General Directions: When asked for EXACT SOLUTIONS, leave answers in fractional or radical form - not decimal form. That is, leave numbers like,, π, and e as part of your answer.. State the domain of each of the following functions. (a) f ( ) = 7 + (b) g ( ) = 4 (c) h ( ) = + 4+ + 7 (d) k( ) = ln + 4 (e) (f) sin m ( ) = e. The table on the right, generated on a graphing calculator, shows some points on the graphs of two functions. Use the table to answer these questions: (a) Where is y s -intercept? (b) Where is y s y-intercept? (c) Give the coordinates of each point where y and y intersect.. Use the graph of function f at the right to answer the following: (a) What are the coordinates of the points P and Q? (b) Evaluate f(). (c) Evaluate f(0). (d) Solve f ( ) = 6 for Assume -scale and y-scale equal y y 0 6 0 4 4 6 6 P (e) Estimate the value when f() = 0. Q Page of 4
4. If f ( + h) f ( ) f ( ) = + 8, evaluate and simplify, h 0. h 4. Given f ( ) =, use your graphing calculator to approimate the following. Round your answers to three decimal places. (a) Find any local minima/maima. (b) Find intervals where f is increasing and/or decreasing. Use interval notation. 6. A media company is going to install cable from a house to their connection bo B. The house is located at one end of a driveway 7 miles back from a road (see diagram). The other end of the driveway and the nearest connection bo are on the same road, miles apart. The cost of installing the cable is $66 per mile off the road and $7 per mile along the road. Let be the distance from where the driveway meets the road to where the cable comes to the road. Develop a function C() that epresses the total installation cost as a function of. Now use your calculator to graph C. Use the graph to determine the value of that will produce the minimum cost. Round to the nearest thousandth of a mile. (Tip: use a window [0, 0, ] [,000, 0,000, 000 ].) State the minimum cost for that installation, rounded to the nearest cent. 7 mi cable mi cable B 7. (a) A partial graph of of f is shown. Complete the graph if f ( ) = f ( ) (b) A partial graph of of g is shown. Complete the graph if g( ) = g( ). 8. On the right is a table of some values of a function f. Answer these questions about f: (a) Tell whether the function is even, odd, or neither. Eplain how you know this. (b) List each -intercept. (c) List each y-intercept. Page of 4 y 7 0 0 9 0
9. The complete graph of f() is shown at the left. On the graphs, scl=. (a) Sketch the graph of f ( ). (b) In terms of f(), what is the formula for the graph shown below? A B C D E 0. Use the functions f and g to evaluate the following. (a) (f + g ) () (b) (f g ) () (c) f ( ) g (d) (f g) (4) (e) ( f o g)( ) (f) ( g o f ) ( ). If f ( ) = +, and ( ) = g, find and simplify y y=f() 4 0 0 4 y = g( ) 0-0 -4 4 - (a) f ( g( f ( ) ) (b) f ( g( ) ) and state the domain (c) ( f ) ( ) (d) g ( ) and state its domain (e) ( f o g )( ) 88. Describe the behavior of the power function ( ) = a,for a < 0 g o and state the domain f as regards each of (a)-(d) below If there is insufficient information to decide, then answer cannot be determined (a) symmetries (b) domain (d) end behaviors: As As (c) range, ( ) f Which end is this? left/right, f ( ) Which end is this? left/right. For each of these functions: I. f ( ) = 7 + 4 + 4 4 II. g ( ) = 0 + 86 + 40. (a) Find the zeros and identify them as rational, irrational, or non-real comple. (b) Factor the polynomial completely over the comple numbers. Page of 4
