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Pre-Calculus

Basic Algebra Review

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

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

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

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

)

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

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

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, ?

Interval Notation--Exercises

- (- ?, -3) ? ( 3, ?)
- (-2, -1) ? (1, 2)
- (1, 3) ? (2, 7)
- (-2, 0) ? -1, 2

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

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)

Rational Exponent/ Radical Conversions

Properties of Radicals

Simplified (Radical) Form

Examples

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

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

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.

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

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.

Functions and Graphs

The Cartesian Coordinate System

Quadrant

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

Distance Between Two Points

y2 y1

x2 x1

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

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

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.

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.

Graphmatica

Setting the view window --gt View,

Grid Range

Click here after zooming or changing view window

to return to original grid size.

Definition of a Function

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.

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

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)

Examples

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 (?, ? )

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, ? )

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).

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.

Six Basic Functions

Square Function

Cube Function

h(x)

m(x)

5

5

x

5

5

x

5

5

5

3.

4.

Six Basic Functions

Square-Root Function

Cube-Root Function

p(x)

n(x)

5

5

x

5

5

x

5

5

5.

6.

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

Linear and Quadratic Functions

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

Slope of a Line

x1 ? x2

(x 2, y 1)

2-1-14

Geometric Interpretation of Slope

2-1-15

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

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

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

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

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

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

The Quadratic Formula

- If ax2 bx c 0, a ? 0, then

Discriminants, Roots, and Zeros

2-5-21

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

Polynomial Rational Functions

Synthetic Division

Quotient

2x3 1x2 2x 5

Dividend coefficients

Quotient coefficients

Remainder

3-1-23

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)

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)

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

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

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

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

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

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

Inverse Functions Exponential and Logarithmic

Functions

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

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

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

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

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.

4-3-35

The Number e

e 2.718 281 828 459

p

e

4-4-36

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.

4-4-37

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

4-4-38(a)

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

4-4-38(b)

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

4-5-39

Properties of Logarithmic Functions

If b, M, and N are positive real numbers, b ?

1, and p and x are real numbers, then

4-5-40

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

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

Change-of-Base Formula

Additional Topics in Analytical Geometry

Conic Sections

Circle Ellipse

Parabola Hyperbola

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)

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.

7-2-70

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.

7-3-71

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

7-4-72

Standard Equations for Translated ConicsII

Ellipses

7-4-73(a)

Standard Equations for Translated ConicsII

Hyperbolas

7-4-73(b)

Projectile Motion

x (v0 cos ?) t y a0 (v0 sin ?) t

4.9 t2

7-5-74