MAT 116 Final Eam Review 1. Determine the equation for the line described, writing the answer in slope-intercept form. The line is parallel to = 8 and passes through ( 1,9). The line is perpendicular to = + 7 and passes through (6, 5). The line passes through the points (,0) and (,6).. Solve for, finding all real and imaginar solutions. Give eact answers in simplified form (no decimal approimations, unless it is specified otherwise). + 1 = 1 ( 6) + 9 = 1 ( 5) + = 0 + + 1 = 8 + + 6 = (f) ( ) = 7 (g) (h) 5 + + 1 = = (i) ( 1) /5 = (j) 6 = 1 (k) (l) 9 1 = = 6 (Find both the eact answer and the approimation to four decimal places). (m) log( + ) = 1 (n) ln ln( + ) = ln5. Solve each inequalit. Epress the answer in interval notation. < 6 and 5 > + 1 5 > + 15 5 1 0
. Consider the functions f () = + and g() = 6. Determine the following. Show all work, simplifing where needed and using proper function notation throughout. Domain and range of f (Epress answer in interval notation.) f (7) g(t + 1) (g + f )( 1) ( f /g)() (f) Domain of f /g (g) (g f )(7) (h) ( f g)() 5. Consider the function = + 5. Use transformations to sketch the graph of the function. Include at least three points on the graph. Determine the domain and range of the function. 6. Write the equation of the graph after the described transformations and determine its domain and range. The graph of = shifted 6 units to the right, reflected across the -ais, and then shifted 10 units upward. The graph of = 7 translated two units to the right, reflected over the -ais, and then translated three units upward. 7. For the given function f below, sketch the graph of f 1. 1 1 1 = 1 = f () 8. Find the inverse of the function f () = 1 7.
9. Consider the quadratic function f () = +. Find the following. Give eact answers onl (that is, no decimal approimations), simplifing as needed. Epress intercepts as ordered pairs. Ais of smmetr Verte -intercept(s) -intercept What if the quadratic function had been given to ou in the form f () = ( + 1) + 5? 10. Consider the rational function f () = +. Find the following. Domain of the function (Epress answer in interval notation.) (f) Equation of the vertical asmptote Equation of the horizontal asmptote -intercept(s) (Epress as ordered pairs.) -intercept (Epress as an ordered pair.) Sketch the graph of the rational function, including asmptotes, intercepts, and at least two additional points. 11. Use appropriate division techniques to find the quotient and remainder when +6 is divided b 5. 1. Let f () = + 6 10. Use appropriate division techniques to find the function value f ( ). 1. Determine whether the binomial + 5 is a factor of the polnomial + 8 + 11 0. If it is a factor, then factor the polnomial completel. 1. Find all real and imaginar roots for the equation + + 6 8 = 0. (This could also be worded Find all real and imaginar zeros of the polnomial function f () = + + 6 8. ) 15. Find a polnomial equation with real coefficients that has the roots 0 and 1 + i. (Epress the answer in the form a n n + a n 1 n 1 + + a 1 + a 0 = 0.) 16. Consider the function f () = ( + 5) ( ). Find the zeros of the polnomial function f () and state the multiplicit of each zero. List the -intercepts as ordered pairs, and discuss the behavior of the graph of f () at each -intercept. State the degree and leading coefficient of f (). Use the leading coefficient test to determine the behavior of the graph of f () as and as. Give a rough sketch of the graph of f () that incorporates the behaviors described in parts and.
