Final Exam Review Sheet
Note: Below is a complete list of the topics we’ve covered and some recommended exercises. If
you want more exercises on a given topic, choose others in the same section or look at the reviews
at the end of each chapter. As I’ve been emphasizing in class, the best way to study is to try each
exercise without looking at the book or your notes for at least a few minutes–if you get stuck, try
to think about the problem a little longer. If you need to look something up at this point, do it,
then try to solve the problem again. Then try a similar problem. Do this until you can do the
problem without looking at anything. This is a good way to ensure that you’ll retain what you’ve
studied.
P.1 Real numbers and their properties: Properties of real numbers, absolute value, exponential expressions
Recommended exercises: 57,60,65,76,83,88,107
P.2 Integral exponents: Negative integral exponents (stuff like 2−3 ), rules of exponents, simplifying expressions with exponents
Recommended exercises: 12,17,24,28,35,40,48,49,50,56 (There was some trouble with this section
and P.3 on the first test–be sure to review them. Exponents come up a lot.)
P.3 Rational exponents and radicals: roots, evaluating expressions involving the exponents
1/n, rational exponents, rules for rational exponents and radicals.
Recommended exercises: 5,8,12,14,16,18,21,25,26,28,33,36,46,67,77
P.4 Polynomials: Definition of polynomials, adding and subtracting polynomials, multiplying
polynomials (FOIL), dividing polynomials
Recommended exercises: 20,22,34,36,38,44,52,78,85,88,97
P.5 Factoring polynomials: Factoring out the greatest common factor, factoring by grouping,
factoring by trial and error, factoring special products (difference of squares, etc.), factoring by
substitution
Recommended exercises: 14,16,22,24,28,32,36,55,56,
P.6 Rational expressions: Definition of a rational expression, reducing rational expressions,
multiplying and dividing rational expressions, adding and subtracting rational expressions, complex
fractions
Recommended exercises: 7,10,16,18,27,30,34,51,53,57,67,72,75
P.7 Complex numbers: Definition of complex numbers, adding, subtracting, and multiplying
complex numbers
Recommended exercises: 15,22,29,33,35,36,66
1.1 Equations in one variable: definition of a linear equation in one variables, solving such
equations, equations involving rational expressions, equations involving absolute value
Recommended exercises: 14,16,18,21,24,37,41,44,63,65,71,74,95
1.2 We skipped this section.
1.3 Equations and graphs in two variables The Cartesian coordinate system, the Pythagorean
theorem and the distance formula, the midpoint formula, the equation for a circle, graphing a circle,
completing the square, equation of a line in standard form, graphing lines, x and y intercepts
Recommended exercises: 19,22,35,42,50,52 (complete the square),67,70,80,82
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1.4 Linear equations in two variables: Slope of a line, point-slope form, slope-intercept form,
finding the equations of a line in any of these forms, finding the equation of a line going through
two given points, graphing a line given any of these forms, parallel and perpendicular lines
Recommended exercises: 10,20,22,24,35,44,48,54
1.5 We skipped this section
1.6 Quadratic equations: Definition of a quadratic equation, the zero factor property, solving
quadratic equations by factoring, solving quadratic equations using the square root property, solving
quadratic equations by completing the square, using the quadratic formula, significance of the
discriminant
Recommended exercises: 5,7,13,16,18,35,40,46,54,58,66
1.7 Linear and absolute value inequalities Interval notation, solving linear inequalities, solving
absolute value inequalities
Recommended exercises: 16,18,60,63,68
2.1 Functions: Definition of a function, the vertical line test, identifying the domain and range
of various functions
Recommended exercises: 19,22,35,48,51,53
2.2 Graphs of functions: Graphing functions by plotting points, graphing various common
functions and equations(y = x2 , x = y 2 , y = x3 , circles and semicircles), tell whether functions are
increasing, decreasing, or constant
Recommended exercises: 10,15,23,29,52
2.3 Families of functions, transformations, and symmetry: Use the graph of y = f (x) to
graph y = af (x − h) + k (horizontal and vertical translations, reflections, stretching and shrinking),
symmetry about the x and y-axes
Recommended exercises: 18,41,47,60,61,68
2.4 Operations with functions: Sums, differences, products, and quotients of fuctions, composition of functions, finding the domains of these
Recommended exercises: 25,37,55,59
2.5 Inverse functions: One-to-one functions, using the horizontal line test to determine whether
a function is one-to-one, definition of the inverse of a function, checking whether something is the
inverse of a given function, finding the inverse of a function
Recommended exercises: 13,49,65,70,83
3.1 Quadratic functions and inequalities: Rewriting quadratics in the form f (x) = a(x −
h)2 + k, graphing quadratics in this form, finding the vertex and x and y intercepts of a quadratic,
solving quadratic inequalities.
