vocabulary: I. Exponential Function: An equation of the form y = ab x where a 0 and the base b is a positive number other than 1. II. Exponential Growth Function: y = ab x where a > 0 and b > 1. III. Growth Factor: b y IV. Parent Function for Exponential Growth Functions: f (x) = b x where b > 1 Asymptote: x-axis (equation: y = 0) (0, 1) (1, b) f (x) = b x (b > 1) Domain: set of all real numbers x Range: y > 0 V. Asymptote: A line that a graph approaches more and more closely. VI. The graph of y = ab x : The graph of y = ab x is a vertical stretch or shrink of the graph of y = b x. The y-intercept of y = ab x occurs at (0, a) rather than at (0,1). VII. Translations: To graph a function of the form y = ab x h + k, begin by sketching the graph of y = ab x. Then translate the graph horizontally by h units and vertically by k units. VIII. y = a(1 + r) t : When a real-life quantity increases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by this equation. Growth factor for this model is : (1 + r)
A = P 1 + r IX. Compound Interest n Formula:, where P = nt r = t = Principal amount rate (as a decimal) time (years) Notes 7.1 page 2 n = A = number of times compounded per year Amount in the account Examples: 1. Graph each of the following. a. y = 5 x 1 x b. y 3 c. 3 x 7 y 2 2. Graph y = 2 4 x + 2 + 1. State the domain and the range. Domain: all real numbers Range: y > 1
Notes 7.1 page 3 3. In 1970, the population of Kern County, California, was about 330,000. From 1970 to 2000, the county population grew at an average annual rate of about 2.4%. Write an exponential growth model giving the population P of Kern County t years after 1970. Graph the model. About how many people lived in Kern County in 1990? P = 330,000(1.024) t 530,290 Population 900,000 750,000 600,000 450,000 300,000 150,000 10 20 30 40 50 Years since 1970 4. You deposit $5500 in an account that pays 3.6% annual interest. Find the balance after 2 years if the interest is compounded with the given frequency. a. semiannually b. monthly $5906.82 $5909.97
Algebra 2 Notes Section 7.2: Graph Exponential Decay Functions Objective(s): To graph and use exponential decay functions. Vocabulary: I. Exponential decay function: y = ab x where a > 0 and 0 < b < 1. y II. Parent Function for Exponential Decay Functions: f(x) = b x where 0 < b < 1 f (x) = b x (0 < b < 1) Asymptote: Domain: Range: y > 0 x-axis all real numbers (0, 1) (1, b) x III. y = a(1 r) t : When a real-life quantity decreases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by this equation. Decay factor for this model is : (1 r) Examples: 1. Graph each of the following. a. x 1 y b. 5 x 2 y 3 c. 3 x 3 y 2 4
Notes 7.2 page 2 x 2 1 2. Graph y 2 1. State the domain and range. 3 Domain: all real numbers Range: y > 1 3. A new car costs $25,000. The value of the car decreases by 15% each year. a. Write an exponential decay model giving the car s value y (in dollars) after t years. y = 25,000(0.85) t 30,000 25,000 20,000 b. Estimate the value of the car after 4 years. Value (dollars) 15,000 10,000 $13,050 5,000 2 4 6 8 10 12 c. Graph the model. Years
Algebra 2 Notes Section 7.3: Use Functions Involving e Objective(s): To study functions involving the natural base e. Vocabulary: I. Natural base e: (Key Concept) The natural base e is irrational. It is defined as follows: As n approaches +, 1 + 1 n approaches e. n e 2.718281828 II. Natural Base Functions: A function of the form y = ae rx. If a > 0 and r > 0, the function is an exponential growth function. If a > 0 and r < 0, the function is an exponential decay function. III. Continuously Compounded Interest (Key Concept): When interest is compounded continuously, the amount A in an account after t years is given by the formula A = Pe rt where P is the principal and r is the annual interest rate (as a decimal). Examples: 1. Simplify the expression. a. e 9 e 6 b. 8 60e 3 12e c. ( 10e 5x ) 3 e 15 5e 5 1000 e 15x 2. Evaluate the expression to the nearest thousandth. a. e 6 b. e 0.28 403.429 0.756
Notes 7.3 page 2 3. Graph the function. State the domain and range. a. y = 4e 0.5x b. y = e 1.5(x + 2) 4 Domain: all real numbers Domain: all real numbers Range: y > 0 Range: y > 4 4. Annual sales of a certain product can be modeled by the function S = 60,000e 0.15t, where S is the number of units sold and t is the number of years since the product went on the market. Estimate the annual sales 6 years after the product went on the market. $24,394.18 5. You deposit $3000 in an account that pays 3.5% annual interest compounded continuously. What is the balance after 3 years? $3,332.13
Algebra 2 Notes Section 7.4: Evaluate Logarithms and Graph Logarithmic Functions Objective(s): Evaluate logarithms and graph logarithmic functions. Vocabulary: I. Definition of Logarithm with Base b: Let b and y be positive numbers with b 1. The logarithm of y with base b is denoted by log b y and is defined as follows: The expression log b y is read as log b y = x if and only if b x = y This definition tells you that the equations are equivalent. log base b of y log b y = x logarithmic form and b x = y exponential form II. Common Logarithm: A logarithm with base 10. It is denoted by log 10 or simply by log. Common Logarithm: log 10 x = log x III. Natural Logarithm: A logarithm with base e. It can be denoted by log e but more often ln. Natural Logarithm: log e x = ln x IV. Parent Graphs for Logarithmic Functions: Domain: Range: Asymptote: x > 0 all real numbers y-axis (equation x = 0) V. Translations: You can graph a logarithmic function of the form y = log b (x h) + k by translating the graph of the parent function y = log b x
