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MATH 1100 – Course Review
SECTION 2.2 - Graphs of Equations – Circles – Text pp. 87-89
Distance Formula: The distance between two points,
d
x1 , y1  and x2 , y2 :
x1  x2 2   y1  y2 2
Standard Form of the Equation of a Circle: Given the center (h, k) and radius r, the
standard form of the equation of a circle:
r 2  x  h    y  k 
2
Diameter of a Circle:
2
A line segment whose endpoints lie on the circle and that
contains the center of the circle.
Midpoint Formula: Given the points
x1 , y1 
and
 x2 , y 2  ,
the midpoint of the line
segment connecting them is given by:
 x1  x2 y1  y 2 
,


2 
 2
SECTION 2.4 - Lines – Text pp. 100-111
Slope:


m  slope 
rise y y 2  y1 y1  y 2



run x x2  x1 x1  x2
Horizontal lines have zero slope.
Vertical lines have undefined slope.
Point-Slope Form of the Equation of a Line:
Equation of a line with slope m, passing through
y  y1  mx  x1 
Slope-Intercept Form of the Equation of a Line:
Equation of a line with slope m and y-intercept
( x1 , y1 ) is
0, b is
y  mx  b
Standard Form of a Linear Equation:
Ax  By  C

Vertical lines are parallel to vertical lines.

Two non-vertical lines are parallel if they have the same slope.

Vertical lines are perpendicular to horizontal lines.

Two lines are perpendicular if the product of their slopes is –1.
SECTION 3.1 – Solutions of Equations – Text pp. 131-135
This section covers algebraic/graphical solutions of equations. Review Section 1.6 if
you have trouble solving basic equations.
SECTION 3.2 - Modeling With Equations – Text pp. 136-145
Guidelines For Modeling With Equations:
1.) IDENTIFY THE VARIABLE:
Identify the quantity that the problem asks you to find. Introduce notation for
the variable (call it x or some other letter).
2.) EXPRESS ALL UNKNOWN QUANTITIES IN TERMS OF THE VARIABLE:
Read each sentence in the problem again, and express all the quantities
mentioned in the problem in terms of the variable you defined in Step 1. To
organize this information, it is sometimes helpful to draw a diagram or make a
table.
3.) SET UP THE MODEL:
Find the crucial fact in the problem that gives a relationship between the
expressions you listed in Step 2. Set up an equation (or model) that expresses
this relationship.
4.) SOLVE THE EQUATION AND CHECK YOUR ANSWER:
Solve the equation, check your answer, and express it as a sentence that
answers the question posed in the problem.
SECTION 3.3 - Quadratic Equations – Text pp. 151-161
Quadratic Equations: An equation of the form ax  bx  c  0 , where a, b, and c are
real numbers with a  0.
2
Zero-Product Property:
“Something Times Something Equals Zero”
AB  0 if and only if A  0 or B  0 .
QUADRATIC FORMULA:
Solutions of ax  bx  c  0 when
2
x
Warning!:
a  0 are:
 b  b  4ac
2a
2
The equation MUST be in the form ax2 + bx + c = 0
The Discriminant
The discriminant of the general quadratic ax  bx  c  0 (a  0) is
2
D  b 2  4ac
1.)
2.)
3.)
If D > 0, then the equation has two distinct real solutions
If D = 0, then the equation has one distinct real solution
If D < 0, then the equation has no real solutions
COMPLETING THE SQUARE
Step 1:
Move all constants to one side.
Step 2:
Make sure the coefficient of x2 is 1.
Step 3:
Take ½ of B and square it and add to
both sides.
B
B
x  Bx     C   
2
2
Step 4:
Use the Perfect Square Trinomial
Formula to factor.
B

B
x   C  
2

2
Step 5:
Solve equation for x using the
Principle of Square Roots.
ax 2  bx  const.
b
const .
x
a
a
2
x  Bx  C
x2 
2
2
2
B

2
2


 C   B  

 2  

B
x 
2
2


 C   B  

 2  

x
SECTION 3.4 - Complex Numbers – Text pp. 164-169
Complex Numbers: A number written in the form
a  bi
where a, b are real numbers, and i  1 .
2
Adding, Subtracting and Multiplying Complex Numbers:
a  bi   c  di   a  c   b  d i
a  bi   c  di   a  c   b  d i
a  bi c  di   ac  bd   ad  bc i
Dividing Complex Numbers:
a  bi  a  bi  c  di  ac  bd   bc  ad i



c  di  c  di  c  di 
c2  d 2
Square Root of a Negative Number:
If –r is negative, then the principle square root of –r is,
r i r
The two square roots of –r are, i r and  i r .
2
2
SECTION 3.6 - Linear Inequalities – Text pp. 181-185
Rule
1.
A  B  AC  B C
2.
A  B  AC  B C
C  0 , then
A  B  CA  CB
4. If C  0 , then
A  B  CA  CB
3. If
RULES FOR INEQUALITIES
Description
Adding the same quantity to each side of an inequality
gives an equivalent inequality.
Subtracting the same quantity to each side of an
inequality gives an equivalent inequality.
Multiplying each side of an inequality by the same
positive quantity gives an equivalent inequality.
A  0 and B  0 , then
1 1
A B 

