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Math Analysis Test #02 Review Sheet
9/18/06
Page 1 of 5
Section 2.1: What is a Function?
1. Definitions: A function f is a rule that assigns to each element x in a set A exactly one element f(x), in a
set B. The set A is the domain of the function. f(x) is the image of x under f. The range is the set of
all possible values of f(x) as x varies through the domain.
2. Know how to evaluate a function at a number and at an expression.
3. Know how to find the domain of a function:
a. Start with the domain being all real numbers.
b. Eliminate x values from the domain where a denominator is equal to zero.
c. Eliminate x values from the domain where a square root has a negative argument.
Section 2.2: Graphs of Functions
1. Definition: If f is a function with domain A, then the graph of f is the set of ordered pairs
{(x, f(x) | x  A}, or, in other words, the graph of f is the graph of the equation y = f(x).
2. Know how to graph a function by calculating and plotting points.
3. Know the graphs of some common functions:
a. Linear function: f ( x)  mx  b
b. Power functions: f ( x)  x n for n = even and odd integers greater than 1.
c. Root functions: f ( x)  n x for n = even and odd integers greater than 1.
d. Reciprocal functions: f ( x)  1 n for n = even and odd integers greater than 0.
x
e. Absolute value function: f ( x)  x
4. The vertical line test: A curve in the coordinate plane is the graph of a function if and only if no
vertical line intersects the curve more than once.
Section 2.3: Increasing and Decreasing Functions; Average Rate of Change
1. Average Rate of Change: The average rate of change of a function y = f(x) between x = a and x = b is
the slope of the secant line connecting the points (a, f(a)) and (b, f(b)):
y f (b)  f (a)

average rate of change =
.
x
ba
2. Increasing and Decreasing Functions:
a. A function f is increasing on an interval I if f(x1) < f(x2) whenever x1 < x2 in I.
b. A function f is decreasing on an interval I if f(x1) > f(x2) whenever x1 < x2 in I.
c. Note: intervals of increase/decrease change wherever a function has an extreme value…
Section 2.4: Transformations of Functions
1. Vertical shifts of graphs:
a. y = f(x) + c (c > 0) shifts the graph of y = f(x) upward by c units.
b. y = f(x) – c (c > 0) shifts the graph of y = f(x) downward by c units.
2. Horizontal shifts of graphs:
a. y = f(x – c) (c > 0) shifts the graph of y = f(x) to the right by c units.
b. y = f(x + c) (c > 0) shifts the graph of y = f(x) to the left by c units.
3. Reflecting graphs:
a. y = –f(x) reflects the graph of y = f(x) in the x-axis.
b. y = f(–x) reflects the graph of y = f(x) in the y-axis.
4. Vertical Stretching and Shrinking of Graphs:
Math Analysis Test #02 Review Sheet
9/18/06
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a. y = af(x) (a > 1) stretches the graph of y = f(x) vertically by a factor of a.
b. y = af(x) (0 < a < 1) shrinks the graph of y = f(x) vertically by a factor of a.
5. Even and Odd Functions:
a. f is even if f(–x) = f(x) for all x in the domain of f. (The graph of f is symmetric with respect to
the y-axis.)
b. f is odd if f(–x) = –f(x) for all x in the domain of f. (The graph of f is symmetric with respect to
the origin.)
Section 2.5: Quadratic Functions; Maxima and Minima
1. If you have a TI calculator, know how to find the extrema of a graph with it.
2. Some facts about the quadratic function f(x) = ax2 + bx + c:
a. Its graph is a parabola.
b. Its standard form is f(x) = a(x – h)2 + k
c. The vertex, or extremum of its graph is at the point (h, k).
d. Its extremum is a maximum if a < 0.
e. Its extremum is a minimum if a > 0.
b
b 2  4ac


