7-1. Exploring Exponential Models. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary. 1. Cross out the expressions that are NOT powers.
|
|
- Alisha Greene
- 6 years ago
- Views:
Transcription
1 7-1 Eploring Eponential Models Vocabular Review 1. Cross out the epressions that are NOT powers. 16 6a 1 7. Circle the eponents in the epressions below a z Vocabular Builder eponential deca (noun) ek spoh NEN shul dee KAY Related Word: deca (verb) Definition: For the function 5 ab, if a. 0 and 0, b, 1, the function represents eponential deca. Math Usage: The equation 5 (0.98) represents eponential deca. Main Idea: Something that decreases as time passes is decaing. Eample: A car loses about 15% of its value ever ear. Its value decas eponentiall. Use Your Vocabular. Circle the equations that represent eponential deca. 5(0.1) 5 (0.) 5 Q R 5 Q 7 5 R P 5 P 0 (1.05) n N 5 N 0 (0.5) t ? ? n. Write numbers to make each equation represent eponential deca. Answers ma var. Samples are given. 5 Q R n m a b Q 5 Q 0 a b t Chapter 7 186
2 Problem 1 Graphing an Eponential Function Got It? What is the graph of 5? 5. Complete the following table of values. 6. Graph the points from our table. Then connect the points with a smooth curve O 16 Summar Eponential Functions For the function 5 ab, if a. 0 and b. 1, the function represents eponential growth. if a. 0 and 0, b, 1, the function represents eponential deca. In either case, the -intercept is (0, a), the domain is all real numbers, the asmptote is 5 0, and the range is Write D if the function represents eponential deca. Write G if it represents eponential growth. G Problem 5 (5 ) G 5 1 Q7 R D 5 5Q 1 R D Identifing Eponential Growth and Deca Got It? Identif 5 () as an eample of eponential growth or deca. What is the -intercept? 8. Compare 5 () to the function 5 ab. Complete each equation. a 5 b 5 Underline the correct word, epression, or phrase to complete each sentence. 9. The value of b is between 0 and 1 / greater than So, the function represents eponential growth / deca. 11. The -intercept is (0, 0) / (1, 0) / (0, ). 5 (0.5) 187 Lesson 7-1
3 Ke Concept Eponential Growth and Deca You can model eponential growth or deca with this function. Amount after t time periods Rate of growth (r. 0) or deca (r, 0) Initial amount A(t) 5 a(1 1 r) t Number of time periods For growth or deca to be eponential, a quantit changes b a fied percentage each time period. 1. Draw a line from the function in Column A to the situation it models in Column B. Column A Column B B(m) 5 00(1 0.0) m You bu 50 mice. Ever month, the population increases b 15%. P(m) 5 50(1.15) m S(t) 5 8(1.05) t You have 1 lb of dog food. Your dog eats a quarter of it the net da, and each da afterward the dog eats a quarter of what s left. You get a job paing $8 per hour. Ever ear, the salar increases b 5% to keep up with the cost of living. A(d) 5 1(0.75) d Your savings account holds $00. For all accounts under $1000, the bank charges a fee of % per month, drawn from the account. Problem Modeling Eponential Growth Got It? Suppose ou invest $500 in a savings account that pas.5% annual interest. How much will be in the account after five ears? 1. An eponential growth model applies to this problem because the amount in the bank increases / decreases b a fied percentage of.5 % each ear. 1. Complete the model. a 5 the initial amount invested 5 r 5 the rate of growth per ear written as a decimal 5 A(t) 5 a(1 1 r) t Q R t 15. Evaluate the model for t A(t) 5 a(1 1 r) t A 5 500( ) 5 A 5 500(1.05) 5 A 5 $ Chapter 7 188
4 Problem Using Eponential Growth Got It? Suppose ou invest $500 in a savings account that pas.5% annual interest. When will the account contain $650? 16. Complete the model. a 5 r the initial amount invested A(t) 5 a(1 1 r) t Q 1.05 R t 17. Use the TABLE feature on our calculator to complete the table. Use for t and for A(t). 18. The account will contain $650 after 8 ears Lesson Check Do ou UNDERSTAND? Vocabular Eplain how ou can tell whether 5 ab represents eponential growth or eponential deca () represents eponential growth. The value of b in this function is Q 1 R 1 represents eponential deca. The value of b in this function is. 1. Now answer the question. Answers ma var. Sample: If b S 1, the function represents eponential growth. If 0 R b R 1, the function represents eponential deca. Math Success Check off the vocabular words that ou understand. eponential function eponential growth eponential deca Rate how well ou can model eponential growth and deca. Need to review Now I get it! 189 Lesson 7-1
5 7- Properties of Eponential Functions Vocabular Review The figure in blue is the image of the figure in black. Identif each transformation as a translation, reflection, or dilation O translation dilation reflection Vocabular Builder eponential growth (noun) ek spoh NEN shul grohth Opposite: eponential deca Definition: For the function 5 ab, if a. 0 and b. 1, the function represents eponential growth. Eample: A savings account earns percent interest annuall. The amount in the account grows eponentiall. The equation 5 0.5(9) represents eponential growth. Use Your Vocabular. Circle the functions that represent eponential growth. 5.5(0.75) 55(1.0) 5 0.8(.7) 5 6(9.) 5. Cross out the functions that do NOT represent eponential growth. 5 (0.5) 5 6( 9.) 5() 5 (7) Chapter 7 190
6 Problem 1 Graphing 5 ab Got It? How does the graph of 50.5? 5 compare to the graph of the parent function? The graphs of 5 5 (black) and 50.5? 5 (blue) are shown at the right. Underline the correct words or value to complete each sentence. 6. Graphing 50.5? 5 involves a reflection / translation of the parent function. 7. The domain of 50.5? 5 is / is not the same as the domain of the parent function. 8. The range of 50.5? 5 is / is not the same as the range of the parent function. 9. The -intercept of 50.5? 5 is (0, 1) / (0, 0.5). 10. Multipling 5 b 0.5 stretches / compresses the vertical scale. Problem Translating the Parent Function 5 b Got It? How does the graph of 5 (1) compare to the graph of the parent function? 11. The graph of 5 (1) is a horizontal / vertical translation of 5 two units to the left / right. 1. The graph at the right shows 5. Sketch the graph of 5 (1) on the same set of aes. Parent Function Stretch A u au. 1B Compression (Shrink) A0, u au, 1B Reflection (a, 0) in -ais Concept Summar Families of Eponential Functions Translations (horizontal b h; vertical b k) 5 b 5 ab 5 b (h) 1 k 6 O All transformations combined 5 ab (h) 1 k 191 Lesson 7-