A B
6. If A  B and C  D , then
AC  B  D
5. If
Multiplying each side of an inequality by the same
negative quantity reverses the direction of the
inequality.
Taking reciprocals of each side of an inequality
involving positive quantities reverses the direction of
the inequality.
Inequalities can be added.
Interval Notation:
Open Interval, (a, b): If a and b are real numbers such that a < b, we define an
open interval (a, b) as the set of all numbers x for which a  x  b , so
x : a  x  b  a, b .

 

Closed Interval, [a, b]: If a and b are real numbers such that a  b, we define an
closed interval [a, b] as the set of all numbers x for which a  x  b , so
x : a  x  b  a, b .
Half-Open Interval, [a, b) or (a, b]:

  
x : a  x  b  [a, b)
x : a  x  b  (a, b]
SECTION 3.7 - Non-Linear Inequalities – Text pp. 187-194
GUIDELINES FOR SOLVING NON-LINEAR INEQUALITIES
1. MOVE ALL TERMS TO ONE SIDE. If necessary, rewrite the inequality so that all
non-zero terms appear on one side of the inequality sign. If the non-zero side of the
inequality involves quotients, bring them to a common denominator.
2. FACTOR. Factor the non-zero side of the inequality.
3. FIND THE INTERVALS.
corresponding equations.
Use the factorization to find all solutions of the
4. MAKE A TABLE OR DIAGRAM. Use test values to make a table or diagram of the
signs of each factor on the interval. In the last row of the table determine the sign of
the product/quotient of these factors.
5. SOLVE. Determine the solution of the inequality from the last row of the sign table.
SECTION 3.8 – Absolute Value – Text pp. 196-198
Properties of Absolute Value Inequalities:
1.)
x c
equivalent to:  c  x  c
2.)
x c
equivalent to: x  c or x  c
SECTION 4.1 - What is a Function? – Text pp. 209-216
Function:
A function is a rule that assigns to each element x in a set A exactly one
element, called f(x), in a set B.



f(x):
Say: “f of x”
Called: “the value of f at x”
The set A:
is called the domain of the function
The range of a function is the set of all possible values of f(x) as x varies
throughout the domain, f ( x) : x  A .

The variable that represents an element in the domain of the function is called
the independent variable, and the variable that represents an element in the
range is called the dependent variable.


Guidelines for Determining the Domain:
 The domain of all linear functions is the set of all real numbers, .
 If there is a variable in the denominator, the domain cannot contain any
value that makes the denominator equal to zero.
 If there is a variable under a square root symbol, the domain cannot
contain any value that creates the square root of a negative number.
SECTION 4.2 - Graphs of Functions – Text pp. 220-227
The Graph of a Function:
If f is a function with domain A, then the graph of f is the set of ordered pairs,
x, f x : x  A .

 

In other words, the graph of f is the set of all points (x, y) such that
that is, the graph of f is the graph of the equation
y  f x  .
y  f x  ;
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 4.3 - Applied Functions : Variation – Text pp. 231-234
Direct Variation/Varies Directly/Directly Proportional:
y  kx
Inverse Variation/Varies Inversely/Inversely Proportional:
y
Joint Variation/Varies Jointly/Jointly Proportional:
k
x
z  kxy
SECTION 4.4 - Average Rate of Change - Text pp. 236-244
Average Rate of Change:
The average rate of change of the function
given by the formula,
average rate of change 
y  f x  between x = a and x = b is
change in y f b   f a 

change in x
ba
Increasing Functions:
f is increasing on an interval I if
f x1   f x2  whenever x1  x2 in I.
Decreasing Functions:
f is decreasing on an interval I if
f x1   f x2  whenever x1  x2 in I.
SECTION 4.5 - Transformations of Functions – Text pp. 247-255
Vertical Shifting:


Upward if the constant is positive
Downward if the constant is negative
Horizontal Shifting:


y  f x   c
y  f x  c 
To the right if the constant is positive
To the left if the constant is negative
Reflective Graphs:
 Reflection in the x-axis:

Reflection in the y-axis:
y  a  f x 
Stretched vertically: a  1
y   f x 
y  f  x
Vertical Stretch/Shrink:


