[ ] 4. Math 70 Intermediate Algebra II Final Exam Review Page 1 of 19. Instructions:


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1 Math Intermediate Algebra II Final Eam Review Page of 9 Instructions: Use our own paper for the review questions. For the final eam, show all work on the eam. This is an algebra class do not guess! Write and solve equations to earn full credit. An approved graphing calculator is required. If ou solve a problem using onl our calculator, ou must give a brief eplanation, including essential kestrokes, of what ou did on our calculator or ou will not receive credit. Completel simplif algebraic answers. Completel reduce numerical answers. Give eact answers (no rounding) unless instructed otherwise, or unless units require rounding to make sense. Include units where appropriate. Plot and label at least points for all nonlinear graphs; plot and label two or more points for linear graphs. Problems ma continue on the net column or page. Evaluate: 9 ) + ) ( ) ( ) + ) 9 ) + ) [ ] ) ( ) + ( ) ) Solve for the specified variable. (a) r = + for a = c for a a b b+ c (c) = for b b c d + f + c (d) = for. g+ h 8) Determine whether each ordered pair is a solution of the given equation or not. (a) = (, ) = (, ) (c) = 9 (, ) (d) = 9, 9) Given the function: f ( ) =, evaluate + the following. If undefined, so state: (a) f ( ) f () (c) f ( ) ) For the line, find the slope, the  intercept, and the intercept:. Then graph. If none, so state. (a) = + = ) Find the slope of the line through the given points. If undefined, so state. (a) (, ) and (,.) (, ) and (, ) ) Parallel, perpendicular, or neither? (a) = and = = and = (c) = + and = + (d) = and + = (e) + = and + = (f) = and = ) Nine less than five times a number is. What is the number? ) Write the equation of the following lines. Use function notation where possible. (a) through (, ) with slope m = Through (, ); vertical (c) Through (, ); slope m = (d) Through (, ) with undefined slope (e) Through (, ), horizontal line
2 Math Intermediate Algebra II Final Eam Review Page of 9 (f) Through the two points (, 9) and (, 9) (g) Through (, ) and parallel to: + = (h) Through (, ) and perpendicular to + = ) Which of the following is the graph of a function? (e) (a) (c) (d) (f) ) Using either the intersection or  intercept method, solve each equation graphicall using our graphing calculator. For each equation answer the following: What function(s) did ou graph in our calculator? Briefl describe the location of the solution on our graph. Round the solutions to four decimal places. (Hint: Do not even TRY to solve these b hand!) (a) = + π = + (c). + = + (d) = + (e) ln( ) + = e (Hint: solutions) (f).( ) =. + (Hint: solutions) (g) π = + (h) log( + ) = + (i). = (Hint: solutions) + = ) Given the sstem: = (a) Solve the sstem using an method ou prefer. Use the graph to solve the sstem. In other words, graph the two linear equations and show that the intersection is the same as the answer ou got in part (a).
3 Math Intermediate Algebra II Final Eam Review Page of 9 8) What would the graph of the sstem = + tell ou about its solution? = (Answer in our own words without actuall solving or graphing the sstem.) 9) Solve the linear sstems below. Write our answer as an ordered pair with integers or fractions, not decimals. If appropriate, write no solution or epress our solution in setbuilder notation. = (a) + 8= + = = ) First, write the augmented matri of the sstem. Second, write the reduced row echelon form of the matri with integers or fractions (not decimals). Third, use matrices to solve the sstem. Write our answer as an ordered triple with integers or fractions, not decimals. If appropriate, write no solution or epress our solution in setbuilder notation. z = (a) z = z = + + z = + z = + + z = + + z =. (c) z =. + + z= + z = + (d) = z 9= z + z = (e) + z =. z =. + z = (f) + z = 8 + z = ) Find three consecutive numbers such that the sum of the first integer, half the second integer, and four times the third integer is. ) How much mone will be in our account if ou invest $ at an annual interest rate of % for ears compounded: (a) quarterl dail (c) continuousl ) A computer was recentl purchased for $., including ta. If the ta rate is.%, find the price of the computer before taes. ) The smallest angle in a triangle has a measure which is onehalf the measure of the largest angle. The measure of the medium sized angle is less than the measure of the largest angle. Find the measures of the three angles in the triangle. ) The perimeter of a rectangle is meters. The length is meters less than twice the width. Find the dimensions of the rectangle. ) A cand shop manager mies M&M s worth $. per pound with trail mi worth $. per pound to make pounds of part mi worth $.8 per pound. How man pounds of each she should use? ) A tree stands straight when it is supported b a foot rope which is tied to the tree and staked to the ground feet awa from the base of the tree. If the rope is tied to the tree trunk feet above the ground, how far is the stake from the tree? Hint: Pthagorean Theorem. 8) Find how man quarts of % butterfat milk and % butterfat milk should be mied to ield quarts of % butterfat milk.