7 1. Each blue figure is a transformation of each black figure. Label the transformation as a vertical stretch, vertical compression, reflection, or translation. 1O O 1 stretch reflection translation translation or translation or compression or compression Problem Using an Eponential Model Got It? The initial temperature of a cup of coffee is 0 F. An eponential model for the temperature of the coffee after minutes is 5 1.5? How long does it take for the coffee to reach a temperature of 100 F? 1. Enter the function into our graphing calculator. Then complete the table It takes about 1.9 minutes to cool to 100 F. Ke Concept Continuousl Compounded Interest The formula for continuousl compounded interest uses the number e: amount in account at time t A(t) P e rt Principal Problem 5 Continuousl Compounded Interest interest rate (annual) time in ears Got It? Suppose ou won a contest at the start of 9th grade that deposited $000 in an account that pas 5% annual interest compounded continuousl. You start college ears later, and spend ears in college. About how much will be in the account after ears of college? Chapter 7 19
8 16. The mone is earning interest over 8 ears. 17. Use the justifications at the right to solve the problem. A 5 P? e r?t 5 000? e Q 0.05 RQ 8 R Substitute values for P, r, and t. < $ 75 Use a calculator. Round to the nearest dollar. Lesson Check Do ou UNDERSTAND? Reasoning Is investing $000 in an account that pas 5% annual interest compounded continuousl the same as investing $1000 at % and $1000 at 6%, each compounded continuousl? Eplain. 18. Cross out the equation(s) that ou could NOT use to solve this problem. A 5 000? e (0.05)t A ? e (0.05)t A ? e (0.0)t A Ae (0.05)t 1 e (0.06)t B A ? e (0.06)t 19. Use a calculator to complete the table. Round to the nearest dollar. t 000 e 0.5t 1000 e 0.t 1000 e 0.6t 1000 e 0.t 1000 e 0.6t 1 $ 10 $ 101 $ 106 $ 10 $ $ 118 $ 1197 $ 5 5 $ Now answer the question. $ 11 $ 150 $ 571 No. One account at 5% earns less than two accounts at % and 6%. Math Success Check off the vocabular words that ou understand. natural base eponential function Rate how well ou can graph eponential functions with base e. Need to review Eplanations ma var. Sample: continuousl compounded interest Now I get it! 19 Lesson 7-
9 7- Logarithmic Functions as Inverses Vocabular Review 1. Circle the base in each power. 5 5 ( ) Vocabular Builder logarithm (noun) LAWG uh rith um Related Words: base, power, eponent Definition: The logarithm base b of a positive number is the eponent to which base b must be raised to get. Eamples: Since 5 16, the logarithm base of 16 is. Similarl, since 5 6, the logarithm base of 6 is. Math Usage: A logarithm is written as log b and is read log base b of. Use Your Vocabular Write each logarithm in words. The first one is done for ou. Sample: log 16 log base of 16. log. log. log 5. log log base of log base of log base of log base of Ke Concept Logarithm A logarithm base b of a positive number satisfies the following definition. log b 5 if and onl if b 5 Chapter 7 19
10 6. Draw a line from each logarithm equation in Column A to its eponential equation in Column B. Column A Column B log log 9 5 b 5 log log b Problem 1 Writing Eponential Equations in Logarithmic Form Got It? What is the logarithmic form of the equation 6 5 6? 7. Use the definition of logarithm: If 5 b, then log b 5. Since 6 5 6, log The logarithmic form of the equation is log Problem Evaluating a Logarithm Got It? What is the value of log 5 15? 9. Use the justifications at the right to complete the equations. 5 log 5 15 Write the logarithm Use the definition of a logarithm to write an eponential equation Write 15 as a power of 5. 5 Since the bases are the same, the eponents must be equal. log Write the value of log A common logarithm is a logarithm with base 10. You can write a common logarithm log 10 simpl as log, without showing the 10. Problem Using a Logarithm Scale Got It? In 1995, an earthquake in Meico registered 8.0 on the Richter scale. In 001, an earthquake of magnitude 6.8 shook Washington state. How man times more intense was the 1995 earthquake than the 001 earthquake? Use the formula log I 1 I 5 M 1 M to compare the intensit levels of earthquakes, where I is the intensit level and M is the magnitude on the Richter scale. 10. Circle the equation that models the problem. log I 1 I log I 1 I log I 1 I Lesson 7-
11 11. Simplif the equation ou circled in Eercise 10. log I 1 I Circle the common logarithm that corresponds to the simplified equation. I 1 I I 1 I I 1 I Use a calculator. The 1995 earthquake was about times more intense than the 001 earthquake. Problem Graphing a Logarithmic Function Got It? What is the graph of 5 log? Describe the domain, range, -intercept, and asmptotes. 1. The graph of 5 is at the right. Circle the graph of its inverse 5 log O 8 16 O O Draw a line from each item in Column A to a corresponding item in Column B. Column A Column B Domain of 5 log 5 0 Range of 5 log. 0 -intercept of graph of 5 log. 0 Vertical asmptote(s) of 5 log none all real numbers Concept Summar Families of Logarithmic Functions Parent function 5 log b, b. 0, b 1 Stretch (u a u. 1) Compression (shrink) (0, u a u, 1) 5 a log b Reflection (a, 0) in -ais Translations (horzontal b h; vertical b k) 5 log b ( h) 1 k All transformations together 5 a log b ( h) 1 k Chapter 7 196
12 16. Circle each equation that is a compression of the parent function. Underline each equation that is a reflection of the parent function. 5 log 5 0. log 8 5 log ( 5) log 10 Problem 5 Translating log b Got It? How does the graph of 5 log b ( ) 1 compare to the graph of the parent function? 17. The graph at the right shows the parent function, 5 log. Underline the correct word or value to complete each sentence. To graph 5 log ( ) 1, translate the graph of the parent function / / units left / right and / / units up / down. The domain becomes. /. The range does / does not change. 8 log O Lesson Check Do ou UNDERSTAND? Vocabular Determine whether each logarithm is a common logarithm. log log 6 log log Circle the base of each logarithm above. Complete. 19. Common logarithms have a base of Logarithms without a base are assumed to have a base of Write Y for es or N for no to indicate whether each logarithm is a common logarithm. N log Y log 6 Y log N log 5 5 Math Success Check off the vocabular words that ou understand. logarithm logarithmic function common logarithm logarithmic scale Rate how well ou can use and graph logarithms. Need to review Now I get it! 197 Lesson 7-
