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Alg Bas Rev

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Solutions: {1,2,3,4,5,6,7}, {4,5}, Interval Notation--Exercises (- , -3) ( 3, ) (-2, -1) (1, 2) ... ab. b. 2. Special Products. 1. Perfect Square. 2. u. 2. 2 ... – PowerPoint PPT presentation

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Title: Alg Bas Rev


1
Pre-Calculus
Basic Algebra Review
2
The Set of Real Numbers
N
Natural Numbers
1
, 2, 3, . . .
Z
Integers
. . . , 2, 1, 0, 1, 2, . . .
__
3
2
Q
Rational Numbers
4
, 0, 8,

,

, 3.14, 5.27
27

5
3
3
I
Irrational Numbers
2

, ?
7
, 1.414213 . . .

3
2

?
R
Real Numbers
7
, 0,
5

,

, 3.14, 0.33
3

,
3
3
Subsets of the Set of Real Numbers
Natural
numbers (N)
Integers (Z)
Zero
Rational
numbers (Q)
Negatives
Real
Noninteger
of natural numbers
numbers (R)
ratios
of integers
Irrational
numbers (I)
N ? Z ? Q ? R
4
Basic Real Number Properties
be the set of real numbers and let x, y, and
z
Let
R
be arbitrary
elements of R.
Addition Properties
Closure
x

y
is a unique element in
R.
Associative
(
x

y
)
z

x
(
y

z
)
Commutative
x

y

y

x
Identity
0
x

x
0
x
Inverse
x
(
x
) (
x
)
x
0
5
Basic Real Number Properties
Multiplication Properties
xy
is a unique element in
R .
Closure
(xy)z x(yz)
Associative
Commutative
xy

yx
Identity
(1)
x

x
(1)
x
æ
ö
1
æ
ö
1
Inverse

ç




ç


x
1
x
?
0
X
x
x
è
ø
è
ø
Combined Property
Distributive
x
z
)
xy
xz
(
y


y
z
xz
yz
(
x
)



6
Interval Notation
Interval Inequality Notation Notation
Line Graph Type

a, b

a
?
x
?
b
Closed

a, b
)
a
?
x
lt
b
Half-open
(
a, b

a
lt
x
?
b
Half-open
(
a, b
)
a
lt
x
lt
b
Open
7
Interval Notation
Interval Inequality Notation Notation
Line Graph Type
x

?

b
,
)
x
?
b
Closed
b
x
(
b, ?
(
)
x
gt
b
Open
b
x

?,
(
a

x
?
a
Closed
a
)
x
?,
(

a
)
x
lt
a
Open
a
8
Sets and Operations
  • Set Builder Notation S x x?Z, 0ltxlt6 So
    S 1,2,3,4,5
  • A ? B x x ? A or x ? B
  • A ? B x x ? A and x ? B
  • Let S 1,2,3,4,5, T 4,5,6,7, V
    6,7,8 Find S ? T, S ? T, and S ? V
  • Solutions 1,2,3,4,5,6,7, 4,5, ?

9
Interval Notation--Exercises
  • (- ?, -3) ? ( 3, ?)
  • (-2, -1) ? (1, 2)
  • (1, 3) ? (2, 7)
  • (-2, 0) ? -1, 2

10
Exponent Properties
Definition of an
1.
For
n
a positive integer
n
a



a

a


…


a
a
n
factors of
2.
For
n
0 ,
0
a


1
a
? 0
0
0

is not defined
3.
For
n
a negative integer,
1
n
a




a
? 0
n
a
11
Definition of b1/n
For n a natural number and b a real
number, b1/n is the principal nth root of
b defined as follows 1. If n is even and b is
positive, then b1/n represents the positive nth
root of b. 2. If n is even and b is negative,
then b1/n does not represent a real number. 3.
If n is odd, then b1/n represents the real nth
root of b (there is only one). 4. 01/n 0
Rational Exponents
For m and n natural numbers and b any real number
(except b cannot be negative when n is even)
12
Rational Exponent/ Radical Conversions
Properties of Radicals
13
Simplified (Radical) Form
14
Examples
15
Foil Method
F
O
I
L
First
Outer
Inner
Last
Product
Product
Product
Product
ß
ß
ß
ß


(2
x
1)(3
x
2)
6
x
2

4
x

3
x
2



Special Products
2
2
a
)

