Review for FINALS. FINAL CULMINATING date FINAL EXAM date

Transcription

1 Date: Name: Review for FINALS FINAL CULMINATING date FINAL EXAM date Success Criteria Ensure our Journals are complete and corrected. These ou ma use on the CULMINATING (but not on the EXAM) Complete the given Review booklet. Check our answers online. FORMULAS GIVEN ON EXAM: FORMULAS:,, r i =, n = Ct C ± x = b b ac a sin A sin B sin C = = a b c c = a + b ab cos C tn+ t tn = a + ( n ) d Sn = r n Sn = a + n d n tn = ar n a( r ) Sn = r general function equations = a( x h) + c = ax + bx + c = a( x r)( x t) = mx + b = ab x p x = ab + c = af [ k( x d )] + c for other functions = x, = x, = x, etc Trig identities [ ( ) ] On Culminating: CAN use Journals word problems Quadratics Exponentials Sinusoidals Trigonometr Finance On Exam: CAN NOT use Journals But formula page is provided Total 9 pages & 6 questions (some with a,b,c )

2 R e v i e w U Date: Name: Transformation of Functions (CH) and Simplifing Rationals (CH). State the transformations, the domain and range and graph each of the following: a. f ( x) = x + d. g. b. f ( x) = ( x ) c. e. f. h. i. f ( x). Which relation is a function? a. {(, ), (, ), (0, ), (-, )} c. {( 7, 7), (, ), ( 7, 6), (, )} b. {(, ), (, ), (, ), (0, )} d. {(, 7), ( 8, ), (, ), ( 9, 0)}. Which relation is a function? a. c. b. d.. For each of the following find f(), f(-), f() f( ) and f - a. b. c.. Find the inverse of each of the following. a. {( 0, ), ( 7, 9), (0, 6), (8, )} b. c. 6. For g( x) = x +, determine a. g(-6) b. g - (x) c. c. graph the function and it s inverse 7. Complete the table for the given transformations. (0, 0) (, ) (, ) (-, -6) = x 6 8. Suppose ou make the changes below to the graph of the functions = f(x). Write the equation of the image function. a. Expand horizontall b a factor of, then reflect in the x-axis b. Compress verticall b a factor of, then reflect about the -axis. c. Translate units right and units up.

3 R e v i e w U Date: Name: 9. Ms. Jackson has 0 m of fencing to make a rectangular pen for her rabbits. The wall of her house will serve as one side of the pen, so she will use fencing for onl three sides of the rectangle. Express the area of the rabbit pen as a function of its width and then determine the domain and range of the area function. 0. Explain each of the transformations. a. = f(-x) b. = f(-x + ) c. = -f(x) -. You are given the function, g(x) = x + x + 6 a) Graph the function. b) State the domain and range of g(x) c) Determine, algebraicall. d) Graph on the same set of axes. e) State the domain and range of g - (x) f) Determine g) Is g - (x) a function?. Simplif each of the following equations, where possible state restrictions. a. x( x + ) x(x ) b. (x ) c. + x 6x h. i. n. o. d. e. + a a + 6 a a a a j. k. p. q x 7x x 8x x x 6 f. l.. g.. m. r.. A rectangle has length (x + ) cm and width (x ) cm. What is the area of the rectangle?. Factor a. b. c. d. e. f. + n n n g.. The area of a rectangle is represented b the function A(x)=. What is the perimeter of the rectangle?

4 R e v i e w U Date: Name: Quadratics (CH) and Exponential (CH). Given the equation of a quadratic function in standard form, ( ) f x = x x + 6, state a. the zero(s) b. the vertex c. whether parabola has a maximum or minimum d. graph the function. What is the equation of the function for the graphs shown? Draw the inverses x x x. The set of ordered pairs {(, 0), (, 0), (, -), (8, -6)} defines a parabola. What is the equation that defines the parabola?. For each of the following find the domain and range, the value of the discriminant, the number of roots, the value of the roots. a. c. b. d.. Given the parabola; a. State the equation of smmetr b. State the Domain and Range c. Find the equation of the parabola d. Graph f (x) e. State the Domain and Range of the inverse. f. Find the equation of the inverse x 6. Simplif each of the following. a) 8 b. c. ( 7 + 8)( 8) d e. + f. 8 g. h. i j. 8 k. ( 6 + )

