Pre-Calculus First Semester Review

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Letitia Murphy
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1 NON CALCULATOR Pre-Calculus First Semester Review Unit 1: 1 37 Unit : 1 18, Unit 3: 19,, 5 6 [1.] Find the domain. Epress the answer in interval notation. 1. f( ) log ( 5) = +. 3 f( ) = [1.] Prove algebraicall whether the function is even, odd, or neither. 3. f () = f () = For question 5-1, find following: (a) Identif the parent function. (b) State the transformation rule or describe the transformation. (c) Graph the function including ke points and an asmptotes. [1.5] 5. f() = + + [1.5] 6. f( ) = log ( ) 4 [1.5] 7. f 1 = + [1.5] 8. f ( ) = ( ) 3 1 1

2 [1.5] 9. f( ) = 3 4 [GrTrg] 1. f( ) = 3sin ( π ) 1 [1.3] Graph the piecewise-defined function. State whether the function is continuous or discontinuous at =. 11. f( ) = if if > 1. if f( ) = if > [.3] For each function below a) Determine the degree. b) Describe the end behavior using limit notation. c) Find the zeros of the function with their multiplicities. d) Sketch a graph of the function including zeros, multiplicities and end behavior f( ) 4( ) ( 3) = f( ) = 36

3 [.7] Find (if it eists) the a) equations of an horizontal or slant asmptote, b) equations of an vertical asmptote(s) and coordinates of an holes, c) -intercept and -intercept, and d) graph the function including additional points in each region of the domain OR using a sign chart. 15. g ( ) = g ( ) = 6 [.7] 17. Use the rational function below, along with the listed attributes, to graph the function. Include additional points in each region of the domain ( + 3)( 3)( + 1) f ( ) = = + 3 ( + 3)( 1) SA: = 1 VA: = 1 -intercepts: (3, ) and ( 1, ) -intercept: (, 3) [.7] 18. Describe the end behavior of the rational function in question 4 using limit notation. [.] For questions 19 &, identif the letter of the graph below that best matches the given function. A B C D E F 19. f( ) =. 1 5 f( ) =

4 Graphing Calculator Allowed Solve algebraicall showing all steps. Check for etraneous roots. Write our answers in interval notation where appropriate. [P3] 1. (5 ) 3(1 ) + 1 [P5]. 3 1 < 7 [P5] [P6] > = [.8] 6. + = + 1 [.6] 5. ( ) [P5] Solve b graphing. Note the procedure used = = [.5] 9. Write in a + bi form: + 4 i 3 i 4

5 [.6] Find a polnomial equation with the given zeros. a) Write the function as a product of linear and irreducible quadratic factors and b) Epress function in standard form. 3. 1, i 31. 3, 4i [.6] Find the zeros of the function and write the function as a product of linear and irreducible quadratic factors all with real coefficients. 3. f () = 3, given zero = 33. f () = , given zeros = 1 and = 4 Solve questions using as sign chart. [P6] [.9] [.9] [3.5] 37. Solve algebraicall and check graphicall. a) ( ) 5 = 6 b) + = c) ln = e 157 d) 5 = 1 log ( 7) c) ( ) ( ) 3 log + log + 1 = d) log ( 1) log ( 3) = 3 5

6 [1.] 38. Find all a) local maima and minima and b) identif intervals on which the function is increasing and decreasing. f () = [1.] Graph the function and tell whether or not it has a point of discontinuit at =. If there is a discontinuit, tell whether it is removable or non-removable. 39. f () = 4. h() = + [1.3] Using the twelve basic parent functions provided in the bo, list the equation of the function(s) that fit the description given. f () = f () = ln f () = e f () = ² f () = f () = f( ) = f( ) = f( ) = sin f( ) = cos f( ) = int( ) f( ) = 1 + e 41. Bounded (3 functions). 4. Increasing on the entire domain (6 functions). 43. Even (3 functions). [1.4] Given f () = ( 4), g () = 3 and h () = + 5. Find and simplif the answer. 44. f h(4) 45. h (g ()) 46. (g f )() 47. (fg)() [1.4] 48. Given: f () = 3 +. Find f -1 (). 6

