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Exponential Functions and Their Graphs Definition of Exponential Function f ( x) ๏ฝ a x where a > 0, a โ 1, and x is any real number Parent function for transformations: f(x) = ๐ โ ๐ ๐ฅ โโ +๐ The graph of f moves to the left if h < 0 or to the right if h > 0. The graph of f moves upward if k > 0 or downward if k < 0. The graph of f is stretched if b > 1 and shrunk if 0 < b < 1. The graph of f is reflected in the x-axis if b is negative. Properties of Exponential Functions The parent graph of the exponential function f(x) = ax has the following properties: 1. 2. 3. 4. 5. The function f is increasing for a > 1 and decreasing for 0 < a < 1. The y-intercept of the graph of f is (0, 1). The graph has the x-axis as a horizontal asymptote. The domain of f is (โโ, โ), and the range of f is (0, โ). The function f is one-to-one. Slide 4-3 Copyright © 2011 Pearson Education, Inc. f(x) = ax for a > 1 y f(x) = ax for 0 < a <1 11 1 x Transformations of Graphs of Exponential Functions Describe the transformations of each function from the parent graph of ๐ ๐ฅ = ๐ ๐ฅ . 1. f ( x) ๏ฝ 3 ๏ญ 7 x 2. ๐ ๐ฅ = 2๐ฅโ3 + 5 Sketch the graphs of #1 and #2, be sure to include accurate points and any asymptotes. 3. 4. Transformations of Graphs of Exponential Functions Describe the transformations of each function from the parent graph of ๐ ๐ฅ = ๐ ๐ฅ . 5. f(x) = 1 ๐ฅโ3 4 Sketch the graph. 7. f(x) = 1 ๐ฅ+1 4 โ2 6. f(x) = โ 1 ๐ฅ+4 4 โ1 One-to-One Property of Exponential Functions: For a > 0 and a โ 1, if a ๏ฝ a , then x1 ๏ฝ x2 . x1 x2 Slide 4-7 Copyright © 2011 Pearson Education, Inc. Solve each equation: 8. 2๐ฅ = 27 10. 2 = ๐ฅ 1 2 9. 2๐ฅ =64 11. 2๐ฅ+3 = 24๐ฅโ9 Compound Interest ๏ฆ r๏ถ A ๏ฝ P ๏ง1 ๏ซ ๏ท ๏จ n๏ธ nt n: number of times compounded in a year A ๏ฝ Pe rt Used when interest is continuously compounded r: interest rate as a decimal t: time in years r: interest rate as a decimal P: principal (initial investment) t: time in years A: amount in the account after time(t) P: principal (initial investment) 12. If you invest $1000 for 25 years and the interest rate is 6%, what is the value of your investment if your interest is compounded quarterly? 13. If you invest $5000 for 18 years and the interest rate is 4.5%, what is the value of your investment if your interest is compounded continuously? Section 4.2 Logarithmic Functions and Their Applications Copyright © 2011 Pearson Education, Inc. Since exponential functions are one-to-one functions, they are invertible. The inverses of the exponential functions are called logarithmic functions. ๏ฝ ๏ฝ If f(x) = ax, then instead of f โ1(x), we write loga(x) for the inverse of the base-a exponential function. We read loga(x) as โlog base a of x,โ and we call the expression loga(x) a logarithm. Copyright © 2011 Pearson Education, Inc. Sli de 412 Definition of Logarithmic Function with Base a Definition: Logarithmic Function For a > 0 and a โ 1, the logarithmic function with base a is denoted as f(x) = loga(x), where y = loga(x) Logarithmic Form y ๏ฝ log a x if and only if ay = x. Exponential Form a ๏ฝx EQUIVALENT! y ๏ฎ ๏ฎ ๏ฎ ๏ฎ Note that loga(1) = 0 for any base a, because a0 = 1 for any base a. There are two bases that are used more frequently than others; they are 10 and e. The notation log10(x) is abbreviated log(x) and loge(x) is abbreviated ln(x). These are called the common logarithmic function and the natural logarithmic function, respectively. There are no logarithms of negative numbers or zero. Slide 4-14 Copyright © 2011 Pearson Education, Inc. Any function of the form k g(x) = b · loga(x โ h) + is a member of the logarithmic family of functions. ๏ฎ ๏ฎ ๏ฎ ๏ฎ The graph of f moves to the left if h < 0 or to the right if h > 0. The graph of f moves upward if k > 0 or downward if k < 0. The graph of f is stretched if b > 1 and shrunk if 0 < b < 1. Copyright © 2011 Pearson Education, Inc. Sli de 415 The graph of f is reflected in the x-axis if b is negative. Properties of Logarithmic Functions The logarithmic function f(x) = loga(x) has the following properties: 1. The function f is increasing for a > 1 and decreasing for 0 < a < 1. 2. The x-intercept of the graph of f is (1, 0). 3. The graph has the y-axis as a vertical asymptote. 4. The domain of f is (0, โ), and the range of f is (โ โ, โ). 5. The function f is one-to-one. 6. The functions f(x) = loga(x) and f(x) = ax are inverse functions. for a > 1 Slide 4-17 Copyright © 2011 Pearson Education, Inc. y = loga x for 0 < a < 1 (1, 0) 1 Slide 4-18 Describe the transformations of each function from the parent graph. Then sketch the graph. One-to-One Property of Logarithms For a > 0 and a โ 1, if loga(x1) = loga(x2), then x1 = x2 . Section 4.3 & 4.4 Properties of Logarithms & Solving Logarithmic Equations Use the Change of Base Formula to evaluate. Round your answers to the nearest thousandth. 