5 R e v i e w U Date: Name: 7. The height, h, in metres of a baseball after it is hit with a bat is described b h(t) = + 7t t, where t is the time in seconds after the ball is struck. a. What is the maximum height of the ball? Find the exact value. b. What time does it reach the maximum height? c. If the ball is caught. metres above the ground, exactl how long will it have been in the air? d. If the ball is caught. metres above the ground, approximatel how long will it have been in the air? 8. A firework was launched from the top of a 0 m building and it reaches a maximum height of 60 m in seconds. a) Determine the equation that represents the height of the rocket over time. b) When does the firework reach the ground? c) What is the height of the rocket when it is in the air for seconds? 9. A biccle maker sold 00 biccles last ear at a profit of $00 each. The maker wants to increase the profit margin this ear, but predicts that each $0 increase in profit will reduce the number of biccles sold b 0. How man $0 increases in profit can the maker add in and expect to make a total profit of at least $00 000? Use the results to find how man increases will give ou a maximum profit? 0. Determine the point(s) of intersection of the functions a. f ( x) = x 8x + 7 and g( x) = x + b. and. For the function, ( ) f x = ( x ) complete the following: a. Determine its inverse b. Graph the function and its inverse. c. State the domain and range of the function. d. State the domain and range of the inverse e. State whether the inverse is a function. Write each as a positive power, then evaluate. a. ( ) ( ) b. c. 8 d. ( ) 6 e. 7 f. ( 7) ( 7). Given the base, what is the new equation given the following transformations. a. A reflection in the -axis and translation of down b. Vertical stretch b and horizontal compression b c. A reflection in the x-axis and translation of to the right x. For the following equations, simplif the equation into = ab + c form, find the equation of the asmptote for the function, state the domain and range and graph the function; a. f x x ( ) x = + b. ( ) f ( x ) = c.

6 6 R e v i e w U Date: Name:. A bucket contains ml of water. The capacit of water in the bucket decreases.8% ever 0 min. What is the equation models the situation? How much water is in the bucket in 0 min? when will the water level be at 900mL? 6. An exponential function with a base of has been compressed horizontall b a factor of and reflected in the -axis. Its asmptote is the line =. Its -intercept is (0, 6). Write an equation of the function and state its domain and range. f x x+ 6 ( ) ( ) 7 7. The function describe the sequence of transformations. = + is the result of transformations. State the base function and 8. What is the difference between exponential growth and exponential deca? 9. Each ear a car loses 8% of its value from the previous ear. A car was bought new for $ 000. Write the equation the represents the cars value after n ears. Find the value of the car after ears. 0. The half-life of a radioactive material is hours. You started with 00 g. a) Write an equation to represent the amount of radioactive material left. b) How much radioactive material is left after hours? c) When will there be approximatel gram of the material left?. The population of the beluga whale is currentl (in0) 000 animals. It s population is increasing b 6% ever ear. a) Write an exponential relation that models the problem. b) What will be the beluga s population in 0? c) What was the beluga s population in 990?

7 7 R e v i e w U Date: Name: Trigonometr (CH) and Sinusoidal Functions (CH6). Solve each of the following triangles. a. b. 0 m x m x m m c. d. 8 m. Determine the value of if a. cot θ = -0. b. cos θ = 0.9 c. sin θ = d. sin θ = -0. e. cos θ = -0. f. tan θ = -. g. csc θ = -0. h. sec θ =. i. tanθ =.8. Determine the corresponding reciprocal ratio that corresponds to.. Tim is standing on top of a cliff and sees a boat in the water. The boat is 00 m from shore and the angle of depression (from Tim s point of view) is 8 o. How tall is the cliff?. A ladder is leaning against a wall, the foot of the ladder is m from the wall. The angle of elevation with the foot of the ladder is 6 o, how long is the ladder? 6. Two cclists leave from the same location with an angle of 6 o between their paths. Jim ccles at a speed of km/h and John at a speed of 0 km/h. How far apart are the after h? 7. Two tracking stations, km apart, track a weather balloon floating between them. The tracking station to the west tracks the balloon at an angle of elevation of o, and the station to the east tracks the balloon at an angle of elevation of 60 o. How far is the balloon from the closest tracking station? 8. Point P(6, -) lies on the terminal arm of an angle in standard position. What is the value of the principal angle to the nearest degree? 9. Determine co-terminal angles with a. 60 o b. o c. 9 o d. 78 o

