College of Charleston Department of Mathematics Math 0: College Algebra Final Eam Review Problems Learning Goals (AL-) Arithmetic of Real and Comple Numbers: I can classif numbers as natural, integer, whole, irrational, rational, and real. I can evaluate and simplif arithmetic epressions without the use of a calculator. I can add, subtract, multipl, rationalize, and divide comple numbers and simplif them into standard form a + bi. (AL-) Polnomials: I can characterize polnomials as monomials, binomials, or trinomials when applicable. I can write polnomials in standard form, identif their degree, and describe the coefficients. I can perform operations on polnomials, including addition, subtraction, multiplication, and (long) division. I can state and appl the formulas for the squares of binomials and the cubes of binomials. I can state and appl the formulas for the difference of two cubes and the sum of two cubes. (AL-) Factoring Polnomials: I can factor polnomials using a variet of techniques. I can complete the square of a polnomial epression. I can factor polnomials in the form A + B or A B. (AL-) Rational Epressions: I can perform algebraic operations on rational epressions, including reducing to lowest terms, addition, subtraction, multiplication, and division. I can simplif comple rational epressions. (AL-) Radicals and Eponents: I can use the laws of eponents, including those involving roots. I can simplif epressions involving nth roots. I can simplif epressions involving rational eponents. I can rationalize the denominator or numerator of a given rational epression. (EQ-) Linear Equations and Linear Models I can solve linear equations in one variable. I can model situations using linear equations and use linear models to form conclusions. (EQ-) Quadratic Equations and Quadratic Models I can find all real and comple solutions to quadratic equations using a variet of methods, including factoring, completing the square, and using the quadratic formula. I can model situations using quadratic equations and use the model to form conclusions. I can describe the discriminant of a quadratic equation and use it to determine its number of solutions over the real numbers. (EQ-) Radical Equations I can solve radical equations and check for etraneous solutions. (EQ-) Factorable Equations and Equations in Quadratic Form I can use factoring to solve quadratic-like equations. I can solve equations in quadratic form. (EQ-) Inequalities, Intervals, and Graphs I can convert between interval notation, inequalit notation, and graphical notation for subsets of real numbers. I can use properties of inequalities to simplif, to solve inequalities, and to combine Please send an corrections to owensks@cofc.edu
inequalities. I can graph solution sets on the real line. I can model real-world problems using inequalities and form conclusions using the model. (EQ-6) Absolute Value Equations and Inequalities I can solve equations involving absolute value. I can solve inequalities involving absolute value. (GR-) Distances, midpoints, and geometric shapes I know the distance formula and can appl it when appropriate. I know the midpoint formula and can appl it when appropriate. I know and can appl geometr formulas related to squares, triangles, circles, boes, spheres, and right circular clinders. I can use the Pthagorean Theorem and its converse. (GR-) Graphs of equations, intercepts and smmetr I can test an equation for smmetr with respect to the -ais, the -ais, and the origin. I can identif smmetr from a graph or complete a graph so that it has a given tpe of smmetr. I can quickl and accuratel graph each of the following basic equations, describing an intercepts or smmetr: =, =, =, =, =, =, =. I can graph functions of the form f() = k, f() =, f() =, f() =, f() =, f() =, and f() =. I can graph piecewise-defined functions and I can determine the equation given the graph of a piecewise-defined function. (GR-) Graphs of lines and sstems of lines I can find the equation of a line given its slope and a point. I can find the equation of a line given its slope and its -intercept. I can find the equation of a line given two points on the line. I can find equations of parallel lines and perpendicular lines. I can write the equation of a line in slope-intercept form, standard form, and point-slope form. I can graph a line. I can identif the slope and -intercept of a line from its equation or graph. I can solve sstems of linear equations in two variables b substitution and I can solve sstems of linear equations in two variables b elimination. I can identif inconsistent sstems of equations in two variables and I can epress the solution of a sstem of dependent equations containing two variables. (GR-) Circles I can convert between standard form and epanded form for the equation of a circle. Given its properties, I can graph a circle and find its equation. I can find -intercepts and -intercepts found on the graph of a