17. Evaluate the following without using a calculator. 7 / 1 log 16 ln e ( ) 6 1 18. Rewrite ln 5 as a sum or difference of multiples of logarithms. 19. Rewrite 1 [log + log] logz as a single logarithm. 0. Solve the following application problems. Define variables, write equations, and solve. Organize all work neatl. (f) (g) Jeff knows that his neighbor Sarah paid $0,0, including sales ta, for a new Buick Park Avenue. If the sales ta rate is 8%, then what is the cost of the car before ta? Raisins sell for $.50 per pound, and bran flakes sell for $.80 per pound. How man pounds of raisins should be mied with 1 pounds of bran flakes to get a miture that sells for $.1 per pound? A mechanic is working on a car with a 0 quart radiator containing a 60% antifreeze solution. How much of the solution should he drain and replace with pure water to get a solution that is 50% antifreeze? A sk diver steps out of an airplane at 5,000 feet. Its height after t seconds is given b the equation S = 16t + 5,000. How long does it take the diver to reach,000 feet? If an archer shoots an arrow straight upward with an initial velocit of 160 feet per second from a height of 8 feet, then its height above the ground (in feet) at time t (in seconds) is given b the function h(t) = 16t + 160t + 8. What is the maimum height reached b the arrow? How long does it take for the arrow to reach the ground? A deposit of $, ( 000 earns.5% annual interest compounded quarterl. Use the compound interest formula A = P 1 + n) r nt to find the amount in the account at the end of 5 ears (rounded to the nearest cent) and the total amount of interest earned during the 5 ears. Use the continuous compound interest formula (A = Pe rt ) to determine the amount of time (to the nearest tenth of a ear) that it would take for $100 to grow to $500 at.7% compounded continuousl. 1. Solve the given sstem of linear equations using algebraic methods. = + = 1 + = 7 + = 5 + = 9 + = 5
. Use graphing calculator techniques to determine the reduced form of the given sstem of linear equations and appropriatel interpret the solution to the original sstem. + z = z = 10 + z = 1 + + z = 8 + + z = 1. For each of the following application problems, set up a sstem of linear equations in two variables, defining the two variables. Solve the sstem using algebraic methods. Mike works a total of 60 hours per week at his two jobs. He makes $8 per hour at Burger King and $9 per hour a West Side Car Wash. If his total pa for one week is $50 before taes, then how man hours does he work at each job? Nora spent $19.80 on postage inviting a total of 60 guests to her graduation part. Each female was sent a picture postcard invitation, while each male was invited with a formal letter. If she put a 5-cent stamp on each postcard and a 7-cent stamp on each letter, then how man guests of each gender were invited?
Answers to MAT 116 Final Eam Review 1. = 1 + 19 = + = + 5. = 10 = or = 1 = 5 ± i 6 = 1 = ± (f) = ± 11 (g) = 8 (h) = 0 or = 7 (i) = 1 ± (j) = 0 (k) = 1 (l) = log6 log.5850 (m) = 5 or = (n) No solution. (1, ) (, ] [, ) (, 5) (, ) (,1) [5, ). Domain: [, ]; Range: [0, ) 5 t 8 + 6 (f) [,6) (6, ) (g) (h) 8
5. Passes through the points: ( 6,1), (, 5), (0,1) 8 6 8 6 6 8 6 8 Domain: (, ); Range: [ 5, ) 6. = ( 6) + 10 Domain: (, ) Range: (,10] = 7 + Domain: (, ) Range: (,] 7. = f 1 () 1 1 1 = 1 = f () 8. f 1 () = + 1 9. = 1 ( 1,5) ( 1 + (0,) 10. (,) (, ) = ) ( ) 10 10,0, 1,0 = ( ),0 (0,1)
(f) Passes through the points: ( /,0), (0,1), (1,5), (, 7), (6, 5) 8 6 8 6 6 8 6 8 11. Quotient: + Remainder: 16 Note: + 6 = ( 5)( + ) + 16 1. f ( ) = 1. + 5 is a factor of + 8 + 11 0. Furthermore, + 8 + 11 0 = ( + 5)( + )( 1). 1. = 1, =, and = ±i 15. + 10 = 0 16. = 5 has multiplicit, and = has multiplicit The graph of f () onl touches the -ais at ( 5,0), and it crosses the -ais at (,0). The degree of f () is 5. The leading coefficienent of f () is 1. as as 9 8 7 6 5 1 1 5 6 7 8 9
17. 9 1 18. ln6 1 ln( 1) ln5 ln 19. log z 0. $7, 50 pounds of raisins 10 = 1 quarts should be drained and replaced with pure water.5 10 7.9 seconds 08 feet; 10+ 10 10.05 seconds (f) $,501.50; $501.50 (g) 1..5 ears ( ) 1, 1 No solutions ( 7,1) An augmented matri solution is below: 1 9 R 1 R 1 1 5 1 1 5 1 9 1 R R 1 1 5 0 1 1 R 1 +R R 1 1 5 0 R +R 1 R 1 1 0 7 0 1 1. ( 1,1,) {(5 z, + z,z) z is an real number}. 8 hours at Burger King, hours at West Side Card Wash 0 females, 0 males