Examples: Notes 7.4 page 2 1. What is the inverse of y = 2 x (solved for y)? y=log 2 x 2. Rewrite the equation in exponential form. a. log 2 32 = 5 b. log 10 1 = 0 c. log 9 9 = 1 d. log 25 2 3. Rewrite the equation in logarithmic form. a. 2 3 = 8 b. m k = h c. e m = k d. 9 = 3 2 1 5 2 5 = 32 10 0 = 1 9 1 = 9 (⅕) 2 = 25 log 2 8 = 3 log m h = k ln k = m log 3 9 = 2 4. Evaluate the logarithm. a. log 3 81 b. log 1 256 c. log 10 0.001 d. log 64 2 4 4 4 3 ⅙ 5. Evaluate the logarithm to the nearest thousandth. a. log 0.85 b. ln 22 0.071 3.091 6. The sales of a certain video game can be modeled by y = 20ln(x 1) + 35, where y is the monthly number (in thousands) of games sold during the x th month after the game is released for sale (x > 1). Estimate the number of video games sold during the 10 th month after the game is released. 79,000 games 7. Simplify the expression. a. e ln 9 b. log 3 27 x 9 3x
Notes 7.4 page 3 8. Find the inverse of the function. a. y = 8 x b. y = ln (x 4) y = log 8 x y = e x + 4 9. Graph the function. a. y = log 2 x b. y = log x 2 3 10. Graph y = log 3 (x 2) + 4. State the domain and range. Domain: x > 2 Range: all real numbers
Algebra 2 Notes Section 7.5: Apply Properties of Logarithms Objective(s): To rewrite logarithmic expressions. Vocabulary: I. Properties of Logarithms: Let b, m, and n be positive numbers such that b 1. Product Property: log b mn = log b m + log b n Quotient Property: log b m n = log b m log b n Power Property: log b m n = nlog b m Rewriting expressions: You can use the properties of logarithms to expand and condense logarithmic expressions. II. Change-of-Base Formula: If a, b, and c are positive numbers with b 1 and c 1, then In particular, log b a log a log a c = log c a = and log c a = log b c log c ln a ln c Examples: 1. Use log 3 12 2.262 and log 3 2 0.631 to evaluate the logarithm. a. log 3 6 b. log 3 24 c. log 3 32 1.631 2.893 3.155 (32 = 2 3 12/3)
2. Expand. a. 4 log 3x b. 5x log 2 c. y 2 2 3x log 7 5y 3 Notes 7.5 page 2 log 3 + 4log x log 2 5 + log 2 x 2log 2 y log 7 3 + 2log 7 x log 7 5 3log 7 y 3. Condense. Write your answer in simplest form. a. log 9 + 3 log x log 3 b. ln 8 + 2 ln 5 ln 10 c. 3log x + 2 log y 3log m log 3x 3 ln 20 log x 3 y 2 m 3 4. Evaluate log 6 24 using common logarithms and natural logarithms. log 24 ln 24 or log 6 ln 6 1.774 5. The Richter scale is used to measure the magnitude of earthquakes. If an earthquake has intensity I, then I its magnitude on the Richter scale, R, is given by the function R(I) log, where I o is the intensity of a barely I o felt earthquake. If the intensity of one earthquake is 50 times that of another, how many points greater is the bigger earthquake on the Richter scale? I R(50I) R(I) = log 50 I I o log I o = log (50 I) log I o (log I log I o ) = log 50 + log I log I o log I + log I o = log 50 1.698970004 about 1.7 points
Algebra 2 Notes Section 7.6: Solve Exponential and Logarithmic Equations Objective(s): To solve exponential and logarithmic equations. Vocabulary: I. Exponential equation: II. Property of Equality for Exponential Equations: Equations in which variable expressions occur as exponents. If b is a positive number other than 1, then b x = b y if and only if x = y. When it is not convenient to write each side of an exponential equation using the same base, you can solve the equation by taking a logarithm of each side. (p. 517) This means that if you are given an equation x = y, then you can exponentiate each side to obtain an equation of the form b x = b y. III. Property of Equality for Logarithmic Equations: If b, x, and y are positive numbers with b 1, then log b x = log b y if and only if x = y. T = (T o T R )e rt + T R IV. Newton s Law of Cooling: where T = T o = t = r = T R = Temperature after t minutes Initial Temperature Time (minutes) Substance's cooling rate Surrounding temperature V. Extraneous Solutions: Because the domain of a logarithmic function generally does not include all real numbers, be sure to check for extraneous solutions of logarithmic equations. Examples: 1. Solve 3 x x 3 1 9 2. Solve 9 x = 35 2 1.62
Notes 7.6 page 2 3. Hot chocolate that has been heated to 90 C is poured into a mug and placed on a table in a room with a temperature of 20 C. If r = 0.145 when the time t is measured in minutes, how long will it take for the hot chocolate to cool to a temperature of 30 C? T = (T o T R )e rt + T R 30 = (90 20)e 0.145t + 20 30 = 70e 0.145t + 20 10 = 70 e 0.145t 0.1428571429 = e 0.145t ln 0.1428571429 = 0.145t ln e t = ln 0.1428571429 0.145 13.42 minutes 4. Solve. a. log 4 (2x + 8) = log 4 (6x 12) b. log 5 (4x 7) = log 5 (x + 5) c. 2 = log 5 (4x + 1) 5 4 6 d. log 7 (3x 2) = 2 e. log 6 (3x) + log 6 (x 4) = 2 f. log (2x) + log(x 5) = 2 17 6 10 ( 5 is extraneous) 5. The population of deer in a forest preserve can be modeled by the equation P = 50 + 200 ln (t + 1) where t is the time in years from the present. In how many years will the deer population reach 500? 8.5 years