circle. (FN-) Functions, domains, and difference quotients I can determine whether a relation represents a function and find the value of a function. I can find and simplif the difference quotient of a function. I can find the domain of a function defined b an equation. I can combine functions using addition, subtraction, multiplication, and division. I can identif the graph of a function and I can use a function s graph to obtain information about the function. (FN-) Linear functions and models I can graph a linear function. I can determine whether a linear function is increasing, decreasing or constant. I can determine the average rate of change of a linear function and use it to identif linear functions. I can build linear models from verbal descriptions and use the models to establish conclusions. (FN-) Quadratic functions and models Given a quadratic equation, I can identif the verte and the ais of smmetr on its graph. I can graph a quadratic function using its equation. I can graph a quadratic function using its verte
and one other point. I can find and identif -intercepts on the graph of a quadratic function. I can use an equation or a graph to find the minimum or maimum value of a quadratic function. (PY-) Properties of Polnomial functions I can identif polnomial functions and their degree. I can graph polnomial functions using transformations. I can analze the graph of polnomial functions. I can identif the real zeros of polnomial functions and their multiplicit. (PY-) Properties of Rational functions I can find the domain of a rational function. I can find the vertical asmptotes of a rational function. I can find the horizontal asmptotes of a rational function. I can find the oblique asmptotes of a rational function. I can find an intercepts appearing on the graph of a rational function. (PY-) Polnomial and Rational inequalities I can produce a completel correct Sign Chart without an errors. I can solve polnomial inequalities (including quadratic). I can solve rational inequalities. (EX-) Composite functions I can combine functions using composition. I can find the domain of a composite function. (EX-) Inverse functions I can determine whether a function is one-to-one. I can obtain the graph of the inverse function from the graph of the function. I can determine the inverse of a function defined b a map or a set of ordered pairs. I can find the inverse of a function defined b an equation. (EX-) Eponential functions I can evaluate eponential functions. I can define the number e and approimate it to five decimal places. I can graph eponential functions. I can solve basic eponential equations. (EX-) Logarithmic functions I can change an eponential equation to a logarithmic equation. I can change a logarithmic equation to an eponential equation. I can evaluate logarithmic epressions without using a calculator. I can determine the domain of a logarithmic function. I can graph logarithmic functions. I can solve basic logarithmic equations. Review Problems. Write each comple number in standard form a + bi. ( i) + (7 i) ( i) ( + 7i) (c) ( + i) ( + 7i) + i (e) + i i (f) ± ( + i). Find all solutions (real and comple) + = 0 = 8 (c) + =. Find all real solutions for and simplif our answers:
= 6 + + = + + 6 (c) + = 0 8 9 = 0 (e) = + + 7 (f) + 6 = 0 ( ) / = (h) 6 + 0 = 0 (i) 7 ( + ) = + (j) + 7 = 9. Simplif each of the following. Write an polnomials in standard form. + ( + ) ( ) + 7 ( )( + ) (c) ( ) (e) + 8 + (f) 6 6. Factor completel: 9 + (c) a 7b + 9 8 (e) a ( ) 9( ) (f) 7 + 8 6. Let f() = and g() =. Find and simplif: (e) (i) (f g) () (g g) () (f g) () (g f) () (c) g ( + h) (f) f() 7. Consider the quadratic function f() = 8 + 0. f( ) (h) f () (f g) () Write in form = a( h) + k. Find the verte. (c) Find the -intercept. Find an -intercept(s).
8. Solve each of the following. Write our answer in interval notation. 7 < 0 8 (c) + > + (e) < (f) + 9 9. Find an equation for each line. Write in slope-intercept form where possible. Through the points (, ) and (,. Through the points (, ) and (, ) (c) Parallel to the line + 6 = 0 and containing the point (, ). Parallel to the line = and containing the point (, ). (e) Perpendicular to the line + 7 = 0 and containing the point (, ). 0. Find the center and radius of each of the following circles: + + 6 + 8 = 0 + 6 + 6 = 0. Find the -intercept of the line + = 0. Find the slope of the line + = 0.. Find the -intercept, the zeros and the multiplicit of each, and sketch a graph of f() = +.. Find and simplif each of the following: a ab when a = and b = a b / when a = and b = 7. Find the domain of each function and write using interval notation: g() = + 6. Simplif completel, reducing all fractions. h() = 6 (c) f() = 9 8 / ( a / b ) ab (c) ( ) / 6 9 7 (e) (f) 8 a + a a 8 6. Find the equation of a polnomial function = f() satisfing the following: Zeros are: 0 of multiplicit, of multiplicit, of multiplicit Leading coefficient: Write our answer in factored form.