13 7- Properties of Logarithms Vocabular Review 1. Circle the logarithms below. log 6 9 log ( 1) a? b (11). Circle the base of each logarithm log log 7 9 log log 5 15 Write T for true or F for false. F T F. If log b 5, then b 5.. If log 5 w, then 10 w If 5, then log 5. Vocabular Builder formula (noun) FAWRM oo luh Other Word Forms: formulate (verb), formulaic (adjective) Definition: A mathematical formula is an equation that ou can use to solve a particular kind of problem. Eample: You can use the quadratic formula to solve quadratic equations. You can use the distance formula to find the distance between two points. Use Your Vocabular 6. Circle the slope formula. A 5 1 (b 1 1 b )h A 5 Pe r t P 5 l 1 w m Chapter 7 198
14 Since 5 log a if and onl if 5 a, logarithms and eponents have corresponding properties. Properties Properties of Logarithms For an positive numbers m, n, and b where b 1, the following properties appl. Product Propert log b mn 5 log b m 1 log b n Quotient Propert log b m n 5 log b m log b n Power Propert log b m n 5 n log b m Use the properties of logarithms to complete each equation. 7. log 0 5 log (5? ) 5 log 5 1 log 8. log 5 ( 9 ) 5 9 log 5 9. log log 5 log 5 log log (5 ) 5 log 5 1 log 5 log 5 1 log Problem 1 Simplif ing Logarithms Got It? What is log 6 log 9 written as a single logarithm? If possible, simplif the single logarithm. 11. Circle the propert ou can use to rewrite log 6. Product Propert Quotient Propert Power Propert 1. Use the propert ou circled above to rewrite the first term of log 6 log 9. log 6 log 9 5 log 6 log 9 1. Circle the propert ou can use to combine the last two terms in Eercise 1. Product Propert Quotient Propert Power Propert 1. Use the propert ou circled above to combine the two terms. 6 log 6 log 9 5 log 5 log Use the definition of logarithm to simplif the epression. log 5 1 Problem Epanding Logarithms Got It? What is log 50 7 epanded? Simplif our answer, if possible. 199 Lesson 7-
15 16. Follow the steps to epand the logarithm. log log 50 log 7 5 log 1 log 15 log 7 Use the Quotient Propert of Logarithms. Use the Product Propert of Logarithms. 5 log 1 log 5 log 7 Write 15 as a power of 5. 5 log 1 log 5 log 7 Use the Power Propert of Logarithms. Properties Change of Base Formula For an positive numbers m, b, and c, with b u 1 and c u 1, log b m 5 log c m log c b. Use the Change of Base Formula to complete each equation. log log log log log log 100 log log 5 log 5 0. Reasoning The base implied in Eercise 17 is / 5 / 10 / 100. Problem Using the Change of Base Formula Got It? What is the value of log 8? 1. Circle the least common factor of 8 and. 8. Complete each equation. Since 5 5, log 5 5. Since 5 8, log log 8 5 log log Got It? What is the value of log 18?. Circle the calculator-read form of log 18. log log 18 5 log 9 log log 9 log 5. The value of log 18 is approimatel.085. Problem 8 5 Using a Logarithmic Scale log 18 log Got It? Chemistr The ph of a substance equals log fh 1 g, where fh 1 g is the concentration of hdrogen ions. Suppose the hdrogen ion concentration for Substance A is twice that for Substance B. Which substance has a greater ph level? What is the greater ph level minus the lesser ph level? Eplain. Chapter 7 00
16 6. If fh 1 bg is the concentration of hdrogen ions for Substance B, circle the ph of Substance B. log fh 1 b g log fh1 b g log 1 fh1 b g log fh1 b g 7. Circle the epression for the concentration of hdrogen ions for Substance A. fh 1 b g? fh1 b g fh 1 b g log fh 1 b g 8. Circle the epression for the ph of Substance A. log f? fh 1 b gg log ffh1 b gg 1 log fh1 b g log fh1 b g 9. Use the Product Propert of Logarithms to epand the epression ou circled above. log f? fh 1 b gg 5 Qlog 1 log fh 1 b gr log fh 1 b g (ph of Substance A) (ph of Substance B) 0. Circle the substance with the greater ph level. Substance A Substance B 1. What is the difference between the ph levels? Eplain how ou know. log, or Eplanations ma var. Sample: The log of twice a quantit is log more than log. Lesson Check Math Success Need to review Do ou UNDERSTAND? Reasoning If log 5 5, what is the value of 1?. Underline the correct epression to complete each sentence. If log 5 5, then > Since > 1, I know > Check off the vocabular words that ou understand. logarithm Change of Base Formula Rate how well ou can use the properties of logarithms Now I get it! 01 Lesson 7-
17 7-5 Eponential and Logarithmic Equations Vocabular Review 1. Underline the base and circle the eponent in each epression. 7 ( 1 ) 1 9 (17). Multiple Choice Which epression is read five to the third power? Vocabular Builder isolate (verb) EYE soh lat Other Word Forms: isolation (noun), isolated (adjective) Definition: When ou isolate something, ou get it b itself. Math Usage: To isolate a variable in an equation means to get the variable alone on one side of the equal sign. Use Your Vocabular. Draw a line from each equation in Column A to the operation(s) needed to isolate the variable in Column B. Column A b 5 15 Column B z Divide both sides b. Multipl both sides b. Then add 5 to both sides. 1 (p 5) 5 Subtract 9 from each side, then divide both sides b. Problem 1 Solving an Eponential Equation Common Base Got It? What is the solution of ?. Circle the common base that ou can use to rewrite 7 and Chapter 7 0
18 5. Rewrite 7 and 81 using the common base. a b 5 6. If two numbers with the same base are equal, their eponents are equal / not equal. 7. Solve for. A B Problem Solving an Eponential Equation Different Bases Got It? What is the solution of ? 8. The equation has been solved below. Write a justification for each step log 5 5 log 10 log 5 5 log 10 5 log 10 log 5 Write the original equation. Take the logarithm of each side. Power Propert of Logarithms Divide each side b log 5 to isolate. 9. Use a calculator to determine the value of to four decimal places. < 1.51 Problem Solving an Eponential Equation With a Graph or Table Got It? What is the solution of ? 10. Write the two equations ou can graph on our graphing calculator to solve this equation. Y Y Use a graphing calculator to find the point of intersection of the two equations. Write the approimate value of. < Lesson 7-5