Recommended exercises: 14,45,49,67,75
3.2 Zeros of polynomial functions: The remainder and factor theorems, the rational zero
theorem, using the rational zero and factor theorems to completely factor a polynomial. Also,
review, polynomial long division from the prerequisites chapter.
Recommended exercises: 9,49, any of 55-78
3.3 The theory of equations: multiplicity of roots of a polynomial, the number of roots (counting
multiplicity) of a polynomial is equal to the degree, using the conjugate pairs theorem to write down
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a polynomial with real coefficients that has given roots, Descartes’ rule of signs
Recommended exercises: 9,14,20,23,28,33,43,45,65
3.4 Miscellaneous equations: Solving equations involving square roots, equations of quadratic
type, equations with exponents, and the other examples that we did in class
Recommended exercises: 11,14,18,38,49,52,71
3.5 Graphs of polynomial functions: Determining behavior at x-intercepts, the leading coefficient test, putting all of this information together to graph a polynomial function, polynomial
inequalities
Recommended exercises: 23,34,55, any of 65-84,88,89
3.6 Rational functions: Definition of rational functions, finding their domains, finding horizontal,
vertical, and slant asymptotes, using this information to graph a rational function
Recommended exercises: 8, any of 33-52, 65,66,82,92
4.1 Exponential functions: Definition of exponential functions, their domain and range, graphing exponential functions, graphing families of exponential functions (vertical and horizontal translations, stretching/shrinking, etc.), solving exponential equations
Recommended exercises: 31, 35, 37, 49, 54, 71, 76, 80
4.2 Logarithms: Definition of logarithms, their domain and range, evaluating logarithmic functions, graphing logarithmic functions, solving logarithmic equations, solving exponential equations
using logarithms
Recommended exercises: 19, 26, 35, 38, 45, 62, 72, 81, 85, 93, 95, 98, 103, 105
4.3 Rules of logarithms: inverse rules of logarithms, logarithm of a product, logaritm of a
quotient, logarithm of a power, using these rules to simplify expressions and solve equations
Recommended exercises: 16, 25, 29, 44, 50, 55, 61
4.4 More exponential and logarithmic equations: Solving various logarithmic equations,
including those with more than one logarithm, solving exponential equations using logarithms, see
strategy for solving exponential and logarithmic equations at the bottom of page 387
Recommended exercises: 5, 10, 16, 19, 23, 29, 42, 45, 47
(See the review exercises on p. 399 for more problems)
5.1 Angles and their measurements Definition of an angle, degree measure of an angle, radian
measure of an angle, converting from degrees to radians and from radians to degrees, quadrants
Recommended exercises: 26, 55, 58, 68, 70, 78, 92
5.2 The Sine and Cosine functions Definition of sine and cosine, evaluating sine and cosine
at multiples of 90 degrees, multiples of 45 degrees, multiples of 30 degrees, using reference angles
to evaluate sine and cosine in other quadrants, the Pythagorean identity (p. 428), using it to find
cosine of an angle if you know sine of that angle and vice versa.
Recommended exercises: 6, 12, 13, 18, 23, 22, 25 (better yet, make a table of sine and cosine
evaluated at all angles we know–try to do this without looking at the book, then check your
answer), 91, 92, 93
5.3 Graphing sine and cosine The graphs of sine and cosine, the definition of a periodic function,
the periods of sine and cosine, definition of amplitude and phase shift, graphing families of trig
functions (horizontal and vertical translations, stretching/shrinking/reflecting, changing the period
(e.g. cos(3x), graphing combinations of these transformations)
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Recommended exercises: 14, 18, 19, 26, 33, 37, 41, 43
5.4 The other trig functions: tan, cot, sec, csc The definitions of these functions in terms
of sine and cosine, evaluating these functions, graphing tan(x), cot(x), sec(x), csc(x) by finding
vertical asymptotes, zeros, and plotting points (to do this, you need to know all solutions to the
equations cos(x) = 0, sin(x) = 0). The periods and fundamental cycles of these functions.