1.
(
a


b
)(

b
a

b


2
2
2
2.
(
a

b
)

a
2
ab

b
2
2
2
3.
(
a


b
)

a

2
ab

b


16
Special Factoring Formulas
Perfect Square
2
2
2
1.
2
(
u



v
uv
v
u
)
2
2
2
2
2.
u
uv

v

(
u

v
)
Perfect Square
2
2
3.
u

v

(
u

v
)(
u

v
)
Difference of
Squares
3
3
2
2


)(
)
4.
v
u
v
(
u
v
u

uv

Difference of
Cubes


3
2
2
5.
3
u
v
u

uv
v
)
u
v
(

)(

Sum of Cubes



17
The Least Common Denominator (LCD)
The LCD of two or more rational expressions is
found as follows 1. Factor each denominator
completely. 2. Identify each different prime
factor from all the denominators. 3. Form a
product using each different factor to the
highest power that occurs in any one
denominator. This product is the LCD.
18
Inequality Properties
For a, b, and c any real numbers
1.
If
a
lt
b
and
b
lt
c
, then
a
lt
c
.
Transitive Property
2.
If
a
lt
b
, then
a

c
lt
b

c
.
Addition Property
3.
If
lt
, then

lt

.
Subtraction Property
a
b
a
c
b
c
4.
If
a
lt
b
and
c
is positive, then
ca
lt
cb
.

Multiplication
Property
ü
ý

(Note
difference
between


þ

4
and
5.
)
5.
If
lt
and
is negative, then
gt
.
a
b
c
ca
cb
a
b

ü
Division
Property
6.
If
a
lt
b
and
c
is positive, then

lt

.
c
c

ý
(Note
difference
between



þ
6
and
7.
)
a
b

7.
If
lt
and
is negative, then

gt

.
a
b
c
c
c
19
Significant Digits
If a number x is written in scientific notation
as x a ? 10n 1 ? a lt 10 , n an
integer then the number of significant digits in
x is the number of digits in a. The number of
significant digits in a number with no decimal
point if found by counting the digits from left
to right, starting with the first digit and
ending with the last nonzero digit. The number of
significant digits in a number containing a
decimal point is found by counting the digits
from left to right, starting with the first
nonzero digit and ending with the last
digit.    Rounding Calculated Values The result
of a calculation is rounded to the same number of
significant digits as the number used in the
calculation that has the least number of
significant digits.
20
Functions and Graphs
21
The Cartesian Coordinate System
Quadrant
22
The Pythagorean Theorem
In a right triangle, the square of the hypotenuse
is equal to the sum of the squares of the other
two sides.
c
a
c2 a2 b2 or
b
Many applications, such as
23
Distance Between Two Points
y2 y1
x2 x1
24
Circle
Standard Equation of a Circle
P(x, y)
Circle with radius r and center at (h,k)
2
2
2
(
x

h
)
(

y

k
)

r


r
gt 0
C(h, k)
With center at (0,0), this simplifies to
x2 y2 r2
25
Examples
  • (-2, 5) is in which quadrant ?
  • Where is the point (10,0) ?
  • What is the distance between the 2 points above?
  • Where is the mid-point of the segment connecting
    the two points above?
  • Find the equation of the circle with the two
    points above as end-points of its diameter.
  • Quadrant II
  • On the positive x-axis

( 4, 2.5 )
(x - 4)2 (y - 2.5)2 6.52
26
Graphing Utility Screens
An image on the screen of a graphing utility is
made up of darkened rectangles called pixels.
The pixel rectangles are the same size, and do
not change in shape during any application.
Graphing utilities use pixel-by-pixel plotting to
produce graphs.
Image Magnification
to show pixels
The portion of a rectangular coordinate system
displayed on the graphing screen is called a
viewing window and is determined by assigning
values to six window variables the lower limit,
upper limit, and scale for the x axis and the
lower limit, upper limit, and scale for the y
axis.
27
Graphmatica
Zooming In
First click on Options and then on AutoSquare to
turn it off.
Then use the mouse to select the area to zoom in
on and click on Zoom in.
28
Graphmatica
Setting the view window --gt View,
Grid Range
Click here after zooming or changing view window
to return to original grid size.
29
Definition of a Function
30
Functions Defined by Equations
In an equation in two variables, if to each value
of the independent variable there corresponds
exactly one value of the dependent variable, then
the equation defines a function.   If there is
any value of the independent variable to which
there corresponds more than one value of the
dependent variable, then the equation does not
define a function. 
The equation y x2 4 defines a
function. The equation x2 y2 16
does not define a function.
31
Vertical Line Test for a Function
An equation defines a function if each vertical
line in the rectangular coordinate system passes
through at most one point on the graph of the
equation.   If any vertical line passes through
two or more points on the graph of an equation,
then the equation does not define a function.
(A) 4y 3x 8 (B) y2 x2 9
32
Agreement on Domains and Ranges
If a function is defined by an equation and the
domain is not indicated, then we assume that the
domain is the set of all real number replacements
of the independent variable that produce real
values for the dependent variable.   The range
is the set of all values of the dependent
variable corresponding to these domain
values.     The symbol f(x) represents the
real number in the range of the function f
corresponding to the domain value x.
Symbolically, f x ? f(x). The ordered pair
(x, f(x)) belongs to the function f. If x is
a real number that is not in the domain of f,
then f is not defined at x and f(x) does
not exist.
The Symbol f(x)
33
Examples
34
Increasing, Decreasing, and Constant Functions
f
(
x
)
g
(
x
)
5
10
g
(
x
) 2
x
2
x