19 Problem Modeling With an Eponential Equation Got It? Resource Management Wood is a sustainable, renewable, natural resource when ou manage forests properl. Your lumber compan has 1,00,000 trees. You plan to harvest 5% of the trees each ear. How man ears will it take to harvest half of the trees? 1. You are harvesting a fied number / percentage of trees each ear, so using a linear / eponential growth / eponential deca model is reasonable. 1. Use the phrases from the blue bo to label each part of the formula. amount after n periods number of time periods T(n) a(1 r) n rate of deca initial amount amount after n periods number of time periods initial amount rate of deca 1. Use the information in the problem to write each amount. a 5 1,00,000 r T(n) 5 600, Use our values for a, r, and T(n) to write an equation. 600, ,00,000(1 0.05) n 16. Solve our equation for n, the approimate number of ears it will take to harvest half the original trees. Problem 5 Solving a Logarithmic Equation Got It? What is the solution of log ( ) 5 1? 17. Circle the equivalent eponential form of log ( ) (1) 10 () log( ) Solve the eponential equation for. 600, ,00,000(1 0.05) n 600,000 1,00,000 5 (0.95)n log n log 0.95 log 0.5 log n n N Chapter 7 0
20 Problem 6 Using Logarithmic Properties to Solve an Equation Got It? What is the solution of log 6 log 5? 19. You can use the Product / Quotient / Power Propert of Logarithms to write an equivalent equation with a single logarithm. 0. Write the single logarithm equation equivalent to log 6 log 5. log 5 1. Solve the single logarithm equation for. log Lesson Check Do ou UNDERSTAND? Error Analsis Describe and correct the error made in solving the equation.. If log z 5 log b, does 5? Eplain. Answers ma var. Sample: Onl when the bases are equal (z 5 b) log = log 9 log = log 9 = 9 = 81 will 5.. Complete the equation below.. Correct the error and solve for. log 9 5 log log 5 log 9 5 log log Math Success Check off the vocabular words that ou understand. eponential equation Need to review logarithmic equation Rate how well ou can solve eponential and logarithmic equations. Now I get it! 05 Lesson 7-5
21 7-6 Natural Logarithms Vocabular Review Write T for true or F for false. T F F 1. The function 5 log b, where b. 0 and b 1 is called a logarithmic function.. A logarithmic equation is an equation that contains onl one logarithm.. The logarithm of a power is the difference of the logarithm and the eponent. Vocabular Builder inverse function (noun) IN vurs FUNGK shun Related Words: function, inverse, input, output Definition: To find the inverse function, switch the order of the elements in the ordered pairs of the function. Eample: function, f(): {(1, ), (, )}; inverse function, f 1 (): {(, 1), (, )} Use Your Vocabular. Complete the table of values for the inverse function, f 1 (), of the function f(). 1 f() f 1 () Chapter 7 06
22 Ke Concept Natural Logarithmic Function If 5 e, then 5 log e 5 ln. The natural logarithmic function is the inverse of 5 e, so ou can write it as 5 ln If 5 e 5, then ln If ln b 5 6, then b 5 e. 1 = e = ln Problem 1 Simplifing a Natural Logarithmic Epression Got It? What is ln 7 1 ln 5 written as a single natural logarithm? 7. The epression is simplified below. Write a justification for each step. ln 7 1 ln 5 ln 7 1 ln 5 ln 7 1 ln 5 ln (7? 5) ln 175 Write the original epression. Power Propert of Logarithms Simplif the second term. Product Propert of Logarithms Multipl. Problem Solving a Natural Logarithmic Equation Got It? What are the solutions of ln 5? 8. Complete: If ln 5, then e 5. Got It? What are the solutions of ln ( 1 5) 5? Check our answers. 9. The equation is solved below. Write a justification for each step. ln ( 1 5) 5 ln ( 1 5) 5 e ln ( e ln ( e ln ( (5 e ) Write the original equation. Write in eponential form. Take the square root of each side. Subtract 5 from each side. Divide each side b. ln (0 1 5 < or <.167 Use a calculator. 07 Lesson 7-6
23 10. Check Substitute our values for in ln ( 1 5) 5. Use a calculator. ln Q? R <.0001 ln Q? R <.9976 Got It? What are the solutions of ln 1 ln 5? Check our answers. 11. Circle the propert of logarithms that justifies writing ln 1 ln as ln 6. Power Propert Product Propert Quotient Propert 1. Use the simplified equation to solve for. 1. Check Substitute our value for in ln 1 ln 5. Use a calculator. ln 6 5 ln? ln < 6 5 e e 6 N 1.15 Problem Solving an Eponential Equation Got It? What is the solution of e 5 1? Check our answer. 1. Use the justifications at the right to solve the equation. e ln 1 Write the original equation. Rewrite in logarithmic form. 5 ln 1 1 Add to each side. <.89 Use a calculator. 15. Check Substitute our value for in e 5 1. Use a calculator. e.89 < Got It? What is the solution of e 5 0? Check our answer. 16. Circle the first step in solving the equation. Underline the second step. Divide each side of the equation b. Write in eponential form. 17. Now solve the equation. Use the Power Propert. 18. Check Substitute our value for in e 5 0. Use a calculator. e. < e 5 0 e ln 10 5ln Write in logarithmic form. Chapter 7 08
24 Problem Using Natural Logarithms Got It? Space A spacecraft can attain a stable orbit 00 km above Earth if it reaches a velocit of 7.7 km/s. The formula for a rocket s maimum velocit v in kilometers per second is v t 1 c ln R. The booster rocket fires for t seconds and the velocit of the ehaust is c km/s. The ratio of the mass of the rocket filled with fuel to its mass without fuel is R. Suppose a booster rocket for a spacecraft has a mass ratio of about 15, an ehaust velocit of.1 km/s, and a firing time of 0 s. Can the spacecraft achieve a stable orbit 00 km above Earth? 19. Identif the values for each variable. R 5 15 c 5.1 t Complete the equation for finding the spacecraft s maimum velocit. Then use a calculator to simplif the equation. v ? 0 1 Q.1 ln 15 R km/s < 5.99 km/s 1. The spacecraft will / will not achieve a stable orbit 00 km above Earth. Eplain. Answers ma var. Sample: The velocit of 5. km/s is less than the 7.7 km/s needed to attain a stable orbit. Lesson Check Do ou UNDERSTAND? Reasoning Can ln 5 1 log 10 be written as a single logarithm? Eplain our reasoning.. The ln 5 1 log 10 can be written as ln 5 5 log e 5 1 log 10.. Can ou use the Product Propert of Logarithms to combine the logarithms? Eplain wh or wh not. No; the two logarithms have different bases.. ln 5 1 log 10 can / cannot be written as a single logarithm. Math Success Check off the vocabular words that ou understand. function logarithm natural logarithmic function Rate how well ou can write and solve equations with natural logarithms. Need to review Now I get it! 09 Lesson 7-6
Practice A ( 1, 3 ( 0, 1. Match the function with its graph. 3 x. Explain how the graph of g can be obtained from the graph of f. 5 x.
