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## Chapter 7 Exponential and Logarithmic Functions

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**Table of Contents**• 7.1 – Exponential Functions, Growth, and Decay • 7.2- Inverses of Relations and Functions • 7.3 – Logarithmic Functions • 7.4- Properties of Logarithms • 7.5 – Exponential and Logarithmic Equations and Inequalities • 7.6 – The Natural Base, e • 7.7 – Transforming Exponential and Logarithmic Functions**Algebra II (Bell work)**• Do Problems # (Skip) (7.1) • Define exponential function and asymptote • Read student to student pg. 491**7.1 Exponential Functions, Growth, and Decay**Algebra II**7-1**Growth that doubles every year can be modeled by using a function with a variable as an exponent. This function is known as an exponential function. The parent exponential function is f(x) = bx, where the baseb is a constant and the exponent x is the independent variable.**7-1**Just Read The graph of the parent function f(x) = 2xis shown. The domain is all real numbers and the range is {y|y> 0}.**7-1**B > 0, exponential growth 0 < b < 1,exponential decay Notice as the x-values decrease, the graph of the function gets closer and closer to the x-axis. The function never reaches the x-axis because the value of 2xcannot be zero. In this case, the x-axis is an asymptote. An asymptoteis a line that a graphed function approaches as the value of x gets very large or very small. as·ymp·tote**7-1**The base , ,is less than 1. This is an exponential decay function. Tell whether the function shows growth or decay. Then graph. Step 1 Find the value of the base. Step 2 Graph the function by using a table of values.**7-1**Tell whether the function shows growth or decay. Then graph. g(x) = 100(1.05)x Step 1 Find the value of the base. The base, 1.05, is greater than 1. This is an exponential growth function. g(x) = 100(1.05)x**7-1**Tell whether the function p(x) = 5(1.2x) shows growth or decay. Then graph. Step 1 Find the value of the base. The base , 1.2, is greater than 1. This is an exponential growth function. p(x) = 5(1.2x) Step 2 Graph the function by using a table of values.**7-1**Math Joke • Q: How did you know that your dentist studied algebra? • A: She said all that candy gave me exponential decay.**7-1**You can model growth or decay by a constant percent increase or decrease with the following formula: In the formula, the base of the exponential expression, 1 + r,is called the growth factor. Similarly, 1 – ris the decay factor.**7-1**Clara invests $5000 in an account that pays 6.25% interest per year. After how many years will her investment be worth $10,000? Step 1 Write a function to model the growth in value of her investment. f(t) = a(1 + r)t f(t) = 5000(1 + 0.0625)t f(t) = 5000(1.0625)t Step 2 When graphing exponential functions in an appropriate domain, you may need to adjust the range a few times to show the key points of the function. Step 3 Use the graph to predict when the value of the investment will reach $10,000. Use the feature to find the t-value where f(t) ≈ 10,000. ** Can also use 2nd , Table to find it****7-1**Table Feature The function value is approximately 10,000 when t ≈ 11.43 The investment will be worth $10,000 about 11.43 years after it was purchased.**7-1**Graph the function. Use to find when the population will fall below 8000. A city population, which was initially 15,500, has been dropping 3% a year. Write an exponential function and graph the function. Use the graph to predict when the population will drop below 8000. f(t) = a(1 – r)t f(t) = 15,500(1 – 0.03)t f(t) = 15,500(0.97)t It will take about 22 years for the population to fall below 8000.**7-1**Graph the function. Use to find when the population will reach 20,000. In 1981, the Australian humpback whale population was 350 and increased at a rate of 14% each year since then. Write a function to model population growth. Use a graph to predict when the population will reach 20,000. P(t) = a(1 + r)t P(t) = 350(1 + 0.14)t P(t) = 350(1.14)t It will take about 31 years for the population to reach 20,000.**7-1**Graph the function. Use to find when the value will fall below 100. A motor scooter purchased for $1000 depreciates at an annual rate of 15%. Write an exponential function and graph the function. Use the graph to predict when the value will fall below $100. f(t) = a(1 – r)t f(t) = 1000(1 – 0.15)t f(t) = 1000(0.85)t It will take about 14.2 years for the value to fall below 100.**7-1**HW pg. 493 • 1-15, 21, 28, • B: 22 • On each graphing problem have a table of at least 5 values. Have points on either side of the y axis**Algebra II (Bell work)**• Turn in all of (7.1) • Define Inverse Relation and inverse function**7.2 Inverses of Relations and Functions**Algebra II**You can also find and apply inverses to relations and**functions. To graph the inverse relation, you can reflect each point across the line y = x. This is equivalent to switching the x- and y-values in each ordered pair of the relation.