4 Math Intermediate Algebra II Final Eam Review Page of 9 9) Two shirts and three pairs of shorts cost $9.. Three shirts and one pair of shorts cost $.8. Find the price of each. ) An airplane takes hours to travel a distance of miles with the wind. The return trip takes hours against the wind. Find the speed of the airplane in still air and the speed of the wind. ) A compan determines that its costs for manufacturing cell phones can be modeled b the cost function: C ( ) = + (in dollars). The compan s revenues for selling cell phones can be modeled b the revenue function: R( ) =. Find the number of cell phones that must be sold in order for the compan to break even. What is the compan s profits at the point when sales are cell phones? ) Suppose that a computer store just announced a sale. The sale is for an 8% discount in the price for a particular computer model. If the discounted price is $, find the original price of the computer. ) Two trains start at the same point going in the same direction on parallel tracks. The faster train travels at mph and the slower train travels at mph. In how man hours will the distance between them be miles? ) Three numbers have a sum of. The middle number is more than the smallest number. The sum of the two smaller numbers is more than the largest number. Find the three numbers. ) In a certain triangle, the sum of the measures of the smallest and largest angles is º more than the measure of the medium angle. The medium angle measures º more than the smallest angle. Find the three angles. ) Lee has some $, $, and $bills. He has a total of bills, worth a total of $9. The number of $bills is less than the number of $bills. Find the number of each tpe of bill. ) Peg walks miles in the same time it takes Ma to walk miles. If Peg walks mile per hour less than twice Ma s rate, what is the rate at which Peg walks? 8) An eperience roofer can roof a house in hours. A beginner roofer needs 9 hours to complete the same job. Find how long it takes for the two to complete the job working together. 9) Two cars start at the same point and travel in opposite directions. One car travels at mph and the other at mph. How far has each traveled when the are miles apart? ) Alan can clean the apartment in hours. When his friend Steve helps, together the can clean the apartment in hours. Find how long it takes Steve to clean the apartment alone. ) The sum of a positive number and twice its square is. Find the number. ) Find the domain of f ( ) = Write our answer using interval notation. ) Find the domain and range for the function shown in the graph below. Write our answers using interval notation. ) Completel factor (over the integers). If it s not possible, write prime. (a) (c) (d) p p + p (e) ( )( + ) ( )( + ) (f) q p+ wp wq (g) (h) + 8 (i) 9+
5 Math Intermediate Algebra II Final Eam Review Page of 9 ) Solve the following polnomial and polnomiallike equations: (a) (+ ) = 8 k = k (c) = (d) ( + )( ) = (e) + ( + ) = (f) + = (g) = ) Solve. (a) = 8= (c) = (d) 9= (e) = (f) 8 = ) A pebble is hurled upward from the top of a skscraper which is 88 feet tall, with an initial velocit of 9 feet per second. The height of the pebble in feet, h(t), after t seconds is given b the polnomial function: h ( t) = t + 9t+ 88. How long does it take for the pebble to hit the ground? 8) Help Mr. Elmer Fudd and his ACME compan break even! Elmer finds that his compan s revenues from the sale of their Wile Coote traps is given b the function: R( ) = + (in thousands of dollars, where is the number of traps sold). He also finds that the costs of producing these traps is given b the function: C ( ) = + (also in thousands of dollars). How man traps does Elmer s ACME compan need to sell in order to break even? Then, calculate the revenue being made at that breakeven point. 9) Divide. + + (a) + 8 ) One of the following problems is an equation to be solved and the other problem is an epression to be simplified. Solve the one that is an equation and simplif the one that is an epression. For the one that is an equation, be sure to check for etraneous solutions. (a) = 8 ) Solve: = + + ) Suppose varies directl as. Suppose that is when is. Find when =. ) Suppose w varies inversel as q. Suppose that w =. when q =. Find w when q =.. ) Suppose B is proportional to the square of C and inversel proportional to D. If B = when C = and D =, find B when C = and D =. ) Solve. (a) + = ++ = (c) + = (d) + = 8 (e) + = ) The hpotenuse of a right triangle is 9cm long, and one leg is cm longer than the other leg. What is the length of both of the legs of the triangle? Hint: Pthagorean Theorem. ) The area of a right triangle is square meters. One leg is. meters longer than the other. What are the lengths of the legs? Round to the nearest hundredth of a meter, if necessar. 8) The width of a rectangular carpet is 9 feet less than its length. If the area of the carpet is square feet, find the width. 9) George and Cind together can stock the shelves of the local market in hours.
