Unit 4 Exponents and Exponential Functions

Transcription

1 Unit 4 Exponents and Exponential Functions Test Date: Name: By the end of this unit, you will be able to Multiply and divide monomials using properties of exponents Simplify expressions containing exponents Differentiate the outcome between a negative sign in the base or in the power of an expression with exponents Understand the relationship between rational exponents and nth roots Use the Power Property of Equality to solve exponential equations Distinguish between a linear and exponential function in the equation, table, and graph Describe the domain and range for an exponential function Graph an exponential growth/decay function

2 Table of Contents Multiplication Properties of Exponents... 3 Division Properties of Exponents... 6 Square Roots as Exponents n th Roots Rational Exponents Solving Exponential Equations Exponential Functions Identifying Exponential Behavior Exponential Growth vs. Decay Exponential Functions Practice Summarize: Graphs of Exponential Functions Exponential Growth and Decay Exponential Growth Compound Interest Exponential Decay

3 4.1 Multiplication Properties of Exponents A monomial is an expression with connected only by multiplication and division. No No in the denominator A constant is a monomial which is a. Examples: Monomial Not a Monomial Try this! Expand and evaluate the following: 1. 2 " 2 $ 2. 4 & 4 " 3. x ( x What do you notice? Product of Powers Property: Examples: 1. 5 " 5 & 2. a(a, )(a & ) 3. xy xy 4. (6n & )(2n 1 ) 5. 6cd ( 5c ( d " 6. ( 4xy " z & )( 6x ( y " z) 3

4 Try this! Expand and evaluate the following: 1. 3 " $ 2. 2 ( " 3. x $ & What do you notice? Power of a Power Property: Examples: 1. 2 & " 2. 3 & $ ( 3. x (, 4. x " & " Try this! Expand and evaluate the following: 1. xy & 2. 2z $ What do you notice? Power of a Product Property: Examples: 1. xy $, 2. 3p ( t, $ 3. 4a $ b : c " 4. 4x " y ( z ; & CHALLENGE: 1. Simplify 5xy & 3x " y " & " 2. Simplify 3x ( $ x " y & (, 4

5 Warm Up: b & b >> = 2 ( & = 2xy " ( = Reminder: When MULTIPLYING powers with the same base, ADD the exponents. When raising a power to a power, MULTIPLY the exponents. When there s a lot going on, follow the order of operations: P: Take care of anything inside parentheses. Start with the innermost set of parentheses. E: Take care of exponents. Raise everything inside parentheses to the power! M: Multiply everything together. o Combine like terms o Add exponents Examples 1. 2a & $ a & & 2. c & " 3c ( " 3. 5x " y " 2xy & z & (4xyz) 4. 2x " y & ( 3y " 5

6 Division Properties of Exponents 1. Quotient of Powers Property Expand and Simplify: " A = C B = " B C D In words: To divide two powers with the same base, the exponents. In symbols: For any nonzero number a, and any integers m and p, Examples: 1. EFF E G 2. HD I J HI K 3. LJ L K 4. MA N FO P M J N D P 2. Power of a Quotient Property Expand and Simplify: & & = E " = $ Q In words: To find the power of a quotient, find the power of the numerator and the denominator. In symbols: For any real numbers a and b not equal to zero, and any integer m, Examples: 1. & ( $ 2. &P D 1 " 3. &R B $ & 4. "S K &R D " 6

7 3. Zero Exponent Property Expand and Simplify: & J & J = Use the Quotient of Power Property: & J & J = In words: A zero exponent is any nonzero number raised to the zero power. It is always equal to 1. In symbols: For any nonzero number a, Examples: 1. T E U 2. RJ S 0 R D 3. "R D S R K S U 4. "TA E O T K 4. Negative Exponent Property Expand and Simplify: E K E J = Use the Quotient of Powers Property: E K E J = In words: For a (a not zero) and n (any number), a WX and a X are reciprocals. In symbols: For any nonzero number a and any integer n, Examples: 1. 2 W$ 2. > Y ZB 3. > & ZK 4. x W>U 5. XZD P B L ZK 6. [ZD \R K \S ZJ 7

8 Directions: Simplify each of the following a 4 b 6 ab c2 d 3 W4c 2 d 4. 4f 3 g 3h W$RK "$R J 6.,\ J 1P j L D " 7. x 3 (y W5 )( x W8 ) 8. & 1 W" 9. ""L D m K >>L K m ZD 10. 6fZ2 g 3 h 5 54f Z2 g Z5 h W>"CZF n J R ZB 12. ("ozk T) ZD "C ZD nr J (o K T B 8