8. Practice A For use with pages 65 7 Match the function with its graph.. f. f.. f 5. f 6. f f Lesson 8. A. B. C. (, 6) (0, ) (, ) (0, ) ( 0, ) (, ) D. E. F. (0, ) (, 6) ( 0, ) (, ) (, ) (0, ) Eplain how
More informationChapter 8 Notes SN AA U2C8
Chapter 8 Notes SN AA U2C8 Name Period Section 8-: Eploring Eponential Models Section 8-2: Properties of Eponential Functions In Chapter 7, we used properties of eponents to determine roots and some of
More information) approaches e
COMMON CORE Learning Standards HSF-IF.C.7e HSF-LE.B.5. USING TOOLS STRATEGICALLY To be proficient in math, ou need to use technological tools to eplore and deepen our understanding of concepts. The Natural
More informationFunctions. Essential Question What are some of the characteristics of the graph of an exponential function? ) x e. f (x) = ( 1 3 ) x f.
7. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A Eponential Growth and Deca Functions Essential Question What are some of the characteristics of the graph of an eponential function? You can use a graphing
More informationwhere a 0 and the base b is a positive number other
7. Graph Eponential growth functions No graphing calculators!!!! EXPONENTIAL FUNCTION A function of the form than one. a b where a 0 and the base b is a positive number other a = b = HA = Horizontal Asmptote:
More information6.4 graphs OF logarithmic FUnCTIOnS
SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS
More information5A Exponential functions
Chapter 5 5 Eponential and logarithmic functions bjectives To graph eponential and logarithmic functions and transformations of these functions. To introduce Euler s number e. To revise the inde and logarithm
More informationExponential and Logarithmic Functions
Eponential and Logarithmic Functions.1 Eponential Growth and Deca Functions. The Natural Base e.3 Logarithms and Logarithmic Functions. Transformations of Eponential and Logarithmic Functions.5 Properties
More informationChapter 9 Vocabulary Check
9 CHAPTER 9 Eponential and Logarithmic Functions Find the inverse function of each one-to-one function. See Section 9.. 67. f = + 68. f = - CONCEPT EXTENSIONS The formula = 0 e kt gives the population
More information8-1 Exploring Exponential Models
8- Eploring Eponential Models Eponential Function A function with the general form, where is a real number, a 0, b > 0 and b. Eample: y = 4() Growth Factor When b >, b is the growth factor Eample: y =
More informationReady To Go On? Skills Intervention 7-1 Exponential Functions, Growth, and Decay
7A Find these vocabular words in Lesson 7-1 and the Multilingual Glossar. Vocabular Read To Go On? Skills Intervention 7-1 Eponential Functions, Growth, and Deca eponential growth eponential deca asmptote
More informationFirst Semester Final Review NON-Graphing Calculator
Algebra First Semester Final Review NON-Graphing Calculator Name:. 1. Find the slope of the line passing through the points ( 5, ) and ( 3, 7).. Find the slope-intercept equation of the line passing through
More information3.1 Exponential Functions and Their Graphs
.1 Eponential Functions and Their Graphs Sllabus Objective: 9.1 The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential or logarithmic
More informationf 0 ab a b: base f
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllabus Objective: 9. The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential
More informationdecreases as x increases.
Chapter Review FREQUENTLY ASKED Questions Q: How can ou identif an eponential function from its equation? its graph? a table of values? A: The eponential function has the form f () 5 b, where the variable
More informationLesson 5.1 Exponential Functions
Lesson.1 Eponential Functions 1. Evaluate each function at the given value. Round to four decimal places if necessar. a. r (t) 2(1 0.0) t, t 8 b. j() 9.(1 0.09), 10 2. Record the net three terms for each
More informationExponential and Logarithmic Functions
7 Eponential and Logarithmic Functions 7.1 Eponential Growth and Deca Functions 7. The Natural Base e 7.3 Logarithms and Logarithmic Functions 7. Transformations of Eponential and Logarithmic Functions
More information( 3x. Chapter Review. Review Key Vocabulary. Review Examples and Exercises 6.1 Properties of Square Roots (pp )
6 Chapter Review Review Ke Vocabular closed, p. 266 nth root, p. 278 eponential function, p. 286 eponential growth, p. 296 eponential growth function, p. 296 compound interest, p. 297 Vocabular Help eponential
More information2. Tell whether the equation or graph represents an exponential growth or exponential decay function.
Name: Date: Period: ID: 1 Unit 9 Review Eponents & Logarithms NO GRAPHING CALCULATOR 1. Under each function, write es if it is an eponential function. If the answer is no, write an eplanation wh not. a)
More informationChapter 4 Page 1 of 16. Lecture Guide. Math College Algebra Chapter 4. to accompany. College Algebra by Julie Miller
Chapter 4 Page 1 of 16 Lecture Guide Math 105 - College Algebra Chapter 4 to accompan College Algebra b Julie Miller Corresponding Lecture Videos can be found at Prepared b Stephen Toner & Nichole DuBal
More information3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS
Section. Logarithmic Functions and Their Graphs 7. LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Ariel Skelle/Corbis What ou should learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic
More informationSections 4.1 & 4.2 Exponential Growth and Exponential Decay
8 Sections 4. & 4.2 Eponential Growth and Eponential Deca What You Will Learn:. How to graph eponential growth functions. 2. How to graph eponential deca functions. Eponential Growth This is demonstrated
More informationLogarithms. Bacteria like Staph aureus are very common.
UNIT 10 Eponentials and Logarithms Bacteria like Staph aureus are ver common. Copright 009, K1 Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations,
More informationLESSON 12.2 LOGS AND THEIR PROPERTIES
LESSON. LOGS AND THEIR PROPERTIES LESSON. LOGS AND THEIR PROPERTIES 5 OVERVIEW Here's what ou'll learn in this lesson: The Logarithm Function a. Converting from eponents to logarithms and from logarithms
More informationf 0 ab a b: base f
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllaus Ojective: 9. The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential
More informationEvaluate Logarithms and Graph Logarithmic Functions
TEKS 7.4 2A.4.C, 2A..A, 2A..B, 2A..C Before Now Evaluate Logarithms and Graph Logarithmic Functions You evaluated and graphed eponential functions. You will evaluate logarithms and graph logarithmic functions.