4. The data below represents the number of students enrolled in grades 9 through in private institutions. Year Enrollment in 000 s 97 00 980 9 98 400 98 400 98 400 984 400 98 6 986 6 987 47 988 06 989 9 990 7 99 99 6 99 9 994 6 99 60 996 97 (a) Using a graphing utility, draw a scatter diagram of the data. Determine if the data appears to be linear, quadratic, cubic, eponential, logarithmic or sinusoidal. (Let 97 correspond to t = 0) (b) Using a graphing utility, find the cubic function of best fit and graph. (c) Use the function found in part (b) to predict the number of students enrolled in private institutions for grades 9 to in the year 000. Round your answer to the nearest thousand students. (d) Use the function found in part (b) to predict the year in which enrollment in private institutions for grades 9 through will reach ½ million students. (e) The function found in part (b) does approimate the data quite nicely for the years that we have information. Eplain why, for this situation, a cubic function might not be a good model over the long term. Include the phrase "end behavior" in your discussion.. Following are some tables for the same rational function f. TABLE A TABLE B TABLE C TABLE D f() f() f() f() -4 8. -. 40 4.00-0 4.48 -. 04. -.9-89.7 400 4.0006-00 4.004 -.0 00004. -.99-98999.7 4000 4.000006-000 4.0000 -.00 00000004 -.999-998999996 40000 4.000000-0000 4.000000 (a) Name each table that suggests the eistence of a horizontal asymptote (end behavior). (b) Name each table that suggests the eistence of a vertical asymptote (local behavior). Page 4 of 4
(c) Use the tables to complete each of the following statements: i. As, f ( ) ii. As, f ( ) iii. As from the right side, iv. As from the left side, f f ( ) ( ) v. Write the equation of any horizontal asymptotes. (Write none for your answer if there isn t one.) vi. Write the equation of any vertical asymptotes. (Write none for your answer if there isn t one.) 6. Consider the rational function (a) State the domain of the function. (b) Complete this statement: As ( + )( ) h ( ) = : ( ), h ( ). (c) Write the equation of any vertical asymptotes. (d) Write the equation of any horizontal asymptotes. (e) Write the coordinates of any holes. (f) List any -intercepts. (g) Graph and label any asymptotes, intercepts, and holes. (h) State the range of this function. 7. The Parks and Wildlife commission introduces 80,000 fish into a large man-made lake. The population of the fish, in thousands, is given by ( 4 + t ) (a) Approimate the population when t is, 0, and. 0 N ( t) =, 0 t, where t is the time in years. + 0.0t (b) What is the limiting number of fish in the lake as time increase? ( ) 8. Find the eact solution of f ( ) = 0.Give the eact values for answer(s) using interval (7 + )( + 4) notation. 9. (a) Solve for : 4 = (b) Simplify: e ln( ) + ln Page of 4
0. The value of an automobile depreciates. It is originally worth $40,000, but then it loses one-tenth of its value every year. (a) How much is it worth at the end of the first year? (b) How much is it worth after years? (c) How much is it worth after t years? (d) How many years would pass before it was worth only half of its original value? (This would be its half-life.) Round this value to the nearest tenth of a year.. A function f is defined by the table below: Complete a table for f 0 f() 0 6 7 f. Graph ( ) g = e. (a) Find all intercepts and asymptotes. Round to two decimal places. (b) State the domain and range of the function.. If y f ( ) = = log( ), find the inverse function f ( ). 