x
5
5
0
5
5
0
10
5
(a)
Decreasing on (?? ? )
(b) Increasing on (?, ? )
35
Increasing, Decreasing, and Constant Functions
h
(
x
)
p
(
x
)
5
5
h
(
x
) 2
x

x

5
5
0
5
5
5
5
(c)
Constant on (?? ? ) (d) Decreasing on
(?, 0 Increasing on 0, ? )
36
Local Maxima and Local Minima
The functional value f(c) is called a local
maximum if there is an interval (a, b)
containing c such that f(x) ? f(c) for
all x in (a, b).
The functional value f(c) is called a local
minimum if there is an interval (a, b)
containing c such that f(x) ? f(c) for
all x in (a, b).
37
Six Basic Functions
Absolute Value Function
Identity Function
g(x)
f(x)
5
5
x
5
5
x
5
5
g(x) x
f(x) x
5
1.
2.
38
Six Basic Functions
Square Function
Cube Function
h(x)
m(x)
5
5
x
5
5
x
5
5
5
3.
4.
39
Six Basic Functions
Square-Root Function
Cube-Root Function
p(x)
n(x)
5
5
x
5
5
x
5
5
5.
6.
40
Graph Transformations
Vertical Translation y f(x) k   Horizontal
Translation   y f(xh)   Reflection   y
f(x) Reflect the graph of y f(x) in the x
axis   Vertical Expansion and Contraction   y
A f(x)
k gt 0 Shift graph of y f(x) up k units k lt 0
Shift graph of y f(x) down ?k? units
h gt 0 Shift graph of y f(x) left h units h lt
0 Shift graph of y f(x) right ?h? units
A gt 1 Vertically expand graph of y f(x) by
multiplying each ordinate value by A 0 lt A lt
1 Vertically contract graph of y f(x) by
multiplying each ordinate value by A
41
Linear and Quadratic Functions
42
Linear and Constant Functions
A function f is a linear function if   f(x)
mx b m ? 0   where m and b are real
numbers. The domain is the set of all real
numbers and the range is the set of all real
numbers. If m 0, then f is called a
constant function   f(x) b   which has the
set of all real numbers as its domain and the
constant b as its range.
2-1-13
43
Slope of a Line
x1 ? x2

(x 2, y 1)
2-1-14
44
Geometric Interpretation of Slope
2-1-15
45
Equations of a Line
Standard form
Ax

