VILLA VICTORIA ACADEMY (07) PREPARATION AND STUDY GUIDE ENTRANCE TO HONORS PRECALCULUS PART FROM HONORS ALGEBRA II ) Simplify. 8 4 ) Evaluate the expression if a ; b ; c 6; d ) Translate each statement into symbols, Six more than the difference of nine and x is Negative 5 is less than less than x. 4) Simplify each expression. a) ab ba b) d a b a) 4 5 b) 98 6 POLYNOMIALS: 5) Simplify. 5e 8 f e f b) a) 6 7 8 v 7w c) 5x y x y d) e)4x x x f) x x x g) x x h) x 7 i) x 5 j) x x x 4 k) x x x 5 l) x m) x n) o) x 8x 6) Factor completely. a) x - 5x + 4 b x - x + c) x - x - 8 d) 9x - 4x + 6 e) ax - 6a + cx - 6c f) (ab + ac ) - ( b + c ) g) 4x - 6 h) a - ( x + ) i) x - 6x - 40x j) x 4-0x + 9 k) 5x - 65
7) Add. a) x - 5x - b) (x + x - 5x - 9 ) + ( x - x - x + 7) -x + x - 8) Subtract. a) 4x - 7x - b) (x - x + 4x + ) (x - x - 5x - 6) (-) x + x - 5 9) Divide using the process of long division. a) x - 4x +_ 45 b) x - x + 8. x - 9 x - RATIONAL EXPRESSIONS 0) Simplify each expression, assuming that no denominator equals zero. a) 0x 7 b) 4x y - 64x 4 y. -5x 0 8x y c) x + 6 d) 8x 4. x - 4 x - 6x e) 0 + x - x f) x + x - 8. x - x - 0 x + x - 0 g) 7x 4 65x x h) a b a b i) 4x y 5z j) a - b a - b. 5z 8xy 4 6 k) b - b b - b l) x - 5 x + 5. b + b b + 5b + 6 x - - x m) - 5 n) x + y. x - - x x - y y - x o) x - - x + - 7. x x - 0
COMPLEX RATIONAL EXPRESSIONS: p) a a b b q) t t RADICALS: ) Simplify. No approximations. Use absolute value where appropriate. a) 80 b) 00x y c) 6.44x d) e) 6 8 x 8x 6 f) 4 cd c g) 98 h) 4 i) x x j) x y 6 7 k) 5 6x 6x 4x l) 5 m) o) x x y n) 08 500 4 x x p) 6 4 6
4 EQUATIONS: ) Solve. a) x - 4-7x = 0 b) x - (4x - 5) = - x - c) 5x ( x - ) = 0 d) y - 5y = 0 e) x - = - x f) x 4 + 6 = 8x g) 8 = + h) = 5. x x - x + 6x - i) x 6 j) x 5 k) 5 x x l) x x m) x + (x + 4) = + ( x + ) n) ( x + 9 ) = - 6 o) x + 6 = p) x 5 8 x + 6 x + 6 q) 7 + x = 4 r) - 5 + x - 5 = SOLVING LITERAL EQUATIONS: s) Solve for a: t) Solve for r: A = wt + ½ a t rs - 4r = 4s - 5 SOLVING QUADRATIC EQUATIONS: ) Use the quadratic formula to find the solution(s) for the following. a) x - x - 5 = 0 b) x = - 4x c) - 4x + 0x + 9 = 0 d) 5x - x + = 0 e) ( x - 5) - x = 0 f) x - x +.5 = 0 DETERMINING NATURE OF SOLUTIONS FOR QUADRATICS; GIVE THE DISCRIMINATE, # OF SOLUTIONS, AND THE SET OF NUMBERS TO WHICH THE SOLUTIONS BELONG. 4) Use the discriminate to determine the number & nature of the solutions. a) x - x + = 0 b) x + x + 6 = 5 c) x - x - 6 = 0 d) x =.8x - 0.8
5 INEQUALITIES: 5) a) - 4 - x b) x - 4 > 5 c) 5 x - d) x + < x - 5 e) (x + )(x - ) < (x + )(x - ) f) x > 4x + 5 g) -5 < x + < 0 h) y - > y or y > 0 i) - y < 6 or y > y - j) 6 - x < k) x/ - l) - x - 5 4 SYSTEMS OF EQUATIONS AND APPLICATIONS. 6) Write an equation(s) for each word problem, solve and label the answer. a) The perimeter of a rectangle is 64 ft. and the length is 7 inches. Find the width. b) Jay has three times as much money as Sue, and Sue has $5 less than Larry. Together they have $5. How much money does each have? c) Adult tickets for a concert were $5 each and student tickets were $ each. A total of 980 tickets, worth $460, were sold. How many adult tickets were sold? d) Pam has some nickels, dimes and quarters worth $6.75. There are four times as many nickels as dimes, and five more quarters than dimes. How many of each kind of coin does she have? e) Sara earns $6 an hour more than her assistant. During an 8 hour day they earn $40. How much does each earn per hour? f) Find three consecutive integers whose sum is 49. g) The perimeter of a rectangle is 4 m. The length of the rectangle is m. less than twice the width. Find the length and width of the rectangle. h) A postal clerk sold some fifteen-cent stamps and some twenty-five cent stamps. Altogether, 0 stamps were sold for a total cost of $.70. How many of each type of stamp were sold? i) Two cars start at the same time from the same point and travel in opposite directions. One car travels 0 mi/h faster that the other. In hours they are 00 miles apart. Find the rate of each car.