8 8 R e v i e w U Date: Name: 0. Find the exact value of; a. tan 0 o b. sec o c.. d.. e. f. sin o 7. Given that tanθ = and if angle is a principal angle that lies in quadrant such that, determine; a. the exact values of x,, and r b. angle θ, to the nearest tenth of a degree c. sec θ. Prove the following trigonometric identities cos θ sin θ = tanθ a. cos θ + sinθ cosθ b. tan θ sin θ = sin θ tan θ c. tan θ cos θ = cos θ d. cos θ sin θ + cos θ = cos θ + sin θ. The angle θ is in Quadrant IV and csc θ = 7 possible coordinates for E and the value of angle θ.. A point E lies on the terminal arm. Determine. For each of the following determine the period, amplitude, domain, range, phase shift, vertical translation. a. b. c. d. e. o f. = cos( x + 60 ) + x g. = sin(x 0). Graph the following sinusoidal functions, where a) = sinx + o b) ( ) = cos x 0 o o 80 x The graph of the function of = sin x is transformed as described below. Determine the equation that causes each of the following transformations a. Vertical stretch b and translate the graph 7 units up b. Reflected on the x axis, compressed the graph horizontall b a factor of and translate left b 0 o.

9 9 R e v i e w U Date: Name: 7. For each of the following graphs state the period, then the equations using sine AND using cosine a. b. dist (m) 8 6 x time (s) c. d. x x e. f. displace x time g. h x x

10 0 R e v i e w U Date: Name: i. j x x 8. The depth of the water in an ocean harbour varies due to the tides and can be modelled using the equation = asin k(x + b) + d. High tide occurs at :00 A.M. with a water depth of 6 m and low tide occurs at :00 P.M. with a water depth of m. (a) Sketch its graph. (b) Determine the equation that models this relationship if t = 0 at midnight. (c) Determine the water depth at :00 P.M. 9. Consider the periodic function f ( x) = sin ( x + 60). State the i. Phase shift iii. Vertical displacement ii. Period iv. Amplitude Graph the function between 80 o to 60 o. v. Domain vi. Range 0. A person who was listening to a siren reported that the frequenc of the sound fluctuated with time, measured in seconds. The minimum frequenc that the person heard was 00 HZ, and the maximum frequenc was 00 Hz. The maximum frequenc occurred at t = 0 and t =. The person also reported that, in s, she heard the maximum frequenc 6 times (including the times at t = 0 and t = ). a. Graph the function b. Write and equation that models the sound. c. What frequenc is heard at. s? d. Find all the different times within ccles when the frequenc is 00 Hz?. The building swas cm back and forth from the vertical. At t = 0s, the building is cm to the right (+) of the vertical. The building swas back to the vertical. At t = 0s, the building swas cm to the left(-) of the vertical. a) Write an equation that models the motion of the building in terms of time. b) What position is the building in at s? c) Find the two times within one ccle that the building is at -0cm?. The number of hours of dalight in Vancouver can be modeled b a sinusoidal function of time, in das. The longest da of the ear is June st (da 7), with.7 h of dalights. The shortest da of the ear is December st (Da ), with 8. h of dalight. a. Graph the function b. Make and equation c. Find the amount of sunlight on Feb th.