Recommended exercises: graph tan, cot, sec, and csc without looking anything up, state the vertical
asymptotes and x-intercepts of each of them, any of 5-30, 55, 61, 64, 77
5.5 The inverse trig functions Definition of arcsin, arccos, arctan (first, review the definition of
an inverse function in 2.5), their domains and ranges, evaluating the inverse trig functions, graphing
the inverse trig functions, evaluating compositions of functions (see example 8), the inverse of a
general sine or cosine functions (see example 9)
Recommended exercises: any of 21-26, 37-52 (choose a good number of these, at least one or two
for each of the inverse trig functions), 70, 73, 75, 81, 99, 100
5.6 Right triangle trigonometry Trigonometric rations (”sohcahtoa”), solving a right triangle
Recommended exercises: 13, 17, 27, 30, 33, 34 (you may need a calculator for these)
(see p. 487 for more exercises from chapter 5)
6.1 Basic trig identities The difference between a trig identity and a (conditional) trig equation,
identities from the definitions, reciprocal identities (if you know the definitions of tangent, secant,
etc., you can deduce these from the definitions), Pythagorean identities, odd and even identities,
verifying that an equation is an identity
Recommended exercises: 10, 15, 29, 33, 47, 93, 100, 102
6.2 Verifying identities various tricks for verifying trig identities (see all of the examples in this
section, e.g. factoring a trigonometric expression, combining fractions, etc.)
Recommended exercises: 28, 30, 38, 48, 55, 58, 61, 63, 64, 74, 81 (really, any of 55-90 are good, get
enough practice with these so that you have some intuition for what sorts of tricks work where)
6.3 Sum and difference identities Sine and cosine of a sum or difference, cofunction identities,
using the sum/difference identities to evaluate the trig functions at angles that are a sum of angles
you know (e.g. 75=45+30, -15=30-45)
Recommended exercises: 37, 40, 43, 46, 47, 48
6.4 Double and half angle identities: The double angle identities, using them to find sine or
cosine or tangent of twice of some angle, the half angle identities, using them to find sine or cosine
or tangent of half of some angle (e.g. find sin(22.5)), using these equations to verify identities
Recommended exercises: 9, 10, 12, 15, 45, 58, 59, 60
6.5 We skipped this section
6.6 Conditional trigonometric equations Finding all solutions to an equation of the form
cos(x) = a, sin(x) = a, or tan(x) = a–find all solutions in the fundamental cycle, then add k times
the period of the function to get all of them), the number of solutions to sin(x) = a, cos(x) =
a, tan(x) = a in the fundamental cycle (may depend on what a is)
Recommended exercises: any of 1-18, 34, 38, 42, 64, 68, 73
(for more exercises from chapter 6, see the review exercises on p.551)
7.1 Law of Sines Types of triangles (AAS, ASA, SSA, SAS, SSS), the law of sines, using the law
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of sines to solve AAS, ASA, and SSA triangles, figuring out how many triangles are possible in the
SSA case and solving them
Recommended exercises: any of 7-20, try to find a few of each kind of triangle and solve them
(especially SSA-try to do one example of each kind of case that can arise)
7.2 Law of Cosines The law of cosines, using it to solve SSS and SAS triangles, general procedure
for solving triangles (see p. 569)
Recommended exercises: any of 7-16, try to do at least two each of SSS and SAS.
7.3 Vectors Definition of vectors, addition of vectors, finding the horizontal and vertical components of a vector given magnitude and direction and vice versa
Recommended exercises: 27,29,32,34,40,41
7.4 Trigonometric form of complex numbers: graphing complex numbers in the complex
plane, finding the trigonometric form of a complex number (i.e. r(cos θ + i sin θ)), taking products
and quotients of complex numbers in trigonometric form
Recommended exercises: 33,35,55,56,58,61 (you may need a calculator for some of these)
7.5 We skipped this section
7.6 Polar equations: Polar coordinates, plotting points in polar coordinates, converting from
polar to rectangular coordinates and vice versa, graphing polar equations
Recommended exercises: any of 23-34, 35-46, 57,58,60,63,68,74,75
7.7 Parametric equations: definition of a parametric equation, graphing parametric equations
Recommended exercises: 8,9,10,13,14
8.1 Systems of linear equations in two variables: Solving systems of linear equations by
graphing, substitution, and elimination; types of systems (one solutions, infinitely many solutions,
no solutions) and how to tell them apart
Recommended exercises: 16,18,24,26,30,38,40,44 (what the book calls addition I called elimination
in class)
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