Math Precalculus Algebra FINAL PRACTICE PROBLEMS Name Please show all work and simplify and bo answers. Non-graphing scientific calculators are allowed. Solve for w. 3 1) m yw tw ) a) Find the equation of the line connecting the points 3, 1 &,5. b) Find the equation of the line perpendicular to this line that goes through the point 1,. Simplify the epression completely. 3 1 1 3) 3 3 1 1 3 3
4) The height of a pole vaulter is approimated by the function H ( t) t 3t 110 where H t 0 and t 0 is the time in seconds. Sketch the graph below. a) How many seconds did it take to reach the maimum height? H(t) b) What was the maimum height reached? t c) How many seconds did it take for the pole vaulter to land on the ground after reaching the ma height? Write as a product of linear factors and find all the zeros. Then sketch the graph below. f 18 8 4 5) 6 4 3 y
6) Find the inverse of ( ) 3 1 f f for 3. Verify together on the same graph. f f 1 f 1 ( ( )) ( f ( )). Graph f and y Solve this equation for. Write eact answer and as a decimal to 3 places. 3 3 7) 6 Solve this inequality and epress solution in interval notation. 1 8) 3 9 3
9) Find the zeros of f, epress f as a product of linear factors, and sketch the graph below. 5 4 3 f 6 14 11 3 y 10) Find f ( a h) f ( a) h for h 0 if 1 f ( ). Simplify any comple fractions. Sketch the graph of f 11) f 1. Include V.A. s, H.A. or O.A., H.A. or O.A. intercept, -intercepts, and y-intercept. y 4
Solve. ln 1 ln 3 ln8 1) Solve. log 4 log 13 log 8 13) Solve. 14) log log log 3 1 Solve. 15) log log log 3 3 3 5
16) Find the focus, directri, and focal diameter of the parabola, and sketch its graph. a) 4y b) 3y 17) Find the vertices and foci of the ellipse, and sketch its graph. y a) 1 b) 4 y 16 c) 4 9 5 6y 30 18) Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. y a) 1 b) y 4 c) 5 9 y 3 6
Solve the following systems of equations algebraically. 19) y 18 y 4 0) y 1 y 7 1) y 16 4 y 4 7
Graph the system of equations, and then solve the system algebraically to find the eact points of intersection. ) y 4 y 4 y 3) y y y 1 4) y 6 9 5) y y 10 8
Word Problems 1) Find out how long it takes a $300 investment to double if it is invested at 8% compounded monthly. r Round to the nearest tenth of a year. Use the formula A P 1 n. nt 0.039t ) The formula A 65e models the population of a particular city, in thousands, t years after 1998. When will the population of the city reach 457 thousand? 3) How long would it take for a bank deposit to increase ten-fold (10 times initial deposit) at today's average continuously compounding interest rate of 0.5%? 4) If Emery has $1500 to invest at 1% per year compounded monthly, how long will it be before he has $100? If the compounding is continuous, how long will it be? (Round your answers to three decimal places.) 5) Cindy will require $19,000 in 4 years to return to college to get an MBA degree. How much money should she ask her parents for now so that, if she invests it at 9% compounded continuously, she will have enough for school? (Round your answer to the nearest dollar.) 9
6) The Feldmans bought their first house for $17,000. Over the years they moved three times into bigger and bigger houses. Now, 41 years later, they are ready to retire and want a smaller house like the first one they bought. If inflation in property values has averaged 3.1% per year during that time, how much will such a house cost them now? (Round your answer to the nearest dollar.) 7) A culture of bacteria obeys the law of uninhibited growth. If 140,000 bacteria are present initially and there are 609,000 after 6 hours, how long will it take for the population to reach one million? 0.40t 8) The amount of a certain drug in the bloodstream is modeled by the function y y0e, where y 0 is the amount of the drug injected (in milligrams) and t is the elapsed time (in hours). Suppose that 10 milligrams are injected at 10:00 A.M. If a second injection is to be administered when there is 1 milligram of the drug present in the bloodstream, approimately when should the net dose be given? Epress your answer to the nearest quarter hour. 9) Assume that the half-life of Carbon-14 is 5700 years. Find the age (to the nearest year) of a wooden ae in which the amount of Carbon-14 is 30% of what it originally had. 10