6 j) Two airplanes leave Chicago at noon, one traveling west at 575 km/h and the other east at 65 km/h. At what time will they be 000 km apart? k) A bicyclist rode up a mountain road at km/h and then back down at 0 km/h. If the round trip took.5 h, how long did the ride up the mountain take? l) A rectangular swimming pool is 6 m longer than it is wide. It is bordered on all sides by a m concrete walk. The area of the pool and walk is 76m greater than the area covered by the pool alone. What are the dimensions of the pool? m) A bicycle rider left town at noon and traveled at a uniform rate of 5mi/h. At :00 PM the same day, a motorcycle rider left for the same place at a uniform rate 0 mi/h greater. At what time did the motorcycle rider overtake the bicycle rider? n) A rectangle is three times as long as it is wide. If the length and the width are each increased by 4, then the area is increased by 76. Find the dimensions of the original rectangle. o) The sum of two numbers is 8 and the sum of their squares is 64. Find the numbers. p) The squares of two consecutive positive integers total 45. Find the integers. q) How many kilograms of water must be evaporated from 0 kg of an 8% salt solution to produce a 5% salt solution. r) One pipe can fill a swimming pool in 8 hours. Another pipe takes hours. How long will it take to fill the pool if both pipers are used simultaneously. s) A new high-speed copier works three times as fast as a regular copier. When both copiers are used, they can copy a group of documents in minutes. How long would each copier require to do the copying alone? t) How many grams of gold alloy that costs $4 per gram must be mixed with 0 g of a gold alloy that costs $7 per gram to make an alloy that costs $5 per gram. u) Flying with the wind, a jet can travel the 400 km distance between San Francisco and New York in 6 hours. The return trip against the wind takes 7 hours. Find the rate of the jet in still air and the rate of the wind. v) A motorboat traveling with the current can go 60 km in 4 hours. Against the current it takes 5 hours to go the same distance. Find the rate of the motorboat in still water and the rate of the current.
7 w) The sum of the digits of a two-digit number is 8. If the digits are reversed, the number is increased by 8. What is the original number. x) Wilma is twice as old as Gary. One year ago, the product of their ages was 0. How old is each? y) The denominator of a fraction is more than the numerator. If 6 is added to the numerator and 6 is subtracted from the denominator, the value of the resulting fraction is equal to to. Find the original fraction. z) Ernest receives $555 per year from his $7000 investment in municipal and co corporate bonds. His municipal bonds pay 6% and his corporate bonds pay 9%. How much money is invested in each type of bond? aa) If y varies directly as x and y = 80 when x = 9, find y when x =. bb) If y varies inversely as x and y = - when x = 6, find x when y = 6. cc) The area of a trapezoid varies jointly as the height of the trapezoid and the sum of the bases. The area of a trapezoid is 48 cm when the height is 8 cm and the sum of the bases is cm. Find the area when the height is cm and the sum of the bases is 8 cm. dd) The power, P, of an electric current varies directly as the square of the voltage, V, and inversely as the resistance, R. If 6 volts applied across a resistance of ohms produces watts of power, how much will volts applied across a resistance of 9 ohms produce? ee) The time needed to fill a tank varies inversely as the square of the radius of he hose. If a hose of radius.5 cm takes 8 min to fill a tank, how long will it take using a hose of radius cm. ff) The price of a diamond varies directly as the square of its mass. If a.4 carat diamond costs $764, find the cost of a similar stone with a mass of.7 carats. gg) The cost of operating an appliance varies jointly as the number of watts, hoursof operation, and the cost per kilowatt-hour. It costs $.45 to operate a 000-watt air conditioner for hours at a cost of $.075 per kilowatthour. Find the cost of operating a 00-watt dishwasher for 40 minutes.
8 VILLA VICTORIA ACADEMY (07) PART REQUIRED SKILLS FOR ENTRANCE TO HONORS PRECALCULUS FROM HONORS ALGEBRA II (INCLUSIVE OF THE SKILLS FROM ALG I TO H ALG II) LINEAR EQUATIONS: 7) Solve each of the following. a) Find the slope of the line which passes through the following points. ) ( -, 4); ( 6, ) ) (-, 5 ) ( -, 8) ) ( -4, 6 ) ( -, 6) b) Find the slope of each line. ) y = x + ) x - y = 7 ) y = - c) Write the equation in standard form of the line that has the given information. ) m =, b = -5 ) m = P ( -, ) ) A (, - ); B( -, ) 4) P( -, ) parallel to y = -x 5 5) P (-, 5) perpendicular to x + y = 6 8) Find the slope for the following: a) x = b) y = GRAPHING SKILLS: LINES: 9) Graph the equations and label the points used to produce your graph. a ) x + y = - b) x + y = - 6 LINEAR INEQUALITIES: 0) Graph the inequalities and label the points used to produce the graph. a) y > - x + b) x - 4y 0 SYSTEMS OF LINEAR INEQUALITIES: ) Graph the solution set of the system and label the points used to produce the graph. a) y 4 b) x > x x + 4y 6 SOLVING LINEAR SYSTEMS: ) Solve the following linear systems using the indicated method. a) by linear combination method b) by substitution method x - y = 5 x + y = 9 x + y = 5 x + y =
9 VILLA VICTORIA ACADEMY (07) PART REQUIRED SKILLS FOR ENTRANCE TO HONORS PRECALCULUS FROM HONORS ALGEBRA II (INCLUSIVE OF THE SKILLS FROM H ALG I TO H ALG II) COMPLEX NUMBERS ) ( 4 ) ( 5 4 ) ) ( 6 ) ( 5 ) ) 4) 09 5) 9 6) 8) 4 7) 4 5 9) 5 0) Graph: 4 LOGARITHMIC & EXPONENTIAL FUNCTIONS Simplify each of the following: ) ) log4 64 6 ) log6 0 6 Solve the following: 5) log55 6) log 8 4) If logb z logb x logb y then write z in terms of x and y. 7) log 4 8) log x 9) log x 7 ) log x 0 ) 9 log 0 x # to 4 EXACT ANSWERS ONLY x log 9 x 4) 5 6 56 ) 0) log x 64 ln x 5 5) 6) x ln e 4 7) ln x ln x ln0 8) ln x ln x 9) x 50 e 0) x x 7 8
0 ) log 9 4 log x log 6 x ) ) log x5 log x5 log 8 4) x a a a log log 4 FUNCTIONS, COMPOSITIONS, INVERSE FUNCTIONS, DOMAIN & RANGE If f(x) = x; g(x) = x + and h(x) = x + Determine the following: 5) f(g(-6)) 6) g(f(-6)) 7) f(h(x)) f x 8) h(f(x)) 9) h(h(x)) 40) 4) h x 4) g(g (x)) 4) Use compositions to show that the following functions are inverses. Use f f x f f x x f ( x) x 7 x 7 gx ( ) 44) If f(x) = 6 x, then f (x) is? 45) Find f (6) and f - 6 46) Find the domain and the range for f(x) and for its f (x). Label each carefully. f x x x CONICS Identify the conic section, transform into the standard form for that conic section, find the center, focus(i), vertex or vertices, equations for asymptotes and directrix, if applicable. Then sketch the graph and label all critical features on the graph that you have identified in the previous list. 47) x + y + x - 0y + 0 = 0 48) 9x + 5y - 8x - 50y + 9 = 0 49) 4x - 9y - 4x - 6y + 6 = 0 50) x = -y + y - 5
DISTANCE AND MIDPOINT Determine the distance and midpoint for each of the following sets of coordinates. 5) A ( -4, -) B(, ) 5) A( -a, b) B( a, 4b) SEQUENCES AND SERIES 5) Expand the simplify the expression ( c d ) 5 using binomial expansion. 54) Find the fourth term in the expansion of the expression ( a - b) 7. 55) Find the term containing x 8 in the expansion of the expression ( x - y ). REMAINDER AND FACTOR THEOREMS 56) Use synthetic division to find P(a) for the given polynomial P(x) and the given value of a. P(x) = 5x + x + x ; a = - 57) Use the factor theorem to determine whether the binomial is a factor of the given polynomial. The binomial is t + & P(t) = t 5 + t 4 + t + t + t + 58) A root of the equation is given. Find all the solutions of the equation. One root is. P(x) = x + 9x + 7x - 6 59) Find the polynomial equation with integral coefficients that has the given numbers as roots. Roots = { 0, -,, - } 60) A root of the equation is given. Find all the solutions of the equation. P(x) = x + x - 0; One root is - +