More informationUse Properties of Exponents
4. Georgia Performance Standard(s) MMAa Your Notes Use Properties of Eponents Goal p Simplif epressions involving powers. VOCABULARY Scientific notation PROPERTIES OF EXPONENTS Let a and b be real numbers
More informationYou studied exponential growth and decay functions.
TEKS 7. 2A.4.B, 2A..B, 2A..C, 2A..F Before Use Functions Involving e You studied eponential growth and deca functions. Now You will stud functions involving the natural base e. Wh? So ou can model visibilit
More informationHonors Algebra 2: Semester 1 Review
Name Block Date Honors Algebra : Semester 1 Review NON-CALCULATOR 6-5 1. Given the functions f ( ) 5 11 1, g( ) 6 ( f h)( ) b) ( g f )( ), and h ( ) 4, find each function. g c) (g h)( ) d) ( ) f -1, 4-7,
More informationc) domain {x R, x 3}, range {y R}
Answers Chapter 1 Functions 1.1 Functions, Domain, and Range 1. a) Yes, no vertical line will pass through more than one point. b) No, an vertical line between = 6 and = 6 will pass through two points..
More information13.2 Exponential Growth Functions
Name Class Date. Eponential Growth Functions Essential Question: How is the graph of g () = a b - h + k where b > related to the graph of f () = b? A.5.A Determine the effects on the ke attributes on the
More informationExponential and Logarithmic Functions
Eponential and Logarithmic Functions 6 Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) CHAPTER OUTLINE 6. Eponential Functions 6. Logarithmic Properties 6. Graphs
More informationCHAPTER 3 Exponential and Logarithmic Functions
CHAPTER Eponential and Logarithmic Functions Section. Eponential Functions and Their Graphs......... Section. Logarithmic Functions and Their Graphs......... Section. Properties of Logarithms..................
More informationMath 121. Practice Problems from Chapter 4 Fall 2016
Math 11. Practice Problems from Chapter Fall 01 1 Inverse Functions 1. The graph of a function f is given below. On same graph sketch the inverse function of f; notice that f goes through the points (0,
More informationSec 5.1 Exponential & Logarithmic Functions (Exponential Models)
Sec 5.1 Eponential & Logarithmic Functions (Eponential Models) 1. The population of the cit Suwanee, GA has consistentl grown b 4% for the last several ears. In the ear 000, the population was 9,500 people.
More information15.2 Graphing Logarithmic
_ - - - - - - Locker LESSON 5. Graphing Logarithmic Functions Teas Math Standards The student is epected to: A.5.A Determine the effects on the ke attributes on the graphs of f () = b and f () = log b
More information6. The braking distance (in feet) for a car traveling 50 miles per hour on a wet uphill road is given by
MATH 34 - College Algebra Review for Test 3 Section 4.6. Let f ( ) = 3 5 + 4. (a) What is the domain? (b) Give the -intercept(s), if an. (c) Give the -intercept(s), if an. (d) Give the equation(s) of the
More information13.1 Exponential Growth Functions
Name Class Date 1.1 Eponential Growth Functions Essential Question: How is the graph of g () = a b - h + k where b > 1 related to the graph of f () = b? Resource Locker Eplore 1 Graphing and Analzing f
More informationSummary, Review, and Test
45 Chapter Equations and Inequalities Chapter Summar Summar, Review, and Test DEFINITIONS AND CONCEPTS EXAMPLES. Eponential Functions a. The eponential function with base b is defined b f = b, where b
More informationChapters 8 & 9 Review for Final
Math 203 - Intermediate Algebra Professor Valdez Chapters 8 & 9 Review for Final SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the formula for
More informationM122 College Algebra Review for Final Exam
M1 College Algebra Review for Final Eam Revised Fall 017 for College Algebra - Beecher All answers should include our work (this could be a written eplanation of the result, a graph with the relevant feature
More information15.2 Graphing Logarithmic
Name Class Date 15. Graphing Logarithmic Functions Essential Question: How is the graph of g () = a log b ( h) + k where b > and b 1 related to the graph of f () = log b? Resource Locker Eplore 1 Graphing
More informationAlgebra II. Chapter 8 Notes. Exponential and Logarithmic Functions. Name
Algebra II Chapter 8 Notes Eponential and Logarithmic Functions Name Algebra II 8.1 Eponential Growth Toda I am graphing eponential growth functions. I am successful toda when I can graph eponential growth
More informationName Date. Work with a partner. Each graph shown is a transformation of the parent function
3. Transformations of Eponential and Logarithmic Functions For use with Eploration 3. Essential Question How can ou transform the graphs of eponential and logarithmic functions? 1 EXPLORATION: Identifing
More informationSAMPLE. Exponential and logarithmic functions
Objectives C H A P T E R 5 Eponential and logarithmic functions To graph eponential and logarithmic functions. To graph transformations of the graphs of eponential and logarithmic functions. To introduce
More informationUnit 8: Exponential & Logarithmic Functions
Date Period Unit 8: Eponential & Logarithmic Functions DAY TOPIC ASSIGNMENT 1 8.1 Eponential Growth Pg 47 48 #1 15 odd; 6, 54, 55 8.1 Eponential Decay Pg 47 48 #16 all; 5 1 odd; 5, 7 4 all; 45 5 all 4
More informationMath 121. Practice Problems from Chapter 4 Fall 2016
Math 11. Practice Problems from Chapter Fall 01 Section 1. Inverse Functions 1. Graph an inverse function using the graph of the original function. For practice see Eercises 1,.. Use information about
More informationab is shifted horizontally by h units. ab is shifted vertically by k units.
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojective: 8. The student will graph logarithmic and eponential functions including ase e. Eponential Function: a, 0, Graph of an
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Practice for the Final Eam MAC 1 Sullivan Version 1 (2007) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the distance d(p1, P2) between the points
More informationThe semester B examination for Algebra 2 will consist of two parts. Part 1 will be selected response. Part 2 will be short answer. n times per year: 1
ALGEBRA B Semester Eam Review The semester B eamination for Algebra will consist of two parts. Part 1 will be selected response. Part will be short answer. Students ma use a calculator. If a calculator
More information= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background
Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic
More information7.4. Characteristics of Logarithmic Functions with Base 10 and Base e. INVESTIGATE the Math
7. Characteristics of Logarithmic Functions with Base 1 and Base e YOU WILL NEED graphing technolog EXPLORE Use benchmarks to estimate the solution to this equation: 1 5 1 logarithmic function A function
More informationThe speed the speed of light is 30,000,000,000 m/s. Write this number in scientific notation.