11 R e v i e w U Date: Name: Series and Sequences (CH7) and Financial (CH8). For the following sequences make the recursive formulas a., 0, 0,.. b., 0,, 0,. c., 6, 8,,,. Consider the series , find each of the following a. The general term, tn. b. The th term c. The sum of the first 0 terms. Determine whether the sequence is geometric, arithmetic, or neither. Tr to come up with general tn formulas a. 8,, 76, 0, b., 7,,, 9,... c.,, 7, 0,, d. e. 6,, 9, 08,06, f. 7,, 9,,,... g. 6, -8,, -6, h. 9, 7,,,. i. 6,,, 8, j. 9,, 9, 6, k., 6, 9,,.... The th term of a sequence is. If the first term is -7, then what is the common difference?. If the 7th term in a geometric sequence is 8 and the common ratio is, then what is the first term? 6. The 6 th term of an arithmetic sequence is 9, and the th term is 7. What is the first term? 7. Calculate the sum of the series: Calculate the sum of the series Expand and simplif the binomial power a. ( x ) b. ( ) c. ( x) 0. How man compounding periods are there in the investment below? Determine the interest rate per compounding period. a. Principal Rate of Compound Compounding Time Interest per Year Period $ 000.8% quarterl 8 ears b. Time of pament Length of Interest rate per Frequenc of annuit ear compounding end of ever 6 months 7 ears.% semi-annuall c. Time of pament Length of annuit Interest rate per ear Frequenc of compounding end of ever 6 months ears 8.% semi-annuall

12 R e v i e w U Date: Name:. A principal of $00 is invested at 6% per ear. Determine the interest earned in 7 ears if the interest is a. Simple interest b. Compound interest, compounded annuall c. Compound interest, compounded quarterl. Determine the amount of an ordinar simple annuit of $00 deposited each month for 7 ears at.% per ear compounded monthl.. Determine the present value of an ordinar simple annuit that pas $00 for 0 ears at.8% per ear compounded semi-annuall.. Determine the monthl pament for a mortgage of $0 000 at.% per ear compounded semiannuall over 0 ears. (Remember to covert semi-annual rate to monthl rate using ( + m) = ( + s) formula). Ra borrows $000 for 7 ears at a fixed rate of simple interest. At the end of 7 ears, he will have paid $,87.0 in total. What is the interest rate on Ra s loan? 6. The future value of an account is $89. in 6 ears. If the rate of compound interest is 7.%/a compounded quarterl, what is the principal investment? 7. George puts $00 down on a new car and finances the rest (makes monthl paments) at an interest rate of 0.8% compounded monthl.. Five ears later, he paid $ for the principal and interest. How much did the car originall cost? 8. Janice is interested in purchasing a television. While shopping she notices a sign that sas Low monthl paments of onl $.9. After speaking to the sales associate she discovers that these paments are on the credit card at an interest rate of 8% compounded monthl for ears. a. How much does the television cost? b. How much interest would Janice have to pa? 9. Tshawn purchases a home of $ , he has a 0% deposit. He finances the rest (makes monthl paments) at.% compounded monthl for ears. How much interest will Tshawn pa over the life of the mortgage? 0. A $000 investment earns % interest each ear. a. Suppose the investment earns simple interest. What will the investment be worth after 8 ears? b. Suppose the investment earns compound interest, where the interest is compounded annuall. What will the investment be worth after 8 ears?. Nicole borrowed $ 000 to purchase her new car. The car dealership offered her a loan at.% compounded monthl for ears. What is Nicole s monthl pament?. You need to borrow $0,000 which option should ou take repa the loan? Option A: The loan paments are monthl and the interest is 0.% compounded monthl over ears Option B: The loan paments are quarterl (ever months) and the interest is 0.8% compounded quarterl over ears

13 R e v i e w U Date: Name:

14 R e v i e w U Date: Name:

15 R e v i e w U Date: Name:

16 6 R e v i e w U Date: Name:

17 7 R e v i e w U Date: Name:

18 8 R e v i e w U Date: Name:

19 9 R e v i e w U Date: Name:

20 0 R e v i e w U Date: Name: