CCSS: F.IF.7.e., F.IF.8.b MATHEMATICAL PRACTICES: 3 Construct viable arguments and critique the reasoning of others

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1 7.1 Graphing Exponential Functions 1) State characteristics of exponential functions 2) Identify transformations of exponential functions 3) Distinguish exponential growth from exponential decay CCSS: F.IF.7.e., F.IF.8.b MATHEMATICAL PRACTICES: 3 Construct viable arguments and critique the reasoning of others 1. Exponential function: a function in which the base is constant and the exponent is the independent variable 2. Characteristics of exponential growth functions: a. Parent function: f(x) = b x, b>1 b. Type: continuous, one-to-one, increasing c. Domain: all real numbers d. Range: all positive real numbers e. Asymptote: y = 0 (the x-axis) f. Y-intercept: (0,1) 3. Transformations of exponential functions: the same transformation rules apply 4. Exponential growth over time: A(t) = a(1+r) t a. A(t) the amount after a given time, t b. a the initial amount c. 1+r the growth factor [r is the growth rate; 1 retains the initial amount] 5. Characteristics of exponential decay functions: a. Parent function: f(x) = b x, 0<b<1 b. Type: continuous, one-to-one, decreasing c. Domain: all real numbers d. Range: all positive real numbers e. Asymptote: y = 0 (the x-axis) f. Y-intercept: (0,1) 6. Exponential decay over time: A(t) = a(1-r) t a. A(t) the amount after a given time, t b. a the initial amount c. 1-r the decay factor [r is the rate of decay; 1 subtracts the decay from the initial amount] 7. Example 1: Graph f(x) = 4 x. State the domain and range. 8. Example 2: State the transformations from the parent graph. Then graph it, state the domain & range. a. f(x) = 3 x 2 b. f(x) = 2 x-1

2 9. Example 3: In 2006, there were 1,020,000,000 people worldwide using the internet. At that time, the number of users was growing by 19.5% annually. Draw a graph showing how the number of users would grow from 2006 to 2016 if that rate continued. 10. Example 4: Graph each function. State the transformations, domain and range. a. f(x) = (1/5) x b. f(x) = -4(1/2) x Solving Exponential Equations and Inequalities 1) Solve exponential equations and inequalities 2) Solve compound interest problems CCSS: A.CED.1, F.LE.4 MATHEMATICAL PRACTICES: 2 reason abstractly and quantitatively 1. Exponential equation an equation in which the variables occur as exponents 2. Property of equality for exponential functions: a. b x = b y if and only if x = y b. Let b > 0 & b 1 3. Compound interest formula: A = P(1 + r/n) nt a. A the amount in the account after t years b. P the principal amount invested c. r the annual interest rate d. n the number of compounding periods per year e. t the time in years 4. Property of Inequality for exponential functions: a. b x > b y if and only if x > y b. b x < b y if and only if x < y c. Let b > 1 5. Example 1: Solve each equation a. 3 x = 9 4 b. 2 5x = 4 2x Example 2: In 2000, the population of Phoenix was 1,321,045. By 2007, it was estimated at 1,512,986. a. Write an exponential function that could be used to model the population of Phoenix. Write x in terms of the number of years since b. Predict the population of Phoenix in Example 3: An investment account pays 5.4% interest compounded quarterly. If $4000 is placed in this account, find the balance after 8 years. 8. Example 4: Solve 5 3 2x > 1/625.