By

C
A
and
B
not both 0
Slope-intercept form


Slope

intercept
y
mx
b
m
y
b
Point-slope form
y

y

m
(
x

x
)
Slope
m
Point (
x
,
y
)
1
1
1
1
Horizontal line

Slope 0
y
b


Vertical line
x

a
Slope Undefined
2-1-16
46
Inequality Properties
An equivalent inequality will result and the
sense will remain the same if each side of the
original inequality   1. Has the same real
number added to or subtracted from it or 2.
Is multiplied or divided by the same positive
number.   An equivalent inequality will result
and the sense will reverse if each side of the
original inequality   3. Is multiplied or
divided by the same negative number.   Note
Multiplication by 0 and division by 0 are not
permitted.
2-2-17
47
Completing the Square
To complete the square of the quadratic
expression   x2 bx   add the square of
one-half the coefficient of x that is,
add   or   The resulting expression
can be factored as a perfect square   x2
bx
2-3-18
48
Properties of a Quadratic Function
Given a quadratic function f(x) ax 2 bx
c, a ? 0, and the form f(x) a (x h) 2
k obtained by completing the square
(h, k)
(h, k)
1. The graph of f is a parabola. 2. Vertex (h,
k) parabola increases on one side of vertex and
decreases on the other. 3. Axis (of symmetry) x
h (parallel to y axis) 4. f(h) k is the
minimum if a gt 0 and the maximum if a lt
0 5. Domain All real numbers Range (?,k if a
lt 0 or k,? ) if a gt 0 6. The graph of f
is the graph of g(x) ax2 translated
horizontally h units and vertically k units.
2-3-19
49
Basic Properties of the Complex Number System
  • 1. Addition and multiplication of complex
    numbers are commutative and associative.
  • 2. There is an additive identity and a
    multiplicative identity for complex numbers.
  • 3. Every complex number has an additive inverse
    (that is, a negative).
  • 4. Every nonzero complex number has a
    multiplicative inverse (that is, a
    reciprocal).
  • 5. Multiplication distributes over addition.

2-4-20
50
Deriving The Quadratic Formula
If ax2 bx c 0, a ? 0, then
Divide by a Complete the square by adding (b/2a)2
to both sides Factor (left) and find LCD
(right) Combine fractions and take the square
root of both sides Subtract b/2a and simplify
51
The Quadratic Formula
  • If ax2 bx c 0, a ? 0, then

Discriminants, Roots, and Zeros
2-5-21
52
Power Operation on Equations
If both sides of an equation are raised to the
same natural number power, then the solution set
of the original equation is a subset of the
solution set of the new equation.       Extr
aneous solutions may be introduced by raising
both sides of an equation to the same power.
Every solution of the new equation must be
checked in the original equation to eliminate
extraneous solutions.
2-6-22
53
Polynomial Rational Functions
54
Synthetic Division
Quotient
2x3 1x2 2x 5
Dividend coefficients
Quotient coefficients
Remainder
3-1-23
55
Left and Right Behavior of a Polynomial
P(x) anxn an1xn1 . . . a1x a0 , an
? 0
1. an gt 0 and n even Graph of P(x)
increases without bound as x decreases to
the left and as x increases to the right.
2. an gt 0 and n odd Graph of P(x)
decreases without bound as x decreases to
the left and increases without bound as x
increases to the right.
y
x
)
(x)
P(x) ? as x ? P(x) ? as x ?
P(x) ? as x ? P(x) ? as x ?
3-1-24(a)
56
Left and Right Behavior of a Polynomial
P(x) anxn an1xn1 . . . a1x a0 , an
? 0
4. an lt 0 and n odd Graph of P(x)
increases without bound as x decreases to
the left and decreases without bound as x
increases to the right.
3. an lt 0 and n even Graph of P(x)
decreases without bound as x decreases to
the left and as x increases to the right.
y
y
)
x)
x
x
P(x) ? as x ? P(x) ? as x ?
P(x) ? as x ? P(x) ? as x ?
3-1-24(b)
57
Fundamental Theorem of Algebra Every polynomial
P(x) of degree n gt 0 has at least one zero. n
Zeros Theorem Every polynomial P(x) of degree n
gt 0 can be expressed as the product of n linear
factors. Hence, P(x) has exactly n zerosnot
necessarily distinct. Imaginary Zeros
Theorem Imaginary zeros of polynomials with real
coefficients, if they exist, occur in conjugate
pairs. Real Zeros and Odd-Degree Polynomials A
polynomial of odd degree with real coefficients
always has at least one real zero.
3-2-25
58
Rational Zero Theorem
If the rational number in lowest terms, is
a zero of the polynomial   P(x) anxn
an-1xn-1 . . . a1x a0 an ?
0   with integer coefficients, then b must be
an integer factor of a0 and c must be an
integer factor of an.
3-2-26
59
Location Theorem
If f is continuous on an interval I, a and b are
two numbers in I, and f(a) and f(b) are of
opposite sign, then there is at least one x
intercept between a and b. Given an
nth-degree polynomial P(x) with real
coefficients, n gt 0, an gt 0, and P(x) divided by
x r using synthetic division 1. Upper Bound.
If r gt 0 and all numbers in the quotient row of
the synthetic division, including the
remainder, are nonnegative, then r is an upper
bound of the real zeros of P(x). 2. Lower
Bound. If r lt 0 and all numbers in the quotient
row of the synthetic division, including
the remainder, alternate in sign, then r is a
lower bound of the real zeros of
P(x). Note In the lower-bound test, if 0
appears in one or more places in the quotient
row, including the remainder, the sign in front
of it can be considered either positive or
negative, but not both. For example, the numbers
1, 0, 1 can be considered to alternate in sign,
while 1, 0, 1 cannot.
Upper and Lower Bounds of Real Zeros
3-3-27
60
The Bisection Method
Approximate to one decimal place the zero
of P(x) x4 2x3 10x2 40x 90 on the
interval (3, 4).
3.625
3.5625
x
(
(
(
)
)
)
3
4
3.5
3.75
Nested intervals produced by the Bisection
Method
3-3-28
61
Zeros of Even and Odd Multiplicity
  • If P(x) is a polynomial with real coefficients,
    then
  •  
  • If r is a zero of odd multiplicity, then P(x)
    changes sign at r and does not have a local
    extremum at x r.
  • If r is a zero of even multiplicity, then P(x)
    does not change sign at r and has a local
    extremum at x r.
  •  