6 Math Intermediate Algebra II Final Eam Review Page of 9 When she works alone, it takes Cind one hour longer to stock than it takes George working alone. How long does it take for George to stock the shelves when he works alone? Round to the nearest tenth of an hour, if necessar. ) Two cars left an intersection at the same time, one heading north, the other heading west. Later the were eactl 9 miles apart. The car heading west had gone 8 miles less than twice as far as the car headed north. How far had each car traveled? Hint: Pthagorean Theorem. ) Solve. Write answers using interval notation. (a) + 9< ( )(+ ) + (c) (d) > + 8 ) Determine whether the equation (or function) is linear or not. Graph it. Give the domain and range using interval notation. (a) = + = (c) = (d) h( ) = (e) g ( ) = + (f) h ( ) = + ) For each quadratic function, find the verte, equation of the ais of smmetr,  and intercepts (if the eist), maimum (or minimum) value and which (ma or min), and then graph it. (a) f ( ) = + f ( ) = + ) Write the equation of a parabola having the same shape (and direction) as (a) f ( ) =, but with verte (, ) g ( ) = +, but with verte (, 9) (c) h ( ) = ( ) +, but with verte (, ) ) Write the equation for a parabola that has the same shape as: f ( ) = but opens in the opposite direction and has a verte of: (, ). ) Find the inverse of: f ( ) = + ) For the following functions, determine if the function is onetoone b graphing it and performing the horizontal line test. If the function is onetoone, find its inverse, and then graph its inverse on the same graph. If it is not invertible, sa so. (a) f ( )= f ( )= (c) f ( )= (d) f ( )= + (e) f ( ) = (f) f ( )= (g) f ( ) = (h) f ( ) = log 8) Solve. (a) = = 9) Draw the graph of the inverse of the function having the given graph. ) Evaluate these logarithms. If undefined, sa so. (a) log 8= log = (c) log = (d) log ( ) = (e) log = (f) log 8 = (g) log =
7 Math Intermediate Algebra II Final Eam Review Page of 9 (h) ln e = ) Solve. If there are no solutions, so state. (a) log = log = (c) log = (d) log ( ) = (e) log = (f) log = (g) log = (h) log = (i) ln e = (j) ln = e + ) For f ( ) = and g( ) =, find the following: (a) ( f o g)( ) ( go f)( ) (c) ( f o g)( ) (d) ( go f)( ) ) Use properties of logs to evaluate log ) Use properties of logs to rewrite the epression as a single logarithm: (a) log + log log + log log ( ) + (c) ln lnw+ lnz ) Write the epression given as the sum or difference of multiples of logarithms: ( + ) (a) log z ln ab ) If log b =.8 and logb =., find the value of log b. ) Approimate log 8 to four decimal places using our graphing calculator. 8) Solve. (a) log + log = log( + ) log( ) = log (c) ln( + ) + ln( ) = ln (d) log ( ) + log = 9) Solve: =. (a) Find the eact solution. Approimate the solution to four decimal places. 8) Solve: e =. (a) Find the eact solution. Approimate the solution to four decimal places. 8) Find the distance between the points (a) (, ) and (, ). (, ) and (, 8) (c) (,) and (,) 8) Find the midpoint of the line segment between: (a) (, ) and (, 8) (,) and (,) (c) (,9) and (,) 8) Find the verte of the parabola = + and indicate whether it opens upward, downward, to the left, or to the right. 8) Identif whether each equation, when graphed, will be a parabola, circle, ellipse, or hperbola. (a) + = 9 = 8) Graph each equation. (a) ( ) ( ) + + = = (c) + ( + ) = (d) = 8) Graph each inequalit. (a) + < ( + ) + ( ) 9 8) Graph the solution for the following sstems. 9 + < (a) + +
8 Math Intermediate Algebra II Final Eam Review Page 8 of 9 88) Write the equation of the circle with the given center and radius: (a) (, ); (, ); 89) Solve. Write solution as an ordered pair. = (a) + = = = = (c) = 9) Find the first three terms of the n sequence with the general term: a n = n + 9) Find the general term for the sequence:,,,,, 9) Use summation (also called sigma) notation to write the sum: (a) ) Evaluate the sum: i= i+ 9) Evaluate the piecewise function + f ( ) = > (a) f () f () (c) f () 9) Evaluate the piecewise function < f ( ) = (a) f ( ) f ( ) (c) f () 9) Graph the piecewise functionf ( ) = > 9) Graph the piecewise functionf ( ) =
9 Math Intermediate Algebra II Final Eam Review Page 9 of 9 ANSWERS ) ) 9 ) ) ) ) 9 + = ; Slope: ;  intercept: (,); intercept: (,) ) 8) 9) ) (a) = or = r r bc a = c (c) b= c c( g+ h) (d) = d+ f h g (a) no no (c) es (d) no (a) undefined (c) (a) = ; slope is undefined,  intercept is (,); intercept: none. ) ) (a) slope is undefined slope m = (zero slope) (a) perpendicular parallel (c) neither (d) perpendicular (e) neither (f) parallel ) ) (a) f ( ) = = (c) f ( ) = (d) = (e) f ( ) = (f) f ( ) = 9 (g) f ( ) = (h) f ( ) = + 8
10 Math Intermediate Algebra II Final Eam Review Page of 9 ) (a) no no (c) no (d) es (e) no (f) es ) For all parts, there are two methods. Method : Intersection of graphs: The solution is the coordinate of the point of intersection, found b (TI8, 8+,8+): nd TRACE (CALC), option (intersection), st curve (enter), nd curve (enter), guess (enter).. Repeat, if necessar, for a second solution. Method : Xintercept : The solution is the  coordinate of the intercept, found b (TI8, 8+,8+): nd TRACE (CALC), option (zero); left bound (enter), right bound, (enter), guess, (enter). Repeat, if necessar, for a second solution. (a) Method : Intersection of graphs = = + Method : Xintercept = + B either method, the solution is.8 (c) Method : Intersection of graphs = π (Close the parentheses before subtracting pi.) = + Method : Xintercept = π + B either method, the solution is.98 Method : Intersection of graphs =. + = + Method : Xintercept =. + (or simplified) (d) (e) (f) (g) (h) B either method, the solution is. Method : Intersection of graphs = (Close the parentheses before subtracting.) = + Method : Xintercept = + (or simplified) B either method, the solution is.88 Method : Intersection of graphs = ln+ (Close the parentheses before adding.) = e Method : Xintercept = ln+ e B either method, the solutions are. 9 and. 9 Method : Intersection of graphs =.( ) =. + Method : Xintercept =.( ) +. B either method, the solutions are. 8 and. Method : Intersection of graphs = π = + (Close the parentheses before adding.) Method : Xintercept = π (or simplified) B either method, the solution is.8 Method : Intersection of graphs = log( ) + = + Method : Xintercept = log( + ) +
11 Math Intermediate Algebra II Final Eam Review Page of 9 ) (i) B either method, the solution is.89 Method : Intersection of graphs =. = Method : Xintercept =. B either method, the solutions are = and. (a) solution: (, ) pressing ENTER to complete the RREF calculation. 8 (a) RREF 8,, RREF no solution 8) These lines are parallel to each other because the slopes are the same but the have different intercepts. The solution to the sstem would be: No Solutions since the lines never intersect. 9) (a) There are infinitel man solutions, so we write the answer in set, = OR: notation: ( ) { } (, ) = no solutions ) On the TI8+, nd is MATRIX. Under MATRIX, use the EDIT menu to input the dimensions of the matri and the coefficients, with coefficients in the same order for each equation and zeros for an missing terms. Eit MATRIX EDIT, then enter MATRIX again. Under MATH, select RREF (for RowReducedEchelonForm), then MATRIXNAMES and the name of the matri used. If fractional answers are requested, press MATH and >Frac before. (c). (or with 8 fractions: ) RREF 8 8,, (d) 9 RREF,, z z = {( ) }
12 Math Intermediate Algebra II Final Eam Review Page of 9 (e). (or with. fractions ) RREF 9 9,, (f) RREF,, ) , , and . ) (a) $.9 8 Use compound r interest formula: A = P =, n n= $.8 Use compound interest formula, n= (c) $. Use continuousl compounded interest formula: rt A= Pe ) $9. ) largest angle 8, smallest angle, middlesized angle ) meters b 8 meters ) pounds of trail mi and pounds of M&Ms nt ) feet (It didn t sa to round!) 8) quarts of % and quarts of % 9) $.9 per shirt and $9.9 per pair of shorts ) plane mph and wind 9 mph ) cell phones to break even. Profits = Revenue Costs. So profits = $ when cell phones are made and sold. ) Original price $ ). hours (or hours, minutes) ) smallest number 9, middlesized number, largest number. ) Smallest angle, middlesized angle, largest angle ) seven $ bills, twelve $ bills, and thirttwo $ bills. ) Peg s speed is mph. (Ma s speed was mph, but that wasn t the question.) 8). hours (or hours, minutes) 9) 8 miles traveled b car going mph and 9 miles traveled b car going mph. ) Steve takes hours if working alone. ) =. (Reject etraneous) ) (, ) (,) (, ) ) domain: (,], range: [, ) ) (a) ( + )( ) + ) ( )( ) (c) ( + )( )( + )( + ) (d) ( p + )( p ) (e) ( )( + ) (f) ( w)( q p) or ( p q)( w ) (g) prime 8 (h) ( )( ) (i) ( )( ) (a) =, =, (c) =,, (d) =, (e) =,
13 Math Intermediate Algebra II Final Eam Review Page of 9 ) (f) = 8, (g) =, (a) =,, = ± i, (c) =, (d) =,, (e) =, ± i 8 8 (f) =, ) seconds 8) traps to break even, $8, revenue 9) 8 (a) ) + (a) epression, ( ) ( ) equation, = (Reject = , etraneous) (d) = ± i, ± (e) = ± + ) shorter leg is centimeters (Did not sa to round.) Longer leg is + centimeters ) shorter leg is approimatel.8 meters, longer leg is approimatel. meters. 8) feet 9) 9. hours or 9 hours and minutes ) northbound car went mph, westbound car went miles. ) (a),, [, ) (c) (, ) [,] (d) (, 8) (, ) ) (a) linear. Domain: (, ), Range:, ( ) ) no solution ) = 99 ) w = =. 8 ) B= ) ± (a) = or (c) = ± =, i = ± not linear. Domain: ( ) Range: (,],,
14 Math Intermediate Algebra II Final Eam Review Page of 9 (c) not linear. Domain: ( ) Range: [, ),, (d) not linear. Domain: (,], Range: [, ) ) 9 (a) Verte,, Ais of smmetr: =,  intercepts,,,, int (,), 9 Minimum value (e) not linear. Domain: [, ), Range: [, ) Verte,, Ais of smmetr: ± =, intercepts,, int (,), Maimum value (f) not linear. Domain: (,], Range: [, )
15 Math Intermediate Algebra II Final Eam Review Page of 9 (c) onetoone, f ( ) =, ) (a) f ( ) = ( ) g ( ) = ( + ) + 9 (c) g ( ) = ( + ) ) f ( ) = ( ) ) f ( ) = or f ( ) = ) A relation is onetoone if its graph passes the horizontal line test. (a) onetoone, f ( ) = + (d) not onetoone; (e) onetoone; f ( ) = = + not onetoone (f) not onetoone;
16 Math Intermediate Algebra II Final Eam Review Page of 9 8) 9) (g) onetoone; log( ) f ( ) = log / ( ) = log (h) onetoone; (a) = = f ( ) = (f) undefined (g) (h). ) (a) = = (c) = (d) undefined epression, has no solution. (e) undefined epression, has no solution. (f) = (g) = (h) = (i) = (j) = ) + (a) ( f o g)( ) = + ( go f )( ) = + (c) ( f o g)() = = + (d) ( gof )() = = = undefined ) ) (a) (c) (d) undefined (e)  ) (a) log (c) ) (a) log + z ln w log( log ln+ lna+ lnb + ) log z
17 Math Intermediate Algebra II Final Eam Review Page of 9 ) log b = logb = logb logb =..8=. log ). log 8 8) (a) = or = 8 8 = (c) = ( = is etraneous) (d) = ( = is etraneous) 9) log (a) = + log. 9 (a) 8) +ln (a) =. 9 8) (a) (c) (c) 8) (a),,. (c) ( ) 8) Verte ( 8, ) 8) (a) ellipse hperbola 8), parabola opens right
18 Math Intermediate Algebra II Final Eam Review Page 8 of 9 8) (a) 8) (d) 88) (a) ( ) + ( + ) = ( + ) + ( ) = (a) 89) (a) (, ) and (, ),,, ( ) ( ) (, ),(, ) (c) (, ) 9) a =, a =, a = 9 9) = n + 9) (a) a n n n n = +
19 Math Intermediate Algebra II Final Eam Review Page 9 of 9 9) 9) ( + n ) (a) n= 8 9) 9 (c) 9) (a)   (c) 9)
9 (0, 3) and solve equations to earn full credit.
Math 0 Intermediate Algebra II Final Eam Review Page of Instructions: (6, ) Use our own paper for the review questions. For the final eam, show all work on the eam. (6, ) This is an algebra class do not
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