Review Topics for Exam 2
4 June, 2012
Format of the Exam
The exam will consist of 10 short answer problems, 5 graphing problems, and two writing
questions. The short answer problems will be designed to test your knowledge of the skills
from Chapters 3 and 4. The writing questions will be designed to test your understanding
of the concepts and your problem-solving skills.
The exam will not specifically test your skills from Chapters 1 and 2, but you may need
to use those skills to solve some problems on the exam.
The exam will take place from 10-11 a.m. and will be followed by a short break and a
“light” lecture.
§ More on Functions
• Find the composition of two functions.
• Find
the implied domain of any function we have studied so far (examples: g(x) =
√
4
1
, h(x) = 2x2 + 4) and use set notation to write it.
x − 7, f (x) = x−2
• Know what the domain, range, implied domain, and target are.
• Know how to solve inequalities.
§ Intro to Graphs
• Determine whether a graph is the graph of a function.
• Given the graph of a function:
– Evaluate the function (e.g. Find f (1).)
– Find the range of the function.
– Find the domain of the function.
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– Find the x- and y-intercepts of the graph.
• Know the graphs of the basic types of functions:
– id(x)
– x2
– x3
–
–
1
x
1
x2
n
– x where n is even
– xn where n is odd
–
–
1
xn
1
xn
where n is even
where n is odd
– constant functions
§ Graph Transformations
• Be able to use, recognize and describe each of the 10 graph transformations from the
section: (d and c are real numbers with d > 0, and c > 1)
– f (x) + d
– f (x) − d
– cf (x)
–
1
f (x)
c
– −f (x)
– f (x + d)
– f (x − d)
– f (cx)
– f ( xc )
– f (−x)
§ Inverse Functions
• Determine whether two functions are inverses.
• Determine whether a function has an inverse.
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• Know the definitions of one-to-one and onto, and how they are related to whether a
function has an inverse.
• Find the inverse of a function.
• If a function has an inverse, and you know its graph, graph the inverse function.
§ nth roots
• Know the definitions of the functions
their targets, and their graphs.
√
n
when n is both even and odd, their domains,
• Find the implied domain of a function involving
• Graph a function involving
√
n
√
n
(either even or odd).
(and possibly some transformations).
§ Basics of Polynomials
• Know the definitions of the terms polynomial, degree, degree k coefficient, leading coefficient, leading term.
• Know how to add, subtract and multiply polynomials.
• Know the relationship between the degree of the product of two polynomials (p(x)q(x))
and the degrees of the polynomials (p(x) and q(x)).
• Find the leading term of a product of polynomials from the leading terms of the
polynomials.
§ Division
• Know how to use long division to find the quotient of two polynomials.
• Know how to write the answer to a division problem (e.g.
4
3
= 1 + 13 ).
• It may be useful to know how to use synthetic division, but you will not be required to
use it. If you do choose to use it, be sure you know which types of problems it works
for, how to set up the problem, how to perform the division, and how to translate your
answer into the language of polynomials.
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§ Roots and Factors
• Know the definitions of the terms root, factor.
• Determine whether one polynomial is a factor of another polynomial.
• Determine whether a real number is a root of a polynomial.
• Given a root of a polynomial, write it as the product of a linear polynomial and a
higher degree polynomial. (This requires you to understand the relationship between
factors and quotients and the original polynomials.)
• Find the roots of a polynomial based on its graph.
• Determine the degree of a product of polynomials.
• Know that a polynomial of degree n has at most n roots. (Important: It may have
fewer. It cannot have more.)
§ Constant and Linear Polynomials
• Know how to graph any constant or linear polynomial.
• Know how to find the roots of any constant or linear polynomial (if they exist).
• Know how to find the slope of a line that passes through two points.
§ Quadratic Polynomials
• Know how to complete the square for any quadratic polynomial.
• Know how to graph any quadratic polynomial in the ”completed square” formula.
• Know what the vertex of a parabola is.
• Know how to determine how many roots a quadratic polynomial has, and how to find
those roots.
§ Factoring Polynomials
• Know the statement of the Fundamental Theorem of Algebra.
• Know how to completely factor any constant, linear, or quadratic polynomial.
• Know how to factor a higher degree polynomial once you know one or more of its roots.
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• Know how to find a root of a higher degree polynomial (by checking the factors of the
constant term.)
• Know how to find the roots, x-intercepts, and factors of a polynomial by looking at its
completely factored form.
§ Graphing Polynomials
• Know how to graph any completely factored polynomial.
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