ALGEBRA 2 CP MIDTERM REVIEW

Transcription

1 ALGEBRA CP MIDTERM REVIEW

2 Algebra II CP MIDTERM REVIEW Name CHAPTER 4 QUADRATICS Add or subtract the following polynomials. (Distribute if necessary, and then combine like terms) x y x y 7 7x 6 x x 7x 9x.. x. x 7 Multiply the following polynomials. Use distribution or FOIL x y 7x y x y x y x. x 6. 4x y x 7y 4. x 8. 4x 4x 7. x 9. x x 4 True or false? 0. a b a b. a ba ab b a b. a ba ab b a b Factor the following polynomials. Look for the greatest common factor, difference of squares, trinomials, and sum/difference of cubes. x x 0 4. y 0. x 8x 4

3 6. 4a a 7. x x 70x 8. m r y 8r y 7r y x. 4x 80y. 8 a 7. c 6c x x 8 Solve by factoring. x x 6. 4c 0c 7. x 4x 0 Simplify 7 8i 4i i 7 i 0. i 8 7i

4 Simplify i i.. 7 6i i. 4i 4i i i. i 4 i Fill in the blanks to complete the square. x 4x x x 9x x m m m 8. Solve by completing the square 9. x x 40. 4x 8x 0 4. x 7x 0 Write each quadratic function in vertex form. Identify the vertex, axis of symmetry and direction of opening. 4. y x 0x 0 4. y x 6x 9 4

5 Use FOIL to write a quadratic equation with the following roots. 44. and 4. -4i and 46. and 7 Find the value of the disciminant and determine the nature of the roots. Choices: real roots, real root, complex roots x 6x x 0x 49. x 4x 0 Disc = Disc = Disc = Nature of roots Nature of roots Nature of roots Solve using the quadratic formula 0. x x 9 0. x 4x b Find the Axis of Symmetry and Vertex for the graph of each function using x. a. y x 8x. f x x x

6 Define a variable, write an equation and solve. 4. A sports team sells about 00 coupon books for $0 each during its annual fund-raiser. They estimate that for each $0.0 decrease in the price, they will sell about 0 more coupon books. How much should they charge for each book in order to maximize the income from their sales? What is the maximum monthly income the team can expect to make from these items?. The sum of two numbers is -, and their product is -6. Write a quadratic equation to find the two numbers, then solve. 6. A rectangular picture frame measures 8 cm by 4 cm. You want to triple the frame s area by adding the same distance x to the length and the width. Write and solve an equation to find the value of x. What are the new dimensions of the picture frame? 6

7 CHAPTER POLYNOMIALS Simplify. Answer all questions using positive exponents! 0 0 x. x x. 0 x 4. y 4 4. a b j j k j k 6. a b a b m n 4 a b 0. ab 4 x y. 6y x. 0 0 x y Divide the following polynomials using synthetic division x 40 x x. x 4. 4x 7 x Divide the following polynomials using Long Division 6 x x x. x 6. 4x x 6 x 7

8 Use direct substitution to find f and f for each function 7. f x x x x 8. f x x 7x 4x 0 Find p and p for each function using synthetic substitution. 9. p x x 4x 0. p x x x Given a polynomial and one of its factors, find the remaining factors of the polynomial.. 6 x x x 8 ; x. 4x x x ; x Solve each equation. State the number and types of roots.. 9x 0 4. x 8x. x 4 x 4 0 Use Descarte s Rule of Signs. State the possible number of positive real zeros, negative real zeros, and imaginary zeros of each function. 4 f x 4x x x 7. f x x 6x 6x 4x List all the possible rational zeros of each function. 4 x x x 8x x 4 8. p 9. g x x 7x x x 0 8

9 Find all of the zeros of each function. x x x 4x 0. g. f x x 9x 0x 8 Find the zeros, relative minimum(s) and relative maximum(s) and graph the following. Your answer must be exact! 4 f x x x. f x 0.x 4x 4. Describe the end behavior for each of the following by completing the statement: as x, f x and as x, f x 4. f x x 4x 4. g x 4x x x Graph the following polynomials, be sure to include the relative maxima and minima. x x 4x g x x 4 x 4x 6. f 7. x y x y 9

10 8. The weight w, in pounds, of a patient during a 7-week illness is modeled by the cubic equation: w n 0.n 0.6n, where n is the number of weeks since the patient became ill. 0 a. Graph the equation, and describe the turning points of the graph and its end behavior. b. What trends in the patient s weight does the graph suggest? Is it reasonable to assume the trend will continue indefinitely? 9. The rainfall r, in inches per month, during a 7-month period is modeled by the equation: r m 0.0m 0.8m 0.67m., where m is the number of months after March. a. Graph the equation, and describe the turning points of the graph and its end behavior. b. What trends in the amount of rainfall received by the town does the graph suggest? 0

11 CHAPTER 6 RATIONAL EXPONENTS AND RADICAL FUNCTIONS Simplify.. 4 6x y 4 x. 6 y 7 x y 6x 6. 4a b a b Rationalize the denominator Express the following using rational exponents x x y