Shrunk vertically:
Even Functions:
0  a 1
f  x  f x
Note: The graph of an even function is symmetric about the y-axis.
Odd Functions:
f  x   f x
Note: The graph of an odd function is symmetric about the origin.
SECTION 4.6 - Extreme Values of Functions – Text pp. 258-266
f x   ax  h   k
Standard Form of a Quadratic Function:
A quadratic function can be expressed in the standard form by completing the
square.
2
Graph of a Quadratic Function:
 Vertex of a Parabola:
Parabola
 b
b2 
vertex  h, k     , c  
4a 
 2a

Minimum Value of f(x):
 b 
minimum  f h   k  f   
 2a 

Maximum Value of f(x):
 b 
maximum  f h   k  f   
 2a 
SECTION 4.7 - Combining Functions – Text pp. 268-275
ALGEBRA OF FUNCTIONS
Let f be a function with domain A, and let g be a function with domain B.

Sum of f and g:
 f  g x  f x  gx

Difference of f and g:
 f  g x  f x  gx

Product of f and g:
 fg x  f x  gx

Quotient of f and g:
f
f x 
 x  
g x 
g

Composition of f and g:

Domain of Sum, Difference & Product:

Domain of Product:
x  A  B : gx  0

Domain of Composition:
x  B : g x  A
 f  g x  f g x
A B
SECTION 5.1 - Polynomial Functions/Graphs – Text pp. 311-323
Px   a n x n  a n 1 x n 1  ...  a1 x  a0
Coefficients:
the numbers
The Constant Term:
ao
The Leading Coefficient:
an
The Leading Term:
an x n
an , an1 ,    , a1 , a0
Guidelines For Graphing Polynomial Functions:
1) ZEROS:
Find all x-intercepts.
2) Find the y-intercept
3) Make a TABLE using the x-intercepts as the “important” (critical) points 
determine whether the graph is positive or negative in these intervals.
4) Use the leading coefficient to determine the end behavior of P(x).
5) GRAPH:
Plot the intercepts and other points found in the table  sketch a
smooth curve through these points that exhibits the proper end behavior.
Zeros of Polynomials:

The set of numbers such that
Pc  0 .
The following are equivalent ways of saying this same thing:
1.)
c is a zero of P.
2.)
x  c is a root of the equation P x  0 .
3.)
x  c is a factor of Px .

SECTION 5.2 - Dividing Polynomials – Text pp. 326-331
   Dx  Qx  Rx
Division Algorithm: P x
Dividend:
Divisor:
Quotient:
Remainder: R(x)
P(x)
D(x)
Q(x)
Polynomial Division:
Review Polynomial Long Division and Synthetic Division.
Remainder Theorem:
If the polynomial P(x) is divided by (x - c) the remainder is P(c).
Factor Theorem:
c is a zero of P if and only if x – c is a factor of P(x).
SECTION 5.3 - Real Zeros of Polynomials – Text pp. 333-341
Rational Zeros Theorem:
If the coefficients of a polynomial
every rational zero of P is of the form
Px are integers, then
p
where, p is a factor of a0 and q is a
q
factor of an.
Guidelines for Factoring a Polynomial P(x):
1) List all possible rational zeros.
2) Use synthetic division to evaluate P(x) at each possible rational zero.
When the remainder is zero, note the quotient you have obtained and
move to Step 3.
3) Repeat Steps 1 & 2 for the quotient until you reach a quadratic that can
be factored using methods from Chapter 3.
UPPER BOUND & LOWER BOUND:
acb
a is a lower bound and b is an upper bound for the roots of
The Upper & Lower Bounds Theorem:
Px .
Let P be a polynomial with real coefficients.
1) If we divide P(x) by x  b (with b  0 ) using synthetic division, and if the row
that contains the quotient and remainder has no negative entry, then b is an
upper bound for the real zeros of P.
2) If we divide P(x) by x  a (with a  0 ) using synthetic division, and if the row
that contains the quotient and remainder has entries that are alternately
nonpositive and nonnegative, then a is a lower bound for the real zeros of P.
[Note: Zero is considered positive or negative as needed for this theorem]
Variation in Sign:
If P(x) is a polynomial with real coefficients, written in descending powers of x
and omitting powers with coefficient of zero, then a variation in sign occurs whenever
adjacent coefficients have opposite signs.
Descarte’s Rule of Signs:
Let P(x) be a polynomial with real coefficients.
(1.)
(2.)
The number of positive real zeros of P(x) is either equal to the number of
variations in sign of P(x) or is less than that by an even whole number.
The number of negative real zeros of P(x) is either equal to the number of
variations in sign in P(-x) or is less than that by an even whole number.
SECTION 5.4 - Fundamental Theorem of Algebra – Text pp. 346-352
The Fundamental Theorem of Algebra:
Every
polynomial
Px   a n x n  a n 1 x n 1  ...  a1 x  a0
n  1,
an  0 with
complex coefficients has at least one complex zero.
Complete Factorization Theorem:
If P(x) is a polynomial of degree n>o, then there exist complex numbers
a, c1 , c2 , ... , cn (with a  0 ) such that
Px   ax  c1 x  c2     x  cn  .
Zeros Theorem:
Every polynomial of degree
multiplicity k is counted k times.
n  1 has exactly n zeros, provided that a zero of
Conjugate Zeros Theorem:
If the polynomial P has real coefficients, and if the complex number z is a zero of
P, then its complex conjugate, z is also a zero P.
SECTION 6.1 - Exponential Functions – Text pp. 379-383
Exponential Function with Base a:
f x   a is 
x
Range of f x   a is 0, 