7. Consider the points A(, ) and B(, ). Find the midpoint of the line segment joining A and B. Find the distance between A and B and simplif. 8. Solve for : = 8 ( ) = 8 (c) e = 7 log (6 ) = 9. A quadratic function = f() has verte at (, ) and passes through (0, ). Find the equation of the function and write in epanded form. Find the domain of the function in interval notation. (c) Find the range of the function in interval notation. Find the -intercepts. 0. Compute and simplif: ( log log ( 0 ) (c) log 7 7 log 8 (e) log ) 9. Consider the function f() =. Find the -intercept. Find an -intercepts. (c) Graph = f().. The graph of = f() is shown below. Draw the graph of = f ().. Solve the sstem of equations { + = + =. Solve for a in the equation = a b k + a. Find the domain of f() = 6. 6
6. Find the equations of the polnomials with lowest possible degree with the graphs shown below. (, 0) (0, 0) (0, ) (, ) 7. Use the graph of the function f shown below to answer parts -(e). (, ) (, ) Find f(). For what value(s) of is f() =? (c) What is the domain of f? What is the range of f? (e) How man times does the graph of = cross the graph of f? 8. For the rational functions given, find an -intercepts, an -intercept, equations of an vertical or horizontal asmptotes, and state the domain using interval notation. f() = + f() = + 9 9. Rationalize the denominator and simplif our answer: 6 0 (c) f() = 0. Find the quotient and remainder when + 7 is divided b +. 7
. For each function f(), find f( + h) f() h and simplif our answer. f() = f() = + (c) f() = +. Which of the following is/are functions? + = 9 (c). Consider the function f() = log ( ). Answers Find an -intercept. Find an -intercept(s). (c) Graph = f().. 8 7i, 0i, (c) + i, 0 i, (e). ±i ± i, (c) ± i + i, (f) + 6 i, + i. /, (c) ±, (e) 8 (f), ± 9/, (h),, (i) 6/7, (j), /. 9/0,, (c) +, + + +, (e) +, (f), ( ). ( )(+), ( +)( )(+), (c) (a b)(a +ab+9b ), (+)( )(+), (e) (a )(a + )( ), (f) ( + )(9 + ), ( )( + ) 6. 7, 8+, (c) + h + h, f () = +, (e) 6 + 6, (f) +, (h) +, (i) + 7. f() = ( + ) + 8, (, 8), (c) (0, 0), (, 0), (, 0) 8. (, ], (, 7 0], (c) (, ) (, ), (, ], (e) (, 9), (f) (, ] [, ) 9. = +, =, (c) = +, =, (e) = 0. Center (, 0), radius r =. Center (, ), radius 7,. (0, /) m = 8
. -int (0, 0), zero = 0 multiplit, zero = multiplicit. 9/, 9/. (, ), (, ], (c) (, ) (, ) (, ). /6, 6b a 7, (c) /, +, (e) 6, (f) a+, 6. f() = ( + ) ( ) 7. midpoint (, ), distance 0 8. 9/,, (c) +ln(7), 7/6 9. f() = 6, (, ), (c) (, ], ( ±, 0 ) 0., 0, (c), 0, (e). (0, ), (, 0), (c) Graph.. = /, =. a = k+b k b or a =. (, ] [, ) 6. = 8 ( + )( )( ), = ( + ) 7. f() =, =, =, (c) <, <, (e) Twice 8. -int (/, 0), -int (0, ), VA =, HA =, domain (, ) (, ) 9
-int (, 0), -int (0, /9), VA = and =, HA = 0, domain (, ) (, ) (, ) (c) -int (, 0) and (, 0), -int (0, ), VA =, HA None, domain (, ) (, ) 9. 0, + 0. Quotient 6, remainder 6 +., 0 + h +, (c) (+h+)(+). not a function, function, (c) not a function f() = log ( ) (, 0). None, (, 0), (c) 0