The bisection method requires that the function
change sign at a zero in order to approximate
that zero. Thus, this method will always fail at
a zero of even multiplicity. Zeros of even
multiplicity can be approximated by using a
maximum or minimum approximation routine,
whichever applies.
3-3-29
62
Analyzing and Sketching the Graph of a
Rational Function
Step 1. Intercepts. Find the real solutions of
the equation n(x) 0 and use these solutions
to plot any x intercepts of the graph of f.
Evaluate f(0), if it exists, and plot the y
intercept.   Step 2. Vertical Asymptotes. Find
the real solutions of the equation d(x) 0 and
use these solutions to determine the domain of
f, the points of discontinuity, and the vertical
asymptotes. Sketch any vertical asymptotes as
dashed lines.   Step 3. Horizontal Asymptotes.
Determine whether there is a horizontal
asymptote, and if so, sketch it as a dashed
line.   Step 4. Complete the Sketch. Using a
graphing utility graph as an aid, and the
information determined in steps 1-3, sketch the
graph.
3-4-30
63
Inverse Functions Exponential and Logarithmic
Functions
64
Operations on Functions
The sum, difference, product, and quotient of
the functions f and g are the functions
defined by (f g)(x) f(x) g(x) Sum
function (f g)(x) f(x)
g(x) Difference function   (fg)(x) f(x)
g(x) Product function   Quotient
function Each function is defined on the
intersection of the domains of f and g, with
the exception that the values of x where g(x)
0 must be excluded from the domain of the
quotient function. The composite of f and g
is the function defined by (f ? g) (x) f
g(x) Composite function The domain of f ? g
is the set of all real numbers x in the domain
of g for which g(x) is in the domain of f.
4-1-31
65
One-to-One Functions
A function is one-to-one if no two ordered pairs
in the function have the same second component
and different first components.
Horizontal Line Test
A function is one-to-one if and only if each
horizontal line intersects the graph of the
function in at most one point.
(a) f(a) f(b) for a ? b
(b) Only one point has ordinate f
is not one-to-one f(a) f is one-to-one
4-2-32
66
Increasing and Decreasing Functions
If a function f is increasing throughout its
domain or decreasing throughout its domain, then
f is a one-to-one function.
(a) An increasing function is always
one-to-one
(c) A one-to-one function is not always
increasing or decreasing
(b) A decreasing function is always
one-to-one
4-2-33
67
Inverse of a Function
If f is a one-to-one function, then the inverse
of f, denoted f 1, is the function formed by
reversing all the ordered pairs in f. Thus, f
1 (y, x) (x, y) is in f To find
the inverse of a function f Step 1. Find the
domain of f and verify that f is one-to-one.
If f is not one-to-one, then stop, since f
1 does not exist. Step 2. Solve the equation y
f(x) for x. The result is an equation of
the form x f 1(y). Step 3. Interchange
x and y in the equation found in Step 2. This
expresses f 1 as a function of x. Step
4. Find the domain of f 1. Remember, the
domain of f 1 must be the same as the range of
f. Check your work by verifying that f 1
f(x) x for all x in the domain of f
, and f f 1 (x) x for all x in
the domain of f 1
4-2-34
68
Exponential Graphs
Basic Properties of the Graph of f(x) bx, b
gt 0, b ? 1   1. All graphs will pass through
the point (0, 1) since b0 1. 2. All graphs
are continuous curves, with no holes or
jumps. 3. The x axis is a horizontal
asymptote. 4. If b gt 1, then bx increases as
x increases. 5. If 0 lt b lt 1, then bx
decreases as x increases. 6. The function f
is one-to-one.
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69
The Number e
e 2.718 281 828 459
p
e
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70
The Exponential Function with Base e
For x a real number, the equation f(x) ex
defines the exponential function with base e.
The graphs of y ex and y e x are shown
in the figure.
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71
Exponential Growth and Decay
Description Equation Graph
Uses
Short-term population growth (people, bacteria,
etc.) growth of money at continuous compound
interest Radioactive decay light
absorption in water, glass, etc. atmospheric
pressure electric circuits
y cekt c, k gt 0 y cekt c, k gt 0
Unlimited growth Exponential decay
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72
Exponential Growth and Decay
Description Equation Graph
Uses
y c(1 ekt ) c, k gt 0
Learning skills sales fads company growth
electric circuits Long-term population
growth epidemics sales of new products
company growth
Limited growth Logistic growth
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73
Logarithmic Function with Base 2
y
10