f. It can be re-arranged into the form: y  a x 

2a 
4a

2
  b b  4ac 

g. Thus, (h, k) = 
,
2
a
4
a


2
Section 2.6: Modeling With Functions
1. Draw a picture.
2. Label the picture with given quantities and the unknown.
3. Write down the equation that relates the unknown to the given information.
4. Solve the equation.
5. Check your answer.
6. Some geometric formulas for areas:
a. Rectangle: Arect  lw
b. Circle: Acirc  r 2
1
c. Triangle: Atri  bh
2
7. Some geometric formulas for volumes:
a. Box: Vbox  lwh
b. Cylinder: Vcyl  r 2 h
4
c. Sphere: Vsphere  r 3
3
1
d. Pyramid: V pyr  Bh
3
1
1
e. Cone: Vcone  Bh  r 2 h
3
3
Math Analysis Test #02 Review Sheet
8. Speed problems: speed 
9/18/06
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distance
time
Section 2.7: Combining Functions
1. Addition: (f + g)(x) = f(x) + g(x)
domain: A  B
2. Subtraction: (f – g)(x) = f(x) – g(x) domain: A  B
3. Multiplication: (fg)(x) = f(x)g(x)
domain: A  B
f
f ( x)
4. Division:  ( x) 
domain: A  B and g(x)  0
g ( x)
g
5. Composition: (f  g)(x) = f(g(x))
domain: all x  domain of g such that their images are in the range
of f
6. Be able to break a function down into the composition of two simple functions.
Section 2.8: One-to-One Functions and Their Inverses
1. One-to-One Function: A function with domain A is called a one-to-one function if no two elements of
A have the same image, that is, f(x1)  f(x2) whenever x1 x2.
2. Horizontal Line Test: A function is one-to-one function if and only if no horizontal line intersects its
graph more than once.
3. Algebraic Test: Start with f(x1) = f(x2). If this reduces to x1 = x2, then f is one-to-one.
4. Definition of the Inverse of a Function: Let f be a function with domain A and range B. Then its
inverse function f-1 has domain B and range A and is defined by f-1(y) = x  f(x) = y for any y in B.
5. Properties of Inverse Functions:
a. f-1(f(x)) = x
b. f(f-1 (x)) = x
6. Know how to find the inverse of a function:
a. Set y = f(x)
b. Solve for x
c. Switch variables x  y
d. y = f-1(x) now.
7. The graph of f-1 is obtained by reflecting the graph of f in the line y = x.
8. A point with coordinates (a, b) on the graph of f corresponds to a point with coordinates (b, a) on the
graph of f-1.
Math Analysis Test #02 Review Sheet
9/18/06
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Section 3.1: Polynomial Functions and Their Graphs
1. Polynomial Functions: A polynomial of degree n is a function of the form
Px   a n x n  a n 1 x n 1  a n 2 x n 2    a 2 x 2  a1 x  a0
2. Know how to determine the end behavior of a polynomial (it’s determined by the anxn term…).
3. Know how to use zeros of a polynomial as a graphing aid (there are at most n of them).
4. Know how roots, zeros, and factors of a polynomial are related.
5. Know how a repeated root affects the graph of a polynomial function.
6. Intermediate Value Theorem for Polynomials: If P is a polynomial function and P(a) and P(b) have
opposite signs, then there exists at least one value c between a and b for which P(c) = 0.
7. Local Extrema of Polynomials: If P is a polynomial of degree n, then the graph of P has at most n – 1
local extrema.
Section 3.2: Dividing Polynomials
1. Division Algorithm: Px  D( x)  Q( x)  R( x) . P is the dividend, D is the divisor, Q is the quotient,
P( x)
R( x)
and R is the remainder. Can also be stated:
.
 Q( x) 
D( x)
D( x)
2. Know how to do long division of polynomials.
3. Know how to do synthetic division of polynomials.
4. Remainder Theorem: If the polynomial P(x) is divided by x – c, then the remainder is the value P(c).
5. Factor Theorem: c is a zero of P if and only if x – c is a factor of P.
Section 3.3: Real Zeros of Polynomials
1. All rational zeros of a polynomial P are of the form
p
, where p is a factor of the constant coefficient a0,
q
and q is a factor of the leading coefficient an.
2. Know how to list all possible zeros of a polynomial and test them using synthetic division.
3. Know how to use Descartes’ Rule of Signs.
4. Know how to use the upper and lower bounds theorem.
Steps for factoring a polynomial:
1. Graph the polynomial on your calculator to guide your work.
2. Make a list of all possible rational zeros using the rational zeros theorem.
3. Order the list and remove any redundant values.
4. Apply Descartes’ Rule of signs to predict how many positive and negative zeros you can expect; if you
find that no roots of a given sign are predicted, then your work is cut in half.
5. Perform synthetic division on the polynomial, starting with the smallest roots.
a. If the remainder is zero, then you found a root; check to see if it is a multiple root, and re-write
the polynomial with that root factored out. Only perform synthetic division on the lower-order
quotient.
b. If the remainder is not zero, look at the signs of the quotient row; the upper and lower bounds
theorem tells you when you can stop looking based on the signs of the numbers in the quotient
row.
6. Use the quadratic formula when you get down to a quadratic quotient.
Math Analysis Test #02 Review Sheet
9/18/06
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Section 3.4: Complex Numbers
9. i   1
10. i 2  1
11. Know how to add and subtract complex numbers.
12. Know how to multiply complex numbers.
13. Know how to divide complex numbers (multiply numerator and denominator by the conjugate of the
denominator).
Section 3.5: Complex Zeros and The Fundamental Theorem of Algebra
1. Fundamental Theorem of Algebra: Every polynomial
Px   a n x n  a n 1 x n 1  a n 2 x n 2    a 2 x 2  a1 x  a0 with complex coefficients has at least one
complex zero.
2. Complete Factorization Theorem: There exist complex numbers such that any polynomial of degree
greater than 1 can be written as P( x)  a  x  c1  x  c2   x  cn 
3. The Zeros Theorem means that you have to keep factoring until you find n zeros  check any zero you
find again with synthetic division until the remainder is not zero.
4. Conjugate Zeros Theorem: If a polynomial has real coefficients and a complex zero z, then z is also a
zero.
5. Linear and Quadratic Factors Theorem: Every polynomial with real coefficients can be factored into
a product of linear and irreducible quadratic factors with linear coefficients.
Section 3.6: Rational Functions
1. To graph rational functions:
a. Factor numerator and denominator
b. Find x and y intercepts
c. Find vertical asymptotes by setting the denominator to zero
d. Find horizontal asymptotes by doing long division; the asymptote is the quotient you get
e. Graph