26. log 3 16 ๏ฝ 27. log 50 5 ๏ฝ Find the exact value of each expression without using a calculator 30. 28. 29. 5 log7 7 ln e ๏ซ ln e 12 5 ๏ญ0.2 log3 81 Expand each logarithmic expression 31. xy log 4 z 4 32. 3 ln 5 x 2 Condense each logarithmic expression 33. 2ln8 ๏ซ 5ln( z ๏ญ 4) 34. 2[3log x ๏ญ log 6 ๏ญ log y] Strategies for Solving Exponential and Logarithmic Equations โข Try to use the one-to-one properties (match them up) โข Condense to single logarithms when possible โข Switch from exponential form to logarithmic form โข Switch from logarithmic form to exponential form Exponential Equations-solve for x, check answers 35. 1. 4(3 ) ๏ฝ 64 x 36. 2. e ๏ญ x2 5 x ๏ซ6 ๏ฝe Exponential Equations-solve for x, check answers 37. 3. 5 ๏ญ 3e ๏ฝ 2 x 38. 4. 6๏จ2 x ๏ซ5 ๏ฉ ๏ซ 4 ๏ฝ 11 Logarithmic Equations-solve for x, check answers 39. 40. log 5 = 2 โ log(๐ฅ) ln ๐ฅ + ln ๐ฅ + 2 = ln(8) Logarithmic Equations-solve for x, check answers 3. log3 (5x ๏ซ 13) ๏ญ log3 6 ๏ฝ log3 3x 41. 6. log x ๏ซ log( x ๏ญ 9) ๏ฝ 1 42. 43. ๐๐๐4 ๐ฅ โ ๐๐๐4 ๐ฅ + 2 = 2 Compound Interest Formula If a principal P is invested for t years at an annual rate r compounded n times per year, then the amount A, or nt r๏ถ ๏ฆ ending balance, is given by A ๏ฝ P๏ง1 ๏ซ ๏ท . ๏จ n๏ธ Continuous Compounding Formula If a principal P is invested for t years at an annual rate r compounded continuously, then the amount A, or ending balance, is given by A = Pert. 44. Melinda invests her $80,000 lottery winnings at a 9% annual interest rate compounded monthly. How long much she invest the money to have $480,000 in her account. Round to the nearest year. 45. At what interest rate would a deposit of $30,000 grow to $2,540,689 in 40 years with continuous compounding? Round to the nearest tenth of a percent. Systems of Equations and Inequalities Section 5.1 Systems of Linear Equations in Two Variables Copyright © 2011 Pearson Education, Inc. ๏ฝ ๏ฝ Any collection of two or more equations is called a system of equations. The solution set of a system of two linear equations in two variables is the set of all ordered pairs that satisfy both equations of the system. โฆ The graph of an equation shows all ordered pairs that satisfy the equation, so we can solve some systems by graphing the equations and observing which points (if any) satisfy all of the equations. THESE SOLUTION WOULD BE WHERE THE GRAPHS INTERSECT. Copyright © 2011 Pearson Education, Inc. โฆ A system of equations that has at least one solution is consistent. Two types of consistent systems are: ๏ Independent - with exactly one solution Two Intersecting Lines ๏ Dependent - with infinitely many solutions Same Lines โฆ A system with no solutions is inconsistent. Parallel Lines Copyright © 2011 Pearson Education, Inc. โฆ In the substitution method, we eliminate a variable from one equation by substituting an expression for that variable from the other equation. โฆ In the addition method, (also called the elimination method) we eliminate a variable by adding the two equations. ๏ It might be necessary to multiply each equation by an appropriate number so that a variable will be eliminated by this addition. Copyright © 2011 Pearson Education, Inc. #46. ๐ฆ = 2๐ฅ + 1 3๐ฅ โ 4๐ฆ = 1 #49. โ2๐ฅ + 5๐ฆ = 14 7๐ฅ + 6๐ฆ = โ2 #47. ๐ฆ โ 3๐ฅ = 5 3๐ฅ โ 3 = ๐ฆ โ 2 #48. 2๐ฅ + ๐ฆ = 9 4๐ฅ + 2๐ฆ = 10 ๏ฝ ๏ฝ #50. Amy has a higher income than Vince, and their total income is $82,000. If their salaries differ by $16,000, then what is the income of each? #51. The Springfield zoo has different admission prices for adults and children. When Mr. and Mrs. Weaver went with their five children, then bill was $33. If Mrs. Wong and her three children got in for $18.50, then what were the individual prices for adult and childrenโs tickets? Matrices and Determinants Section 6.1 Solving Linear Systems Using Matrices Copyright © 2011 Pearson Education, Inc. ๏ฝ ๏ฝ ๏ฝ A matrix is a rectangular array of real numbers. The rows of a matrix run horizontally, and the columns run vertically. A matrix with m rows and n columns has size m ๏ด n (read โm by nโ). โฆ The number of rows is always given first. Columns Rows This is a 2 x 3 matrix. Both equations in the system need to be in standard form: Ax + By = C 2x โ 3y = 6 y= x +4 Already in standard form. Put into standard form. 2x โ 3y = 6 -x + y = 4 2x โ 3y = 6 -x + y = 4 Put the values of A, B and C into the matrix. 2 โ3 6 โ1 1 4 *See handout for specific instructions on using the calculator to now solve the system of equations. It is on the CAS website Check solutions to #46 โ 49 using the matrix capabilities of your calculator.