10.2 Parabolas
-Conic sections: curves resulting from the
intersection of a right circular cone and a plane
-Parabola: the collection of all points P in the plane
that are the same distance from the point F
(focus) as they are from the line D (directrix)
F
D
P
-Axis of symmetry: the line through F and
perpendicular to line D
-Vertex: the point of intersection of the parabola with
its axis of symmetry
P
vertex
D
F
Axis of symmetry
-Equations of a Parabola:
-Vertex (h,k) and Axis x = h (vertical)
(x – h)2 = 4a(y – k)
Focus: (h,k+a)
Directrix: y = k–a
opens:
UP
Directrix: y = k+a
DOWN
(x – h)2 = -4a(y – k)
Focus: (h,k–a)
-Vertex (h,k) and Axis y = k
(horizontal)
(y – k)2 = 4a(x – h)
Focus: (h+a,k)
Directrix: x = h–a
RIGHT
Directrix: x = h+a
LEFT
(y – k)2 = -4a(x – h)
Focus: (h–a,k)
When writing equations of parabolas,
REMEMBER:
a = distance from vertex to focus
(and focal width = 4a)
and (h,k) = vertex
a
a
10.3 Circles
-Conic sections: curves resulting from the
intersection of a right circular cone and a plane
-Parabola: the collection of all points P in the plane
that are the same distance from the point F
(focus) as they are from the line D (directrix)
F
D
P
10.3 Circles
-Conic sections: curves resulting from the
intersection of a right circular cone and a plane
-Parabola: the collection of all points P in the plane
that are the same distance from the point F
(focus) as they are from the line D (directrix)
F
D
P
10.4 Ellipses
-Ellipse: the set of all points in a plane, the sum
of whose distances from 2 fixed points (foci) is a
constant
(x,y)
d1
focus
minor axis
d2
focus
vertex
d1 + d2 is a constant
major axis
vertex
center
-Standard Equation of an Ellipse:
center (h,k)
major axis = 2a
minor axis = 2b
(𝒙−𝒉)𝟐
major axis horizontal:
𝒂𝟐
+
(𝒚−𝒌)𝟐
major axis vertical:
𝒂𝟐
(𝒙−𝒉)𝟐
𝒃𝟐
+
(𝒚−𝒌)𝟐
𝒃𝟐
When writing equations of ellipses REMEMBER:
• Foci are c units from the center
• Vertices are a units from the center
• c2 = a2 – b2
V
c
F
F
a
V
10.5 Hyperbolas
-Hyperbola: the set of all points (x,y) in a plane,
the difference of whose distances from 2
distinct fixed points (foci) is a constant
(x,y)
d1
d2
center
vertex
d 2  d1
is constant
transverse axis
focus
-Standard Equation of a Hyperbola:
center (h,k)
(𝒙−𝒉)𝟐
transverse axis horizontal:
𝒂𝟐
−
(𝒚−𝒌)𝟐
transverse axis vertical:
𝒂𝟐
(𝒙−𝒉)𝟐
𝒃𝟐
−
(𝒚−𝒌)𝟐
𝒃𝟐
When writing equations of hyperbolas REMEMBER:
• Foci are c units from the center
• Vertices are a units from the center
• But now c2 = a2 + b2
Asymptotes of a hyperbola:
transverse axis
(horizontal)
b
y  k   ( x  h)
a
transverse axis
(vertical)
a
y  k   ( x  h)
b
Conic Sections
-Classifying conic sections given General Form:
Ax2 + Cy2 + Dx + Ey + F = 0
1. If A = C  Circle
2. If AC = 0  Parabola
3. If AC > 0  Ellipse
4. If AC < 0  Hyperbola
-Examples: Identify the conic section.
1. 3x2 + 6y2 + 6x – 12y = 0
2. 2x2 – 3y2 + 6y + 4 = 0
3. y2 – 2x + 4 = 0
4. 4x2 + 4y2 – 16y + 15 = 0
5. 5x2 – 5y + x + 3 = 0
11.1 Combinations and
Permutations
-Multiplication Principle of Counting:
If a procedure P has a sequence of S stages that can
occur in R ways, then the number of ways that the
procedure P can occur is the product of the R ways S
stages can occur
-Combinations: the unordered selection of objects from a set.
- Permutations : the ways that a set of n objects can be
arranged in order.
-The number of combinations of n objects taken r at a
time is:
-The number of permutations of n objects taken r at a
time is:
- n factorial = n! = n  (n  1)  (n  2)...4  3  2  1
special case: 0! = 1
11.2 Probability
-Experimental Probability:
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠
-Theoretical Probability:
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
12.1 Adding and Subtracting
Matrices
-An m x n matrix
has:
m rows
n columns
-The order is
determined by:
mxn
D.Adding/Subtracting Matrices:
A = a1i + b1j
B = a2i + b2j
A + B = (a1 + a2)i + (b1 + b2)j
A – B = (a1 – a2)i + (b1 – b2)j
12.2 Matrix Multiplication
Multiplying matrices by scalars:
kv  ka1i  kb1 j
Multiplying Matrices:
(m x n)(m x n) = The n value of the first matrix must
equal the m value of the second matrix
A = a1i + b1j
B = a2i + b2j
Multiplying Matrices
12.3 Determinants and
Inverses
-Identity Matrix
12.3 Determinants and
Inverses
-Inverse Matrix:
if AB = BA = I, then
12.3 Determinants and
Inverses
-The determinant of Matrix A
12.3 Determinants and
Inverses
-The determinant of Matrix A
12.6 Vectors
Vector—a quantity having both magnitude and
direction (a directed segment)
P – initial point
Q – terminal point
Magnitude – length of PQ
B. Geometric Vectors:
v
-v
P
Q
w
w–v
-v
w+v
Algebraic Vectors
v = <a,b>
v = ai + bj
<horizontal change, vertical change>
(1,3)
(-3,-2)
-Adding/Subtracting Vectors:
v = a1i + b1j
w = a2i + b2j
v + w = (a1 + a2)i + (b1 + b2)j
v – w = (a1 – a2)i + (b1 – b2)j
-Multiplying vectors by scalars:
kv  ka1i  kb1 j
F. Magnitude of a vector:
v  a1  b1
2
2
G. Finding Unit Vectors:
Given vector v,
v
u
v
is a unit vector in same direction as v
Dot Product:
Given v = a1 i + b1 j and w = a2 i + b2 j
v  w  a1a2  b1b2
Determining whether vectors are parallel or
orthogonal (normal or perpendicular):
1. v and w are parallel if
v = kw
2. v and w are orthogonal if
vw  0