3 7.3 Logarithms and Logarithmic Functions 1) Convert equations from exponential to logarithmic form and vice versa. 2) Evaluate logarithmic expressions 3) Identify properties of logarithmic graphs 4) Graph logarithmic functions 5) Describe transformations of logarithmic graphs CCSS: F.IF.7.e, F.BF.3 MATHEMATICAL PRACTICES: 6 attend to precision 1. Logarithm the exponent, y, in the inverse of the exponential function a. For the function y = b x, its inverse would be x = b y. Y is known as the logarithm of x. b. Log b x = y if and only if b y = x. 2. Graphs of logarithmic functions a. Parent graph: f(x) = log b x b. Type: continuous, one-to-one c. Domain: all positive real numbers d. Range: all real numbers e. Asymptote: y-axis f. x-intercept: (1,0) 3. Transformations of logarithmic functions: the same transformation rules apply 4. Example 1: Write each equation in exponential form. a. log 3 9 = 2 b. log 10 1/100 = Example 2: Write each equation in logarithmic form. a. 5 3 = 125 b. 27 1/3 = 3 6. Example 3: Evaluate log Example 4: a. Graph f(x) = log 3 x b. Graph log 1/4 x 8. Example 5: Describe the transformations of each graph. Then graph the function a. f(x) = (1/3)log 6 x 1 b. f(x) = 4log 1/3 (x+2) 7.4 Solving Logarithmic Equations and Inequalities 1) Convert equations from exponential to logarithmic form and vice versa. 2) Evaluate logarithmic expressions 3) Identify properties of logarithmic graphs 4) Graph logarithmic functions 5) Describe transformations of logarithmic graphs CCSS: A.SSE.2, A.CED.1 MATHEMATICAL PRACTICES: 4 Model with mathematics

4 (7.4) 1. Property of Equality for Logarithmic Functions: log b x = log b y if and only if x = y 2. Property of Inequality for Logarithmic funcions: a. If b>1, x>0, and log b x > y, then x > b y b. If b>1, x>0, and log b x < y, then 0 <x < b y c. If b >1, then i. log b x > log b y if and only if x > y ii. log b x < log b y if and only if x < y 3. Example 1: Solve log 8 x = 4/3 4. Example 2: Solve log 4 x2 = log 4 (-6x 8) 5. Example 3: Solve log 6 x > 3 6. Example 4: Solve log 7 (2x + 8) > log 7 (x + 5) 7.5 Properties of Logarithms 1) Approximate values of logarithms 2) Expand and condense logarithms using properties of logarithms 3) Apply properties of logarithms to solve logarithmic equations CCSS: A.CED.1 MATHEMATICAL PRACTICES: 8 Look for and express regularity in repeated reasoning 1. Product Property: log x ab = log x a + log x b (keep the base, add the exponents) 2. Quotient Property: log x a/b = log x a log x b (keep the base, subtract the exponents) 3. Power Property: log b m p = plog b m (keep the base, multiply the exponents) 4. Example 1: Use log to approximate the value of log Example 3: Given log , approximate the value of log Example 4: Solve the equation. 4log 2 x log 2 5 = log Common Logarithms 1) Define common log 2) Solve exponential equations and inequalities using logarithms 3) Apply the Change of Base Formula CCSS: A.CED.1 MATHEMATICAL PRACTICES: 4 Model with mathematics

5 (7.6) 1. Common logarithms logarithms using base The Change of Base Formula: log a n = log b n log b a 3. Example 1: Use a calculator to evaluate each expression to the nearest ten-thousandth. a. log 6 b. log Example 3: Solve 5 x = Example 4: Solve 3 7x > 2 5x 3 6. Example 5: Express log in terms of common logarithms. Then round to the nearest ten thousandth. 7.7 Base e and Natural Logarithms 1) Write exponential equations in logarithmic form and vice versa 2) Simplify expression with e and the natural log 3) Solve base e equations 4) Solve natural log equations and inequalities 5) Solve problems involving continuous compounding CCSS: A.SSE.2 MATHEMATICAL PRACTICES: 7 Look for and make use of structure 1. e an irrational number, that naturally occurs in situations involving continuous exponential growth and decay 2. The function f(x) = e x models continuous exponential growth 3. The function f(x) = e -x models continuous exponential decay 4. Natural logarithm the inverse of the natural base exponential function, written log e x, and abbreviated as ln x. 5. Example 1: Write each exponential equation in logarithmic form a. e x = 23 b. e 4 = x 6. Example 2: Write each logarithmic equation in exponential form. a. ln x b. ln 25 = x 7. Example 3: Write each expression as a single logarithm a. 4 ln 3 + ln 6 b. 2 ln 3 + ln 4 + ln y 8. Example 4: Solve 3e -2x + 4 = 10. Round to four decimal places. 9. Example 5: solve each equation or inequality. Round to the nearest ten-thousandth. a. 2 ln 5x = 6 b. ln (3x + 1) 2 > 8