1
f
y
5
x
2

or

y
log2x

x

5
10
5
5

1
DOMAIN of
f
(? , ? ) RANGE of
f
1

RANGE of
f
(0, ? ) DOMAIN of
f
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74
Properties of Logarithmic Functions
If b, M, and N are positive real numbers, b ?
1, and p and x are real numbers, then
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75
The Decibel Scale
The decibel level D of a sound of intensity I
, measured in watts per square meter (W/ m2) is
given by     where I0 1012 W/ m2 is the
intensity of the least audible sound that an
average healthy person can hear.   Sound
Intensity, W/ m2 Sound   1.0 ? 1012 Threshold
of hearing 5.2 ? 1010 Whisper 3.2 ?
106 Normal conversation 8.5 ? 104 Heavy
traffic 3.2 ? 103 Jackhammer 1.0 ? 100
Threshold of pain 8.3 ? 102 Jet plane with
afterburner
76
The Richter Scale
The magnitude M on the Richter scale of an
earthquake that releases energy E , measured in
joules, is given by where E0 104.40
joules is the energy released by a small
reference earthquake.
Magnitude on Richter scale Destructive power
M lt 4.5 Small 4.5 lt M lt
5.5 Moderate 5.5 lt M lt 6.5 Large 6.5 lt M
lt 7.5 Major 7.5 lt M Greatest
77
Change-of-Base Formula
78
Additional Topics in Analytical Geometry
79
Conic Sections
Circle Ellipse
Parabola Hyperbola
80
Standard Equations of a Parabola with Vertex at
(0, 0)
1. y2 4ax Vertex (0, 0) Focus (a,
0) Directrix x a Symmetric with
respect to the x axis. Axis the x axis
a lt 0 (opens left) a gt 0 (opens right)
2. x2 4ay Vertex (0, 0) Focus (0,
a) Directrix y a Symmetric with
respect to the y axis. Axis the y axis
a lt 0 (opens down) a gt 0 (opens up)
81
Standard Equations of an Ellipse with Center at
(0, 0)
Note Both graphs are symmetric with respect to
the x axis, y axis, and origin. Also, the major
axis is always longer than the minor axis.
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82
Standard Equations of a Hyperbola with Center at
(0, 0)
2
2
x
y
1.



1
2
2
a
b
intercepts
(vertices)
x
a
y
intercepts none
Foci
(

, 0)
(
, 0)
F'
c
F
c
2
2
2
c

a

b
Transverse axis length 2
a
Conjugate axis length 2
b
2
2
y
x
2.



1
2
2
a
b
intercepts none
x
y
intercepts
a
(vertices)
Foci
(0,

)
(0,
)
F'
c
F
c
2
2
2
c

a

b
Transverse axis length 2
a
Conjugate axis length 2
b
Note Both graphs are symmetric with respect to
the x axis, y axis, and origin.
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83
Standard Equations for Translated ConicsI
(x h)2 4a(y k) Vertex (h, k) Focus (h, k
a) a gt 0 opens up a lt 0 opens down
Parabolas
Circles (x h)2 (y k)2 r2 Center (h,
k) Radius r
(y k)2 4a(x h) Vertex (h, k) Focus (h a,
k) a lt 0 opens left a gt 0 opens right
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84
Standard Equations for Translated ConicsII
Ellipses
7-4-73(a)
85
Standard Equations for Translated ConicsII
Hyperbolas
7-4-73(b)
86
Projectile Motion
x (v0 cos ?) t y a0 (v0 sin ?) t
4.9 t2
7-5-74
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