Chapter 1 Section 1.1 Scientific Notation Powers of Ten 1 1 1.1.1.1.1 Standard Scientific Notation N n where 1 N and n is an integers Eamples of numbers in scientific notation. 8.17 11 Using Scientific
More informationCHAPTER 3 Exponential and Logarithmic Functions
CHAPTER Eponential and Logarithmic Functions Section. Eponential Functions and Their Graphs......... Section. Logarithmic Functions and Their Graphs......... Section. Properties of Logarithms..................
More information2 nd Semester Final Exam Review Block Date
Algebra 1B Name nd Semester Final Eam Review Block Date Calculator NOT Allowed Graph each function. Identif the verte and ais of smmetr. 1 (10-1) 1. (10-1). 3 (10-) 3. 4 7 (10-) 4. 3 6 4 (10-1) 5. Predict
More informationTest # 33 QUESTIONS MATH131 091700 COLLEGE ALGEBRA Name atfm131bli www.alvarezmathhelp.com website MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More informationLesson Goals. Unit 5 Exponential/Logarithmic Functions Exponential Functions (Unit 5.1) Exponential Functions. Exponential Growth: f (x) = ab x, b > 1
Unit 5 Eponential/Logarithmic Functions Eponential Functions Unit 5.1) William Bill) Finch Mathematics Department Denton High School Lesson Goals When ou have completed this lesson ou will: Recognize and
More informationExponential and Logarithmic Functions
7 Eponential and Logarithmic Functions In this chapter ou will stud two tpes of nonalgebraic functions eponential functions and logarithmic functions. Eponential and logarithmic functions are widel used
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More information9-1. The Function with Equation y = ax 2. Vocabulary. Graphing y = x 2. Lesson
Chapter 9 Lesson 9-1 The Function with Equation = a BIG IDEA The graph of an quadratic function with equation = a, with a 0, is a parabola with verte at the origin. Vocabular parabola refl ection-smmetric
More informationReteaching (continued)
Zero and Negative Eponents Eercises Write each epression as an integer, a simple fraction, or an epression that contains onl positive eponents. Simplif...3 0. 0-0,000 3. a -5. 3.7 0 a 5 5. 9-6. 3-3 9 p
More informationChapter 11 Exponential and Logarithmic Function
Chapter Eponential and Logarithmic Function - Page 69.. Real Eponents. a m a n a mn. (a m ) n a mn. a b m a b m m, when b 0 Graphing Calculator Eploration Page 700 Check for Understanding. The quantities
More informationSolving Systems Using Tables and Graphs
3-1 Solving Sstems Using Tables and Graphs Vocabular Review 1. Cross out the equation that is NOT in slope-intercept form. 1 5 7 r 5 s a 5!3b 1 5 3 1 7 5 13 Vocabular Builder linear sstem (noun) LIN ee
More informationAlgebra 1B Assignments Exponential Functions (All graphs must be drawn on graph paper!)
Name Score Algebra 1B Assignments Eponential Functions (All graphs must be drawn on graph paper!) 8-6 Pages 463-465: #1-17 odd, 35, 37-40, 43, 45-47, 50, 51, 54, 55-61 odd 8-7 Pages 470-473: #1-11 odd,
More informationUnit 7 Exponential Functions. Mrs. Valen+ne Math III
Unit 7 Exponential Functions Mrs. Valen+ne Math III 7.1 Exponential Functions Graphing an Exponen.al Func.on Exponen+al Func+on: a func+on in the form y = ab x, where a 0, b > 0 and b 1 Domain is all real
More informationExponential Growth and Decay Functions (Exponent of t) Read 6.1 Examples 1-3
CC Algebra II HW #42 Name Period Row Date Section 6.1 1. Vocabulary In the eponential growth model Eponential Growth and Decay Functions (Eponent of t) Read 6.1 Eamples 1-3 y = 2.4(1.5), identify the initial
More information2.0 % annual 3.0 % Quiz. CD Specials
6.1 6. Quiz Tell whether the function represents eponential growth or eponential deca. Eplain our reasoning. (Sections 6.1 and 6.) 1. f () = (.5). = ( 3 8) Simplif the epression. (Sections 6. and 6.3)
More informationThe formulas below will be provided in the examination booklet. Compound Interest: r n. Continuously: n times per year: 1
HONORS ALGEBRA B Semester Eam Review The semester B eamination for Honors Algebra will consist of two parts. Part will be selected response on which a calculator will not be allowe Part will be short answer
More informationSTANDARD FORM is a QUADRATIC FUNCTION and its graph is a PARABOLA. The domain of a quadratic function is the set of all real numbers.
EXERCISE 2-3 Things to remember: 1. QUADRATIC FUNCTION If a, b, and c are real numbers with a 0, then the function f() = a 2 + b + c STANDARD FORM is a QUADRATIC FUNCTION and its graph is a PARABOLA. The
More informationExponential and Logarithmic Functions, Applications, and Models
86 Eponential and Logarithmic Functions, Applications, and Models Eponential Functions In this section we introduce two new tpes of functions The first of these is the eponential function Eponential Function
More information1. For each of the following, state the domain and range and whether the given relation defines a function. b)
Eam Review Unit 0:. For each of the following, state the domain and range and whether the given relation defines a function. (,),(,),(,),(5,) a) { }. For each of the following, sketch the relation and
More informationis on the graph of y = f 1 (x).
Objective 2 Inverse Functions Illustrate the idea of inverse functions. f() = 2 + f() = Two one-to-one functions are inverses of each other if (f g)() = of g, and (g f)() = for all in the domain of f.
More information7-1 Practice. Graphing Exponential Functions. Graph each function. State the domain and range. 1. y = 1.5(2) x 2. y = 4(3) x 3. y = 3(0.
7-1 Practice Graphing Eponential Functions Graph each function. State the domain and range. 1. = 1.5(2) 2. = 4(3) 3. = 3(0.5) 4. = 5 ( 1 2) - 8 5. = - 2 ( 1 4) - 3 6. = 1 2 (3) + 4-5 7. BILGY The initial
More informationis on the graph of y = f 1 (x).
