1. 4 2y 1 2 = x = x 1 2 x + 1 = x x + 1 = x = 6. w = 2. 5 x

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2 .... VII x + x + = x x x 8 x x = x + a = a + x x = x + x x Solve the absolute value equations.. z = 8. x + 7 =. x =. x =. y = 7 + y VIII Solve the exponential equations.. 0 x = x+ = 00. x+ = 8. x (9 x ) = X Solve the radical equations.. y = 0. x + + = 8. x x + = 0. x x + = 0.. XI x + + = w + 7 = Solving Inequalities Solve the following inequalities. Express the answer graphically and using interval notation.. x +. (x + ) (x ). (x + ) > (x ) +. x 0. x +. x + + > 9 7. x + 8. x < 9. x x < 0 (. x x ) = 8 IX Solve the logarithmic equations.. log (x + ) = log ( x). log (x + ) = log (x). log (x + ) + log (x ) =. ln x + ln(x + ) = 0. ln x + ln(x + ) = ln. x = x+ 0. x + x.. x + x 0 (x + )(x ) x + 7 0

3 XII Lines and Regions. Given the equation x + y = 0, draw the graph and find the x and y intercepts and the slope.. Given the equation x =, draw the graph and find the x and y intercepts and the slope.. Given the equation y =, draw the graph and find the x and y intercepts and the slope.. Write, in slope-intercept form, the equation of the line that passes through the points (, ) and (, ). XIV Exponents and Radicals. 8x ( )... ( x / y / x / 0x y 9 ( a b 9a b 8 ) ). Write, in slope-intercept form, the equation of the line perpendicular to the line x y = and through the point (, ).. Graph the solution set of x y. 7. Graph the solution set of x + y < a a b x x + XIII Relations, Domain, and Range For each relation, state if it is a function, state the domain and the range, and sketch a graph.. y = x +. y = x. y = x x +. f(x) = x + +. f(x) = x x 9. x = y + 7. y = x + 8x XV Complex Numbers Perform the indicated operations and simplify ( i)( + i). ( i). i 7. i + i 8. y = x 9. y = x 0. h(x) = x x x

4 XVI Exponential Functions and Logarithms. Graph f(x) = x +. Graph g(x) = x. Express 8 = in logarithmic form.. Express log = in exponential form.. Solve log x =. Solve log x 9 = 7. Graph h(x) = log x 8. Use the properties of logarithms to completely expand log y 9. How long will it take $80 to be worth $000 if it is invested at % interest, compounded quarterly? XVII Systems of Equations and Matrices { x + y = 7. Solve the system x y = x + y + z =. Solve the system x + y + z = x + y + z = 0 [ ] [ ]. Simplify + 0. Multiply [ ]. Find the determinant of [ ]. Find the inverse of XVIII Applications. Sam made $0 more than twice what Pete earned in one month. If together they earned $70, how much did each earn that month?. A woman burns three times as many calories running as she does when walking the same distance. If she runs miles and walks miles to burn a total of 770 calories, how many calories does she burn while running mile?. A vertical pole is standing in a small lake. If one-sixth of the length of the pole is in the sand at the bottom of the lake, feet of the pole is in the water, and two-thirds the total length is in the air above the water, what is the length of the pole? XIX Conic Sections Graph the following. Find the center, foci, and, if they exist, asymptotes.. (x ) + y =.. (x + ) + (x + ) (y ) 9 (y ) 9. (x ) + y = = = Identify the conic section and write in standard form.. x x + y = 0. 9x + 8x + y y = 7. 9x + 8x y y = 99. x + y x = 0

5 XX Sequences and Series. Write out the first four terms of the sequence whose general term is a n = n.. Write out the first four terms of the sequence whose general term is a n = n.. Write out the first four terms of the sequence whose general term is a n = n +.. Find the general term for the following sequence:,, 8,,, 7,.... Find the general term for the following sequence:,,,,,.... Find the sum: (k ) k=0 7. Expand the sum: XXI Functions k=0 [( ] )x k y k k Let f(x) = x + 9 and g(x) = x. Find the following.. f( ) + g(). f() g(). f( ) g( ). f() g(). (g f)( ). f (g(x)) XXII Fundamental Counting Rule, Factorials, Permutations, and Combinations 8!. Evaluate!(8 )!. A particular new car model is available with five choices of color, three choices of transmission, four types of interior, and two types of engine. How many difference variations of this model car are possible?. In a horse race, how many different finishes among the first three places are possible for a ten-horse race?. How many ways can a three-person subcommittee be selected from a committee of seven people?. How many ways can a president, vice-president, and secretary be selected from a committee of seven people? XXIII Trigonometry. Graph f(x) = sin(x) through one period.. Graph g(x) = cos(x) through one period.. A man whose eye level is feet above the ground stands 0 feet from a building. The angle of elevation from eye level to the top of the building is 7. How tall is the building?. A man standing at the top of a meter lighthouse observes two boats. The angle of depression to the farther boat is 8. The angle of depression to the closer boat is 0. What is the distance between the two boats? 7. f () 8. f ( f () ) st boat nd boat lighthouse

6 Answers I II Factor the following polynomials.. a b a b 0a b = a b ( ab b ). Start by factoring out the common term 7x y + x y 0x y 0x y = x y [ 7xy + y 0xy 0 ] Factor [ this result using grouping. ] x y 7xy + y 0xy 0 = x }{{}}{{} y [7y(xy + ) 0(xy + )] = x y [(7y 0)(xy + )]. x y x y + 8xy 8 = ( x y + ) ( xy ). x 7xy + y = (x y)(x y). y + y = ( y ) ( y + ). 7x + y = 7 ( x + 8y ) = 7(x + y) ( x xy + y ) 7. 8r s = ( 9r s ) ( 9r + s ) = (r s)(r + s) ( 9r + s ) 8. Consider the polynomial u + u + = (u + ). If we let u = x + y, then (x + y) + (x + y) + = (x + y + ). Expand the following polynomials.. (x + )(x )(x ) = ( x ) (x ) = x x x +. (x + y) = x + xy + 9y. ( x + ) ( x ) = 8x + 8x 8x 8 = x. ( x x + ) = x x + 0x x + 9. (x + ) = x + x + 0x + 0x + x +. (x ) = x x + x 0x + x x + III Simplifying Rational Expressions Simplify the following. Assume all variable are greater than zero = = = + 7 = 8. 8 x = 9 x = = 9

7 IV V VI. x = Solve the linear equations.. x = =. y =. x = Solve the polynomial equations.. y = 8, y =. x = 0, x =, x =. x =, x = ± i. x = 0, x =. t = ± i. x =, x = ± i 7. x =, x = 8. y = ± 0. y = Solve the rational equations.. Working the problem, we get x =. However, x = causes the denominators to be zero in the original equation. Hence, there is no solution.. x =. x =. a =, a =. x =, x = VII Solve the absolute value equations.. z =, z = 7. x = 0, x = 0. No solution. An absolute value can not equal a negative number.. x =, x =. y = VIII IX. x =. x =. x =. x = Solve the exponential equations.. x =, x = Solve the logarithmic equations. Rules of logarithms: log b (x) = y x = b y log b (x y) = log b (x) + log b (y) log b (x y) = log b (x) log b (y) log b (x y ) = y log b (x) The argument and base of a logarithm must be positive.. x =. x =. x = ; Note that x = is not a solution since the argument of a logarithm must be positive.. x =. x = ln. x = ln ln 7

8 X Solve the radical equations.. y = 8. x < or x > or Interval: (, ) (, ). x = XI. x =. No solution. Substituting x = into the original equation fails.. x =. w =, w = Solving Inequalities Solve the following inequalities. Express the answer graphically and using interval notation.. x 0 or Interval: (, 0] 0 7. x 0 or x 0 or Interval: (, 0] [0, ) < x < 0 or Interval: (, 0) 9. (, ) 0. x 7 or Interval: (, 7]. x > or Interval: (, ) 7 0. (, ] [ ),. x or Interval: [, ]. [, ). 7 x [ or Interval: 7, ]. (, 7 ) [, ] 7 7 8

9 XII Lines and Regions. x y. x-intercept: (, 0) y-intercept: (0, ) slope: x + y < x-intercept: (, 0) y-intercept: none slope: none x-intercept: none y-intercept: (0, ) slope: XIII Relations, Domain, and Range For each relation, state if it is a function, state the domain and the range, and sketch a graph.. y = x + Domain: [, ) Range: [0, ) y = x +. y = x + 7 9

10 . y = x Domain: [0, ) Range: [, ) -. f(x) = x x 9 Domain: All real numbers except ± Range: (, ) y = x x + Domain: All real numbers except Range: All real numbers except 0. x = y + Domain: [, ) Range: (, ) f(x) = x + + Domain: (, ) Range: (, ] y = x + 8x Domain: (, ) Range: [, )

11 8. y = x Domain: (, 0] Range: [0, ) y = x Domain: (, ) Range: (0, ) 9 x 0. h(x) = x x Domain: All real numbers except and Range: (, 0] ( ), XIV Exponents and Radicals. 8x = x. 7 8 = 9. ( ) =.. ( ) x / y / = x7 x / 0x y 9 = x x y y ( a b ) = a b 9a b 8 7a a b = ab b = + x = x ( x ) x + x 9

12 XV Complex Numbers Perform the indicated operations and simplify.. 9 = 8i. 9 =. = 9. ( i)( + i) =. ( i) = 7 i. i = i 7. XVI i + i = i Exponential Functions and Logarithms. f(x) = x +. g(x) = x ( ). log 8 = =.. 7. h(x) = log x log log y ( 9. Use A = P + r ) kt ( k 000 = t.7 years. ) t XVII Systems of Equations and Matrices ( ).,, k +. 8 k + for k natural numbers. k [ ]

13 XVIII Applications. Let x = money Pete earns, x + 0 = money Sam earns (x + 0) + x = 70 x = 0 so Pete earns $0 and Sam earns $0.. Let x = calories burned walking, x = calories burned running (x) + x = 770 x = 70 so 0 calories burned running.. Let x = length of the pole. x + + x = x x = 0 feet. XIX Conic Sections Graph the following. Find the center, foci, and asymptotes if possible.. (x ) + y = Circle: center (, 0), radius (x + ) (y ) + = 9 Ellipse: center (, ), foci ( ± 7, ) (x + ) (y ). = 9 Hyperbola: center (, ), foci (, ), (, ), asymptotes y = x, y = x (x ) + y = - Parabola: vertex (, ), focus directrix y = Identify the conic section and write in standard form.. Circle: (x ) + y =. Ellipse: (x + ) + (y ) 9 (, ), = Hyperbola: (x + ) (y ) 9. Parabola: y = (x ) + =

14 XX Sequences and Series.,, 7, 0. 0,, 8, XXIII Trigonometry. Graph f(x) = sin(x) through one period..,, 9, 7. a n = n. a n = ( ) n = n - Π Π Π Π = 7. y + xy + x y + x y + x XXI Functions Let f(x) = x + 9 and g(x) = x. Find the following.. f( ) + g() = + =. Graph g(x) = cos(x) through one period. - Π Π Π. x = + 0 tan 7 9. ft.. Distance between the boats = tan tan 0.9 meters. Π. f() g() = 9 0 = 9. f( ) g( ) = 7 = 8. f() g() = 9 9 = 9 9. (g f)( ) = g (f( )) = g() = 9. f (g(x)) = f ( x ) = ( x ) + 9 = x + 7. f (x) = x 9 ; f () = 9 = 7 8. f ( f () ) = XXII. Fundamental Counting Rule, Factorials, Permutations, and Combinations