**7-2**Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. Just Watch ● ● ● ● Graph each ordered pair and connect them. Switch the x- and y-values in each ordered pair.**7-2**Just Watch • Reflect each point across y = x, and connect them. Make sure the points match those in the table. • • • • • Domain:{x|0 ≤ x ≤ 8} Range :{y|2 ≤ x ≤ 9} • • Domain:{x|2 ≤ x ≤ 9} Range :{y|0 ≤ x ≤ 8} Blue Line Red Line**7-2**# 2 pg. 501 Graph the relation. Graph the Inverse. Identify the domain and range of each relation**7-2**When the relation is also a function, you can write the inverse of the function f(x) as f–1(x). This notation does not indicate a reciprocal. Functions that undo each other are inverse functions. To find the inverse function, use the inverse operation. In the example above, 6 is added to x in f(x), so 6 is subtracted to find f–1(x).**7-2**1 2 1 1 f(x) = x – 2 2 f–1(x) = x + Use inverse operations to write the inverse of f(x) = x – if possible.**7-2**1 1 1 = 2 2 1 1 2 1 1 1 f(x) = x – f(1) = 1 – 2 2 2 2 2 f–1(x) = x + Substitute for x. f–1( ) = + The inverse function does undo the original function. Check Use the input x = 1 in f(x). Substitute 1 for x. Substitute the result into f–1(x) = 1**7-2**Math Joke • Q: How did the chicken find the inverse? • A: It reflected the function across y = eggs**7-2**x 3 x 3 Use inverse operations to write the inverse of f(x) =. f(x) = f–1(x) = 3x**7-2**1 x 3 3 1 1 1 = 1 3 3 3 3 f–1( ) = 3( ) Substitute for x. The inverse function does undo the original function. CheckUse the input x = 1 in f(x). f(x) = f(1) = Substitute 1 for x. Substitute the result into f–1(x) f–1(x) = 3x = 1**7-2**f(x) = x + 2 2 2 3 3 3 f–1(x) = x – Use inverse operations to write the inverse of f(x) = x + .**7-2**f(1) = 1 + 5 5 5 = 3 5 3 3 2 2 2 2 f(x) = x + 3 3 3 3 3 Substitute for x. f–1(x) = x – f–1( ) = – The inverse function does undo the original function. CheckUse the input x = 1 in f(x). Substitute 1 for x. Substitute the result into f–1(x) = 1**7-2**1 1 3 3 f–1(x) = x + 7 f–1(6) = (6) + 7= 2 + 7= 9 Use inverse operations to write the inverse of f(x) = 3(x – 7). f(x) = 3(x – 7) Check Use a sample input. f(9) = 3(9 – 7) = 3(2) = 6**7-2**f–1(x) = x +7 10 3 + 7 5 5 5 f(2) = 5(2) – 7 = 3 f–1(3) = = = 2 Use inverse operations to write the inverse of f(x) = 5x – 7. f(x) = 5x – 7. Check Use a sample input.**Graph f(x) = – x– 5 . Then write the inverse**and graph. y = – x – 5 x = – y – 5 1 1 1 1 2 x + 5= – y 2 2 2 –2x –10 = y y =–2(x + 5) f–1(x) = –2(x + 5) f–1(x) = –2x – 10**1**f(x) = – x – 5 2 f–1(x) = -2x - 10**7-2**Graph f(x) = x+ 2. Then write the inverse and graph. f–1(x) = y = x + 2 x = y + 2 2 2 3 3 2 2 3 3 3 3 2 2 x – 2 = y x – 3 x – 3 = y 3x – 6 = 2y**7-2**f–1(x) = 3 2 x – 3 Graph f(x) = x + 2**7-2**L = c – 2.50 = L 13.70 – 2.50 0.80 0.80 Juan buys a CD online for 20% off the list price. He has to pay $2.50 for shipping. The total charge is $13.70. What is the list price of the CD? c = 0.80L + 2.50 Step 1 Write an equation for the total charge as a function of the list price. c–2.50 = 0.80L Step 2 Find the inverse function that models list price as a function of the change. Step 3 Evaluate the inverse function for c = $13.70. = 14 The list price of the CD is $14.**7-2**To make tea, use teaspoon of tea per ounce of water plus a teaspoon for the pot. Use the inverse to find the number of ounces of water needed if 7 teaspoons of tea are used. 1 1 1 6 6 6 t = z + 1 Step 1 Write an equation for the number of ounces of water needed. t– 1 = z Step 2 Find the inverse function that models ounces as a function of tea. 6t – 6 = z Step 3 Evaluate the inverse function for t = 7. z = 6(7) – 6 = 36 36 ounces of water should be added.**7-2**HW pg. 501 • 7.2- • 3-17 (Odd), 20-25, 41-46, • 38 (Good Bell work)**Algebra I (Bell work)**• Look over 8.4/Packet for any questions • Then we will cover quiz**Quiz 8.1-8.3**• If you got an A on the last quiz, you automatically get 5 (QEC/TEC) points • For everyone else, improve your letter grade by at least one on the new quiz, you get 5 (QEC/TEC) • Must have passed the first quiz to be eligible to get extra credit • We will take the Quiz 8.1-8.3 Version B tomorrow • You will get a worksheet as an assignment tomorrow after quiz**Algebra II (Bell work)**• Turn in all of (7.2) • Have your books/notebooks open to 7.3**7.3 Logarithmic Functions**Algebra II**7-3**Reading Math Read logba=x, as “the log base b of a is x.” Notice that the log is the exponent. You can write an exponential equation as a logarithmic equation and vice versa.**7-3**1 1 log6 = –1 6 6 1 1 2 log255 = 2 Write each exponential equation in logarithmic form. log3243 = 5 log1010,000 = 4 logac =b**7-3**1 1 3 3 8 = 2 1 4–2 = 16 Write each logarithmic form in exponential equation. 91 = 9 29 = 512 1 16 b0 = 1**7-3**A logarithm with base 10 is called a common logarithm. If no base is written for a logarithm, the base is assumed to be 10. For example, log 5 = log105.**7-3**Math Joke • Q: Why are you drumming on your algebra book with two big sticks? • A: Because we’re studying log rhythms