Domain of

f x   a x ,
where
a  0.
x
SECTION 6.2 - The Natural Exponential Function – Text pp. 387-393


The irrational number e is defined as the value that 1 
The Number, e:
approaches as
n   . Correct to 5 decimals, the number is, e  2.71828 .
1

n
n
The Natural Exponential Function:
The natural exponential function is the
exponential function with base e. It is often referred to as the exponential function.
r

At   P1  
 n
Compound Interest:
where,
nt
At   amount after t years
P  principal
r  annual in terest rate
n  number of times in terest is compounded per year
t  number of years
At   Pe rt
Continuously Compounded Interest:
Exponential Growth:
A population that experiences exponential growth increases according to the
model,
nt   no e rt
where,
nt   population at time t
no  initial size of population
r  relative rate of growth ( proportion of population )
t  time
SECTION 6.3 - Logarithmic Functions – Text pp. 397-405
The Logarithmic Function with Base a (a>0):


log a x  y

Domain of f x   log a x is 0, 
Range of f x   log a x is 
ay  x
Properties of Logarithms:
log a 1  0
log a a  1
log a a x  x
a loga x  x
Common Logarithm:
log x  log 10 x
Natural Logarithm:
ln x  log e x
SECTION 6.4 - Laws of Logarithms – Text pp. 407-411
Laws of Logarithms:
1.)
Product:
log a  AB  log a A  log a B
2.)
Quotient:
 A
log a    log a A  log a B
B
3.)
Power:
log a AC  C  log a A
 
Change of Base Formula:
log b x 
log a x
log a b
SECTION 6.5 - Exponential & Logarithmic Eqns – Text pp. 412-417
Guidelines For Solving Exponential Equations:
1.)
Isolate the exponential expression on one side of the equation.
2.)
Take the logarithm of each side, then use the Laws of Logarithms to “bring
down” the exponent.
3.)
Solve for the variable.
Guidelines For Solving Logarithmic Equations:
1.)
Isolate the logarithmic term on one side of the equation – you may need to first
combine the logarithmic terms.
2.)
Write the equation in exponential form.
3.)
Solve for the variable.
SECTION 7.1 - Systems of Equations – Text pp. 447-453
Systems of Equations:
When solving a problem with more than one unknown,
you must have more than one equation. Note: The number of equations should equal
the number of unknowns.
Solving Systems of Equations:

Graphical Method

Substitution Method

Elimination Method
SECTION 7.2 - Pairs of Lines – Text pp. 456-461
Solutions of Linear Systems:
solving a linear system of equations:
There are three possible situations when
1.
One Solution: The lines intersect at one point
2.
3.
4.
No Solution: The lines are parallel – no intersection
Infinite Solutions:
The lines are the same – intersect at all points
SECTION 7.7 - Systems of Inequalities – Text pp. 507-510
Graphing Inequalities:
1.
Graph the Corresponding Equation:
Use a dashed curve for > or <. Use a solid curve for  or .
2.
Test Points: If the test point does not satisfy the inequality, then the
region is not part of the graph.
Solution Region:
problem.
A solution region is the shaded area a graphical inequality
Vertex:
A vertex of a solution region is a point in the solution region that is the
intersection of two graphs.
Bounded Solution Region: A solution region of a system of inequalities is bounded if
a sufficiently large circle can enclose it.
Unbounded Solution Region:
A solution region of a system of inequalities is
unbounded if it cannot be enclosed by a circle.
Systems of Inequalities:
The solution of a system of inequalities can be found by
graphing each inequality in the system on one graph. The intersection of all
graphs is the solution of the system. In other words, the solution of the system
of inequalities is only the regions on the graph by all inequalities in the system.