Objective 2 Inverse Functions Illustrate the idea of inverse functions. f() = 2 + f() = Two one-to-one functions are inverses of each other if (f g)() = of g, and (g f)() = for all in the domain of f.
More informationChapter 12 and 13 Math 125 Practice set Note: the actual test differs. Given f(x) and g(x), find the indicated composition and
Chapter 1 and 13 Math 1 Practice set Note: the actual test differs. Given f() and g(), find the indicated composition. 1) f() = - ; g() = 3 + Find (f g)(). Determine whether the function is one-to-one.
More information15.2 Graphing Logarithmic
Name Class Date 15. Graphing Logarithmic Functions Essential Question: How is the graph of g () = a log b ( h) + k where b > 0 and b 1 related to the graph of f () = log b? Resource Locker A.5.A Determine
More informationA function from a set D to a set R is a rule that assigns a unique element in R to each element in D.
1.2 Functions and Their Properties PreCalculus 1.2 FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1.2 1. Determine whether a set of numbers or a graph is a function 2. Find the domain of a function
More informationExplore 1 Graphing and Analyzing f(x) = e x. The following table represents the function ƒ (x) = (1 + 1 x) x for several values of x.
1_ 8 6 8 Locker LESSON 13. The Base e Teas Math Standards The student is epected to: A.5.A Determine the effects on the ke attributes of the graphs of ƒ () = b and ƒ () = log b () where b is, 1, and e
More informationNCC Precalculus Partnership Program Final Examination, 2004
NCC Precalculus Partnership Program Final Eamination, 2004 Part I: Answer onl 20 of the 25 questions below. Each question is worth 2 points. Place our answers on the answer sheet provided. Write the word
More information13.2 Exponential Decay Functions
6 6 - - Locker LESSON. Eponential Deca Functions Common Core Math Standards The student is epected to: F.BF. Identif the effect on the graph of replacing f() b f() + k, kf(), f(k), and f( + k) for specific
More informationTest #4 33 QUESTIONS MATH1314 09281700 COLLEGE ALGEBRA Name atfm1314bli28 www.alvarezmathhelp.com website SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
More information4.5 Practice B. 4.5 Practice A. Name Date. Possible zeros: Possible zeros: 5. Justify. your answer. your answer. In Exercises 1 6, solve the equation.
Practice A Practice B In Eercises, solve the equation.. q q 0q 0. k + k + 9k 0.. p p. 8u u n + n 9n 8 0 In Eercises 7 0, find the zeros of the function. Then sketch a graph of the function. 7. f + 8. g
More information13.3 Exponential Decay Functions
6 6 - - Locker LESSON. Eponential Deca Functions Teas Math Standards The student is epected to: A.5.B Formulate eponential and logarithmic equations that model real-world situations, including eponential
More informationMATH 021 UNIT 1 HOMEWORK ASSIGNMENTS
MATH 01 UNIT 1 HOMEWORK ASSIGNMENTS General Instructions You will notice that most of the homework assignments for a section have more than one part. Usuall, the part (A) questions ask for eplanations,
More informationLESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More informationItems with a symbol next to the item number indicate that a student should be prepared to complete items like these with or without a calculator.
HNRS ALGEBRA B Semester Eam Review The semester B eamination for Honors Algebra will consist of two parts. Part is selected response on which a calculator will NT be allowed. Part is short answer on which
More informationReady To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions
Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte
More informationChapter 4. Introduction to Mathematical Modeling. Types of Modeling. 1) Linear Modeling 2) Quadratic Modeling 3) Exponential Modeling
Chapter 4 Introduction to Mathematical Modeling Tpes of Modeling 1) Linear Modeling ) Quadratic Modeling ) Eponential Modeling Each tpe of modeling in mathematics is determined b the graph of equation
More informationP.4 Lines in the Plane
28 CHAPTER P Prerequisites P.4 Lines in the Plane What ou ll learn about Slope of a Line Point-Slope Form Equation of a Line Slope-Intercept Form Equation of a Line Graphing Linear Equations in Two Variables
More information3.1 Graphing Quadratic Functions. Quadratic functions are of the form.
3.1 Graphing Quadratic Functions A. Quadratic Functions Completing the Square Quadratic functions are of the form. 3. It is easiest to graph quadratic functions when the are in the form using transformations.
More informationLESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More informationName Period Date. Practice FINAL EXAM Intro to Calculus (50 points) Show all work on separate sheet of paper for full credit!
Name Period Date Practice FINAL EXAM Intro to Calculus (0 points) Show all work on separate sheet of paper for full credit! ) Evaluate the algebraic epression for the given value or values of the variable(s).
More informationGlossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression
Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important
More informationWe all learn new things in different ways. In. Properties of Logarithms. Group Exercise. Critical Thinking Exercises
Section 4.3 Properties of Logarithms 437 34. Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one
More informationFunctions. Essential Question What are some of the characteristics of the graph of a logarithmic function?
5. Logarithms and Logarithmic Functions Essential Question What are some o the characteristics o the graph o a logarithmic unction? Ever eponential unction o the orm () = b, where b is a positive real
More informationModeling with Exponential and Logarithmic Functions 6.7. Essential Question How can you recognize polynomial, exponential, and logarithmic models?
.7 Modeling with Eponential and Logarithmic Functions Essential Question How can ou recognize polnomial, eponential, and logarithmic models? Recognizing Different Tpes of Models Work with a partner. Match
More informationGoal: To graph points in the Cartesian plane, identify functions by graphs and equations, use function notation
Section -1 Functions Goal: To graph points in the Cartesian plane, identify functions by graphs and equations, use function notation Definition: A rule that produces eactly one output for one input is
More informationWriting Equations in Point-Slope Form
. Writing Equations in Point-Slope Form Essential Question How can ou write an equation of a line when ou are given the slope and a point on the line? Writing Equations of Lines Work with a partner. Sketch
More informationMath Review Packet #5 Algebra II (Part 2) Notes
SCIE 0, Spring 0 Miller Math Review Packet #5 Algebra II (Part ) Notes Quadratic Functions (cont.) So far, we have onl looked at quadratic functions in which the term is squared. A more general form of
More informationMaintaining Mathematical Proficiency
Name Date Chapter 3 Maintaining Mathematical Proficienc Plot the point in a coordinate plane. Describe the location of the point. 1. A( 3, 1). B (, ) 3. C ( 1, 0). D ( 5, ) 5. Plot the point that is on
More information