4. The unlabeled sketch at right shows the graph of y = log ( + ). The sketch correctly shows the shape of the graph, but it is not to scale. (a) (b) (c) Find the equation of any asymptotes and sketch. Find the -intercept and label. Find the y-intercept and label.. Solve for : log (8 8 ) log = 6. Approimate the value of y to three decimal places: y = log 40 7. Eponential Growth Applications: P t = Ce describes the population P of a certain country where the time is measured in t years. Assume the function ( ) kt (a) What is the growth rate constant (k) if the population has tripled in years? Approimate k to a tenth of a percent. (b) Use the eact value of k to find the population in 0 years if the initial population was 0,000, 8. Given sin θ = 4 7 and π < θ < π, find the EXACT values of the following: π (a) tanθ (b) secθ (c) cos θ + Page 6 of 4 (d) csc( θ )
9. Graph two full periods of each of the following functions. F ( ) = cos( 4) + G( t) = sin ( πt π ) For each function determine the following: (a) period (b) range (c) amplitude (d) phase shift 0. Find a function of the form f ( ) = Acos( B ) + D for the graph below.. Household electrical power in the US is provided in the form of alternating current. Typically voltage fluctuates smoothly between +.6 volts and.6 volts. The voltage cycle 60 times per second. Use a cosine function to model the alternating voltage. cscθ cscθ. Verify that = sec θ + cscθ cscθ. Use trigonometric identities to find the eact value of sin( α β) and cos( α β) π π tan α =, π < α < ;sin β =, π < β < + +, given: 4. Use trigonometric identities to find the eact value of tan(θ), where (, 4) is a point on the terminal side of θ.. Find, to the nearest degree, all angles, 0 θ 60 θ, such that sin ( θ ) = 0. 8 6. Find an algebraic epression in (not involving trig functions) that is equivalent to csc( tan ( ) for >0. 7. Givencos = sin : (a) Solve the equation algebraically for the eact value of the solution(s) on the interval [0, π). (b) Verify the answer(s) in part (a) using the ZERO or INTERSECT features of your graphing calculator. Round answers to the nearest tenth of a radian. Page 7 of 4 t
8. Write TRUE or FALSE in the blank before each statement. [Recall that if a statement in mathematics is not always true, then it is considered false.] If it is false, write a corrected true statement in the blank. If it is already true, write Identity in the blank. (a) sin (A + B) = sin(a) + sin (B) (b) sin ( A ) = ( sin A) (c) (sin A + cos A) = sin A + cos A (d) (e) cos (A B) = cos A cos B sin (A) = sin A (f) tan ( ) A = tan( A) y 9. The graph at right shows two cycles of the graph of y = cos t + cost. Algebraically find all zeros to the equation over the domain of all reals. Epress your answer in terms of π. -π -π π π 40. In traveling across flat land, you notice a mountain directly in front of you. The angle of elevation [to the peak] is.. After you drive miles closer to the mountain, the angle of elevation is 9. Approimate the height of the mountain to the nearest foot. (80 feet = mile) 4. Sketch the curve described by the parametric equations by hand. Verify using a graphing calculator : = cos t, y= sin t; 0 t π (-., 0) Does this curve represent y as a function of? 4. This question concerns the three graphs. (a) Find possible formulas for the three functions. (i) f (ii) g (iii) (., ) (4, 0) 40 0 0 0 0-4 - - 0 4 (b) Estimate the average rate of change for function h between = and = 0? Page 8 of 4 - -4 - h 4 - - - 4 - (c) Where is function h increasing faster, between = 4 and =, or between = and = 0? -
