1. Simplify. Assume all variables represent positive numbers.

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1 MATD 090, INTERMEDIATE ALGEBRA Review for test 4 Test 4 covers all cumulative material, in addition to sections , 7., 7., 7.4, 7.. Bring a non-graphing calculator and something to write with. Be prepared to show an appropriate amount of work on all test problems.. Simplify. Assume all variables represent positive numbers a y 0 7 z 8y 6 7 4a b e. 48 y. Multiply and simplify. Assume all variables represent positive numbers. 0 0 ( + )( ) 4 0 ( 7 ). Simplify Rationalize the denominator. Assume all variables represent positive numbers. 7 z + y. Simplify. Epress answer in the form a+ bi. ( i) ( + i) ( 4+ i)( 9i) ( + i)( i) e. i i 0 i + i page of 7

2 6. Find the missing lengths. Give an eact value and a decimal approimation to three places, if appropriate.? 0? 6 7. Epress in terms of i, and simplify i 80 i 8. Solve by completing the square. Give an eact value and a decimal approimation to three places, if appropriate. You must show your work using this metho + = 9. Find the -intercepts of the function. 6= f ( ) = g( ) = 4 0. Solve. Give eact (simplified) values only. + = 6 ( 4) = ( ) = 6 + = ( )( ) e. ( + 7) = f.! = 4 g.! 7 = h. + +=. Find the discriminant for the quadratic equation. Then, without solving the equation, determine the number and type of solutions (rational, irrational, or imaginary) = 0 0+ = = 0 0 = 0 page of 7

3 . For the quadratic function f ( ) =!(! ) Describe the transformation from the graph of y =. Graph the function. For the quadratic function g ( ) = ( + )! Describe the transformation from the graph of y =. Graph the function f = For the quadratic function ( ) find the verte graph the function. For the line described by the equation y= 0 What is the -intercept? What is the y-intercept? What is the slope? Graph the line. 6. Solve. Epress your solution in interval notation. < 9 7. For the function g( ) = 7+ 0 Evaluate g ( 6) What point is on the graph of g? Find the domain 8. Find the domain f ( ) = g( ) =!4 h( ) = + 9. The profit, in dollars, from sales of grand pianos for a manufacturer is given by the function ( ) P = , where is the number of pianos produced and sold in a given month. How many pianos should the manufacturer produce to maimize profit? What is the maimum profit attainable? page of 7

4 0. Choose the appropriate formula for the quadratic function whose graph is drawn below. f = ( ) ( ) f = ( + ) ( ) f = ( ) ( ) 6 7 f = ( + ) ( ). Is the discriminant positive, negative, or zero?. The number of inmates in custody in US prisons and jails can be modeled by the quadratic function p( ) = + 94+, where ( ) is the number of years after 990. [ In what year will the number of inmates reach its maimum? What is the maimum number of inmates? p is the number of inmates (in thousands) and. Harvey wants to create 0 oz of iodine solution of strength %. He has on hand some iodine solution of strengths 8% and 0%. How much of each should she use? 4. A rectangular garden has area 90 square feet. Its length is feet less than twice its width. What are the dimensions?. The still water speed of a motorboat is miles per hour. The motorboat travels upstream, to a point 8 miles away, and then back to its starting point. The round trip takes. hours. What is the speed of the current? 6. The current of a river flows at a rate of miles per hour. A rowboat travels 6 miles downstream in the same amount of time it takes to travel 6 miles upstream, what is the still water speed of the rowboat? 7. A large pump can empty a swimming pool in hours less time than a small pump. The two pumps working together can empty the pool in hours. How much time does it take each pump working alone? Round your answer to one decimal place Joe rides his bicycle 40 miles to a vista point at the top of a hill. His average biking speed up the hill is miles per hour slower than his average biking speed down the hill. The total time for his round trip is 7 hours. Find Joe s average speed in each direction. page 4 of 7

5 ANSWERS:. 9a. 7 8i 4 y y b a b z y z 0i 6 i e. 4 7i. e. 6 y i. i or i 4 i =, = = = z 9. ( 0,0 ), (,0 ) + 44,0,,0 y y page of 7

6 0. + i, i { +, }. Reflect about ais Shift right units. { + 4 i, 4i}, e. { 7+, 7 } f. {,} g. { } h. { } Note: 0 is etraneous.. Shift left unit Shift down units.. Discriminant = 4 Two imaginary solutions Discriminant = 96 Two irrational solutions - Discriminant = 0 One rational solution 89 = 7 Discriminant = Two rational solutions 4. (, ) page 6 of 7

7 . (,0 ) ( 0, 0) 6. [,) g ( 6) = ( 6, ) (because inde is odd) [,+ ) R { }. Verte = ( 47,4) 07,4,000 inmates. Equations: + y= y= 0.. oz of strength 8%, and 7. oz of strength 0% 4. L w ( )( ) = Equation: ( ) Width = 7. feet, and Length = feet. Equation: w w = =. + c c Answer: The current speed is mph 6. Equation: 6 6 = s+ s The still water speed is miles per hour 7. Equation: +! = Large pump:.0 hours Small pump:.0 hours 8. Equation : = Joe s average speed is 8 mph uphill and 0 mph downhill d 700 pianos $4,7,000. negative positive zero page 7 of 7