7 [1.4] 49. Verif that f and g are inverses of each other. Use correct notation and show all algebraic steps. f ( ) 8 = + and g( ) = 8 [P4] 5. Write the equation of a line passing through the point (3, 4) that is a) parallel to 5 = 7, and b) perpendicular to 5 = 7 [1.6] 51. The height of a right circular clinder equals its diameter. Write the volume of the clinder as a function of its radius. [.] 5. Write the statement as a power function equation and answer the question. The electrical resistance of a wire varies directl as its length and inversel as the square of the diameter of the wire. a) Write a model for this situation. b) Suppose 5 mm of a wire of diameter 3 mm has a resistance of 8 Ω. Use this information to find the constant k. c) What is the resistance of 4 mm of the same tpe of wire if the diameter is 4 mm? [.] 53. The table below gives the weight and pulse rate of selected mammals. Use the power regression equation to determine the pulse rate of a human weighing 1 pounds. Mammal Bod Weight Pulse Rate (beats/min) Rat. 4 Guinea Pig.3 3 Rabbit 5 Small Dog 5 1 Large Dog 3 85 Sheep 5 7 Human 7 7 NOTE: An of the following regressions ma be tested on the final: linear, quadratic, cubic, quartic, eponential, logarithmic, logistic, power, sinusoidal. 7

8 [.1] 54. The Sweet Drip Beverage Compan sells cans of soda in machines. The marketing director finds that sales average 6, cans per month when the cans sell for $.5 each. For ever $.5 increase in the price, the sales per month drop b 1 cans. a) Write an equation to model the total revenue realized b Sweet Drip, where is the number of $.5 increases in the price of a can of soda. b) How much should Sweet Drip charge per can of soda to realize their maimum revenue? [3.] 55. Fruit flies are placed in a container with a banana and east plants. Suppose the fruit fl population after 3 t das is given b Pt () =..37t e a) What is the maimum number of fruit flies the container can hold? b) How man fruit flies were originall placed in the container? c) How long does it take for the number of fruit flies to reach one-half of the maimum flies that the container can hold? For questions 56-57, write a model for the situation. Be sure to clearl define our variables. Then use our model to answer the question. Solve algebraicall AND graphicall. [3.] 56. Shan invested $1 at 3.5% interest compounded monthl. Determine how long it will take for Shan to save up $7 assuming to additional deposits or withdraws. [3.] 57. A radioactive isotope decas at a rate of 3% per da. A scientist has an initial amount of 5 g. Determine approimatel how man das it will take for half the isotope to deca and then write a half-life model. 8

9 [3.] 58. Use the TVM Solver: a) Find the monthl pament for a mortgage on a $75, house paid at the end of each month if the interest rate is 3.5% on a 3 ear loan. b) After graduating from college and getting our first job, ou decide to open an individual retirement account (IRA) using $5 ou got for graduation. The account pas 6.% interest at the beginning of each month and ou have budgeted to invest $1 each month. If ou retire in 5 ears, what will IRA be worth? [3.5] 59. The wind speed s (in miles per hour) near the center of a tornado can be modeled b s = 93 log d + 65 where d is the distance (in miles) that the tornado travels. a) In 195, a tornado traveled miles through three states. Estimate the wind speed near the tornado s center. b) An F5 tornado has wind speeds of more than 3 mph, estimate the distance an F5 tornado with wind speed of 31 mph will travel. [4.8] 6. When a spaceship is fired into orbit from a site such as Cape Canaveral, which is not on the equator, it goes into an orbit that takes it alternatel north and south of the equator. Its distance from the equator can be approimatel modeled b a sinusoid function. Suppose that the spaceship is fired into orbit from Cape Canaveral. Ten minutes after it leaves the Cape, it reaches its farthest distance north of the equator, 4 kilometers. Half a ccle later it reaches its farthest distance south of the equator (on the other side of the Earth), also 4 kilometers. The spaceship completes an orbit once ever 9 minutes. Let be the number of kilometers the spaceship is north of the equator (ou ma consider south of the equator to be negative). Let be the number of minutes that have elapsed since liftoff. a) Sketch a complete ccle of the graph of distance versus time. b) Write the sinusoidal model. c) Use our equation to predict the distance of the spaceship from the equator when = 41 and = 163 mins. d) Find the number of kilometers Cape Canaveral is from the equator b calculating when =. Since we just finished Unit 4: Conic Sections, there are no review problems included on these sections but Conic sections will be on the final. Go back and use our Unit 4 Review to stud. 9

1.2 Functions and Their Properties PreCalculus

1.2 Functions and Their Properties PreCalculus 1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given