4. For each function, select its most specific function type from the name bank below. Name Bank = {Eponential, linear, logarithmic, polynomial, quadratic, rational, sinusoidal} a. f ( ) = b. c. + f ( ) = d. f ( ) = e. f. + 44. The figure below shows the complete graphs of the functions f and g. Based on the graphs, the equation f ( ) g( ) = 0 has how many solutions? 4. Use the graphs of the functions h and k below to answer the following: a. For what value(s) of is h ( ) k( )? b. Solve the inequality h ( ) < k( ) k() h() Page 9 of 4
MA 80 COURSE REVIEW Answer Key. a) { and 7 } d) (, 4 ) (, ) b) { < < } e) { < or < } c) 7 or f) (, ). a) at = b) at y = c) (-,) and (4,6). a) P(,6) Q(-,-) b) - c) - d) e). 4. 6+ h+. a) local minima: (-0.98, -4.464) and (0.98, -4.464) local maimum: (0, 0) b) Decreasing: (-, -0.98) (0, 0.98) Increasing: ( -0.98, 0) (0.98, ) C = + + $,4.74 6. ( ) 66 49 7( ) 7. a) b) 8. a) f is not odd because f ( ) f ( ) for all f is not even because f ( ) f ( ) for all b) (-,0) (,0) c) (0,) Revised Fall 006 Page 0 of 4
9. a) Translated points A B C D E A (0,-) B (,0) C (4,) D (,0) E (6,) b) f ( ) + 0. a) - b) 4 c) undefined d) - e) 4 f). a) 6 d) ( ) b) 4, e) 8 c) 4, or. a) symmetric with respect to the y-ais b) all reals yy 0 c) { } d) as, f ( ) right end as, f ( ) left end 7. I. a) rational: = ; irrational: = ± b) ( ) 7 f = ( + ) ( ) II. a) rational: =,4 ; nonreal comple: = ± i b) g( ) = ( 4)( )( i + ) i 4. a) The data appears to fit a cubic or a sinusoidal model. Revised Fall 006 Page of 4 g = + domain:{ 0} b) If you elect to use year 97 as = 0, and rounding coefficients to 6 decimal places, the regression function is: f ( ) = 0.077 9.790 + 7.98866 + 77. 7787. c),746,000 students d) The closest year is 004. e) Because the end behavior of this function is increasing as, the enrollment at these institutions would increase without bounds as time goes by. Realistically, this is unlikely to happen. Enrollment in 000s # Students Enrolled in Grades 9- in Private Institutions 00 400 00 00 00 000 97 98 99 00 Year
. a) C, D b) A, B c) (i) as, f ( ) 4. (ii) as, f ( ) 4. (iii) as (iv) as (v) y = 4. (vi) = +, f ( ), f ( ) 6. a) { 0 and } b) as, h( ) c) = 0 d) y = e) (,4 ) f),0 g) See graph at right y y and y 4 h) { } y = (,0) = 0 (,4 ) 7. a) t = 04,000 fish b),00,000 fish t = 0 4, fish t = 70, fish, 4,0, 7 8. ( ) 9. a) = b) 0. a) $6,000 c) 40,000( 0.9) t dollars b) $9,60 d) 6.6 years. - - 0 6 7 f - - 0 ( ). a) (.9,0), (0,0.7) y = b) domain: all reals range: { yy< } Revised Fall 006 Page of 4
. f ( ) = or f ( ) = ( ) 0 0. 0 4. a) No horizontal asymptote vertical asymptote at = b) -intercept at c) y-intercept at = ( 0, ) (,0). = 0.4 = 6..9 7. a) 4.8% b) 08,948 8. a) 4 b) 7 c) 4 d) 7 49 49 = 8 64 π 9. F a) b) [,8 ] c) d) none π π G a) b), c) d) Revised Fall 006 Page of 4
π 0 0. f ( ) = 0cos + 0. solutions vary. f ( t) =.6cos( 0π t). a) 4. 4 7 + 6, 6. 4 o, 06 o 6. + 9 7. a) π, 7π 6, π 6 b).6,.7,.8 8. a) False ( ) b) True sin A+ B = sinacosb+ cosasinb c) False ( ) d) False ( ) e) False ( ) = sin A+ cosa = sin A+ sin AcosA+ cos A= + sin Acos A cos A B = cosacosb+ sinasin B sin A sinacosa f) False cot A tan A = π π 9. + nπ, + nπ, π + nπ 40. 689 feet 4. y is not a function of 4. a.i.) ( ) f = + ( 4) - a.ii.) g( ) = ( + ) ( ) 6-6 a.iii.) h( ) = OR h( ) 4 = + 4. b) The rate of change is c) Function h is increasing faster between = and =0. 4. a) rational b) polynomial c) polynomial d) none e) eponential f) sinusoidal 44. two 4. a) (, ] [, ) b. (,) or < < Revised Fall 006 Page 4 of 4