9 Directions: Simplify each of the following. 1. m5 np m 4 p 2. 5c2 d 3 W4c 2 d 3. 8y7 z 6 4y 6 z f 3 g 3h W$RK "$R J 6.,\ J 1P j L D " 7. x 3 (y W5 )( x W8 ) 8. & 1 W" 9. ""L D m K >>L K m ZD 10. 6fZ2 g 3 h 5 54f Z2 g Z5 h W>"CZF n J R ZB "C ZD nr J 12. NZK X ZJ N B X D ZF 13. jz1 k3 Z4 ("ozkt)zd 14. j 3 k 3 (o K T B 15. "R D S K x &R B Sx ZK W" 9

10 Square Roots as Exponents Do Now: Use your calculator to evaluate the following. 16 = (16) F K = (100) F K = 100 = Calculator Tutorial #1 Use parentheses to evaluate expressions involving rational exponents on a graphing calculator. For example, to find 125 F D, press 125 [^] [ ( ] 1 [ ] 3 [ ) ] [ENTER]. What do you notice? Why is this happening? Check it out: b F K " = Definition: Examples: Write each expression in radical form, or write each radical in exponential form. Example 1: 25 F K Example 2: 18 Example 3: 5x F K Example 4: 8p Example 5: 49 F K Example 6: 22 Example 7: 7w F K Example 8: 2 x 10

11 n th Roots Use your calculator to evaluate the following. 6 & D = 216 2, j = 64 What do you notice? = = Calculator Tutorial #2 To use exponents, press the caret symbol (^) to raise a number to a power. Calculator Tutorial #3 To find n th roots, enter your number n, then press [MATH] and choose. (5) We know that if 8 " = 64, then 64 = 8. Similarly, if 2 $ B = 16, then 16 = 2. Definition: For any real numbers a and b and any positive integer n, if a X = b, then a is an nth root of b. Examples: Evaluate. D Example 1: 27 J Example 2: 32 D Example 3: 64 B Example 4: 10,000 Like square roots, nth roots can be represented by rational exponents. Definition (Part 2): Examples: Use the n th root definition to convert forms and evaluate. Example 1: 125 F D Example 2: 1296 F B Example 3: 27 F D Example 4: 256 F B 11

12 Rational Exponents Simplify these expressions using Multiplication Properties: Simplify these expressions using the n th root definition: 36 F K & = 36 F K & = 32 $ F J = 32 $ F J = Definition: Examples: Convert forms and evaluate the following expressions. Example 1: 8 K D Example 2: 64 K D Example 3: 36 D K Example 4: 27 K D Example 5: 256 J B Example 6: 81 J K Example 7: 7w D J K Example 8: 2 x & Challenge Problems: 1. 8 K D WJ B x " y $ W F K

13 Solving Exponential Equations Warm Up: Answer the following questions to what power is 32? 2. 6 to what power is 216? 3. 5 to what power is 625? Find a solution to the following equations R = R = R = 625 The Power Property of Equality As long as b is a real number greater than zero and not equal to 1, then b R = b S if and only if x = y. Examples: 1. If 5 R = 5 &, then x = If n = > ", then 4X = 4 F K R = &R > = 81 This property helps us when solving more complicated exponential equations (like example 4). Another Example: 25 RW> = 5 13

14 Examples: Solve each equation for x &R = "R = 9 R > RW> = $R = 32 R > R = > " 6. > &, R > = > "1 R = R = > >"( 1. The sun protection factor (SPF) of a sunscreen indicates how well it protects you from the sun s harmful rays. Sunscreen with an SPF of f absorbs about p percent of the UV-B rays, where p = 50f U.". Find the SPF that absorbs 100% of UV-B rays. 2. The population p of a culture that begins with 40 bacteria and doubles every 8 hours is modeled by p = 40 2 G, where t is time in hours. Find t if p = 20,

15 Exponential Functions The zombies are here Each night, every zombie will infect a new person How many nights do you think it will take to infect the whole room? Night # of zombies Write a function that represents this scenario: An exponential function has the form The following restrictions apply: Note: The base is a. The exponent is a. Directions: Use your table above to graph the function. 1. What is the y-intercept of the function? What does it represent in this scenario? 2. What is the domain of the function? 3. What is the range of the function? Summarize: How do you find the y-intercept? How do you find the domain and range? 15