12 Express the following in simplest radical form. No decimal answers r 6s 6 0. x y Solve Each Equation m 9. 4 x 0. t 4 t 80 0 Solve the following equations.. x 7. a 4. r r 4. 8x x 7 x 6. x 6 x

13 7. 6 x 8. x x Find the inverse of each function. Then graph the function and its inverse. 40. g x x 4. f x x 4. y x Determine whether each pair of functions are inverse functions. 4. f g x x x x 44. f g x x 6x x 6

14 Find the inverse. 4. f x x 46. gx x 7; x 0 Given f x x x and gx x f x gx 48. f x gx find the following and state the domain. 49. x gx f 0. f g x x. f g7. f gx. g f x 4

15 CHAPTER 7 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Determine whether each function represents exponential growth or decay.. y. 6 x. y. x. y 4 x Describe the shifts of the following exponential equations from its parent function and then graph. Then, identify the functions domain and range. 4. x x y. y.4 Write each expression in logarithmic form Write each equation in exponential form. 8. log 9. log 4 Write an equivalent exponential or logarithmic equation. x 0. e x. ln 0 x. e 0.. ln x Evaluate each expression. 4. log. log log 7 x 7 7. ln e 8. ln y e Solve each equation. Check your solutions.

16 9. log x 0. logb. log 6 6 x -, use log and log to evaluate each expression.. log. 8 log 4. 9 log. log 44 6 Expand the following logarithms. 6. log 4x 7. x ln y Condense the following logarithms. 8. log 4 log 6 9. log8 6 log8 Solve each equation or inequality. Check your solution. 0. x 8. n n. 8 9 x 7 x4 x x. e 4. e Solve each equation. Check your solutions. 6

17 . 4x 6. x log x log 6 log 7. log4 log4 x log log y log 8 log 9. log 8 log 8 4 log 8 b 40. ln 8x 4. ln x Find the inverse. x 4. y 4 4. lnx y 44. log x y 8 4. A computer system depreciates at a rate of 6.% per year. If the computer system originally cost $4000, how long would it take for it to be worth half its value? 46. Suppose you deposit $000 in an account paying % annual interest compounded continuously. a. What is the balance after 0 years? b. How long will it take for the balance in your account to reach $00? 7

18 47. Suppose you deposit $00 in an account paying.% interest compounded continuously. How long will it take for your money to double? 48. A cup of coffee contains 0 milligrams of caffeine. If caffeine is eliminated from the body at a rate of % per hour, how long will it take for half of this caffeine to be eliminated from a person s body? 49. You re off to college! You buy a computer for $00. It is expected to depreciate at a rate of 0% per year. What will be the value of the computer in years? 0. A computer system depreciates at an average rate of 4% per month. If the value of the computer system was originally $000, in how many months is it worth $70?. A piece of machinery valued at $0,000 depreciates at a fixed rate of % per year. After how many years will the value have depreciated to $00,000?. The Miller s bought a condominium for $8,000. Assuming that the value of the condo will appreciate at approximately % a year, how much will the condo be worth in 7 years?. The population of a city of one million people is increasing at a rate of % per year. If the population continues to grow at this rate, in how many years will the population have doubled? 8

19 CHAPTER STATISTICS. Of the 4 employees at Speedy Pizza, sixteen make $4.7 an hour, four earn $.0 and hour, three earn $6.8 an hour, six earn $4.8 an hour and thirteen earn $. an hour. Find the mean, median and mode of the hourly wages. Mean: Median:. Given the following set of data:, 4, 67, 9, 8,, 8, 4,, 4 Mode: Stem Leaf a. Make a stem and leaf plot of this data. b. What is the minimum? b. c. What is the maximum? c. d. What is the range? d. e. What is the mean? (to nearest tenth) e. f. What is the mode? f. g. What is the median? g. h. Construct a box plot of the data. j. What is the lower quartile? j. k. What is the upper quartile? k. l. What is the inter-quartile range? (IQR) l. m. What is the outlier number? m. n. What is the lower fence? n. o. What is the upper fence? o. p. Name the outlier(s) if any. p. 9

20 From the box plot, answer the following questions. a. What is the lower quartile? a. b. What is the upper quartile? b. c. What is the inter-quartile range? (IQR) c. d. What is the outlier number? d. e. What is the lower fence? e. f. What is the upper fence? f. g. Name the outlier(s) if any. g. 4. Below are the weights of 4 people boarding a sail boat. 4, 67, 7, 8, 0, 4, 4,,, 9, 64,, 44, 6 a. What is the mean weight? a. b. What is the standard deviation of the weights? b. c. Construct a normal distribution of the weights. d. Between what numbers does 68% of the data lie? d. e. Between what numbers does 9% of the data lie? e. f. Above what number can we find weights that are or more standard f. deviations above of the mean? g. Calculate the z-score for a person weighing pounds. g. h. What is the probability for a person to have a z-score of -.? h. 0