16 Identifying Exponential Behavior Up until now, we have been working with linear functions. The graph of a linear function is, and a linear function has a. There are 2 methods we can use to determine whether a function is linear vs. exponential: 1. Graphing Example: Graph the data in the table. Determine whether the relationship is linear or exponential. x y Looking for a constant ratio Example: Exponential functions have constant ratios instead of a constant rate of change. This means that if the x- values are at regular intervals and the y-values differ by a common factor, the data is probably exponential. In this example, the constant ratio is. Summarize: How can you determine whether a function is linear or exponential? 16

17 Exponential Growth vs. Decay After the zombie outbreak, our class is now full of zombies. The school administration figures out what s going on and sends Principal Wayne to clear our class of the zombie epidemic. Principal Wayne can cure one half of the remaining zombies each day with a vaccine created in Mr. Benters Biology Lab. When will our entire class be cured? Write a function that represents this scenario: Day # of zombies Use your table to graph the function below. 1. What is the y-intercept? What does that represent in this scenario? 2. What is the domain? 3. What is the range? A slightly more realistic biology example: A certain bacteria population doubles in size every 20 minutes. Beginning with 10 cells in a culture, the population can be represented by the function B = 10 2 C, where B is the number of bacteria cells and t is the time in 20 minute increments. How many bacteria cells will there be after 2 hours? 17

18 Exponential Functions Practice Create a table and graph the function. You will need to choose which values to use in your table. Identify the y-intercept, domain, and range of each function. Also identify whether the function represents exponential growth or decay. USE PENCIL! 1. y = 2 R Growth or decay? (circle one) y-intercept: Domain: Range: x y 2. y = 2 R 1 Growth or decay? (circle one) y-intercept: Domain: Range: x y 3. y = 2 R + 3 Growth or decay? (circle one) y-intercept: Domain: Range: x y Class Discussion: 18

19 4. y = > " RW> x y Growth or decay? (circle one) y-intercept: Domain: Range: 4. y = > " R " x y Growth or decay? (circle one) y-intercept: Domain: Range: 5. y = > " RW" + 6 x y Growth or decay? (circle one) y-intercept: Domain: Range: Class Discussion: 19

20 Summarize: Graphs of Exponential Functions Exponential Growth Functions Equation: Exponential Decay Functions Equation: Domain: Domain: Range: Range: Intercepts: Intercepts: End behavior: End behavior: Sketch of graph: Sketch of graph: 20

21 Exponential Growth and Decay Exponential Growth The number of online blogs has rapidly increased in the last 15 years. In fact, the number of blogs increased at a monthly rate of about 13.7% over 21 months, starting with 1.1 million blogs in November The average number of blogs per month from can be modeled by the equation y = C or y = C where y represents the total number of blogs in millions and t is the number of months since November Label the diagram below with what each variable or constant represents. Calculator Tutorial #4 When solving exponential equations, you will often encounter unfriendly decimals. If you round these before your final answer, you may get a slightly incorrect answer. On your calculator, use the [2 nd ] [(-)] keys to get [Ans], your EXACT previous answer. y = C In general, the equation for exponential growth is as follows: y = a 1 + r C Example 1: The prize for a radio station contest begins with a $100 gift card. Once a day, a name is announced. The person has 15 minutes to call or the prize increases by 2.5% for the next day. a. Write an equation to represent the amount of the gift card in dollars after t days with no winners. b. How much will the gift card be worth if no one wins after 10 days? Example 2: A college s tuition has risen 5% each year since If the tuition in 2000 was $10,850, write an equation for the amount of the tuition t years after Predict the cost of tuition for this college in

22 Compound Interest Compound interest is a special kind of exponential growth. It is interest earned or paid both on the initial investment and previously earned interest. In general, the equation for compound interest is as follows: A = P 1 + r n XC Example 3: Maria s parents invested $14,000 at 6% per year compounded monthly. How much money will there be in the account after 10 years? Example 4: Determine the amount of an investment if $300 is invested at an interest rate of 3.5% compounded every other month for 22 years. Exponential Decay In general, the equation for exponential decay is as follows: y = a 1 r C Example 5: A fully inflated child s raft for a pool is losing 6.6% of its air every day. The raft originally contained 4500 cubic inches of air. a. Write an equation to represent the loss of air. b. Estimate the amount of air in the raft after 7 days. Example 6: The population of Campbell County, Kentucky has been decreasing at an average rate of about 0.3% per year. In 2000, its population as 88,647. Write an equation to represent the population since If the trend continues, predict the population in