Trigonometric Form and deMoivre's Theorem

A complex number can be expressed in three ways.

The relationship between the three can be summarized with a diagram:

Converting from one form to the other involves using the right triangle to relate the sides to the angle.
Sometimes complex numbers are easier to work with in trigonometric form. For example, to multiply, divide, or take a power, it is often easier to convert to trig form and apply:


The third property above can be used to take the nth root of a complex number.

For
For example:

Trig Identities - Day 2

And now it's time to learn how to verify trig identities! - a.k.a. TRIG PROOFS! (insert ominous music here)

To do this you are given two things that are already equal to each other but are still forced to prove that they are equal through the use of tiny baby steps. The end goal of any trig identity verification is to make the left side of the equation and the right side of the equation look the same.

HOWEVER, this must be done with extreme caution, for ONLY ONE SIDE OF THE EQUATION CAN BE CHANGED. If you ever even think of messing around with the other side of the equation, or even going so far as to CROSS-MULTIPLY, you will be promptly cast into the fearsome pit of trigonometric DOOM!!!!

In order to become a trig proof master, one must follow THE RULES (Actually, they're more like guidelines):

1. Don't lose track of what you are trying to prove. Randomly swapping in trig expressions for other trig expressions doesn't usually work. A better plan would be to look at what the end result should be before diving in. Verifying trig identities is not guess and check. (For an example of guess and check, see 'finding the roots of a polynomial.')

2. Work on making the more complicated side less complicated (or rewrite everything in terms of sine and cosine). Making something more complicated than it already is is generally a waste of time.

3. Work from minor functions (csc, sec, cot, tan) to major functions (sin and cos). This makes sense when you realize that sin and cos are a lot less complicated. (See rule #2)

4. Use the common denominator. When you find a common denominator, you can put two fractions together to form one fraction. And, believe it or not, one fraction is in fact less complicated than two fractions (See rule #2). I know, it's weird, but 1 really is less than 2.

5. Use conjugates. Remember how, in the first section of this post, sin θ was equal to 1 / csc θ? Guess what? It still is! And guess what else? In general, sin θ is a lot less complicated than 1 / csc θ. (Rule #2) Is this starting to sound familiar?

6. Use

Did you notice that the left hand side is one fraction and the right hand side is two? And did you notice that they all have a common denominator? Does anyone remember rule #4?

7. Use the Force... just checking to see if anyone was paying any attention.

8. Try not to take the square roots of things or raise them to powers. Remember how we're only supposed to change one side of the equation? If you suddenly make one side exponentially bigger or smaller, what are you supposed to do with the other side to make them match up?

9. Remember:

Last time I checked, trying to put two fractions together by finding a common numerator was not considered to be a very good idea. People who do this are getting way too overexcited about rule #4.

10. A trig proof is like an Oreo cookie. There is always more than one way to verify a trig identity. If your first attempt fails miserably, don't be afraid to go back and try something new!

11. DO NOT CROSS-MULTIPLY! DO NOT ASSUME THE LEFT HAND SIDE EQUALS THE RIGHT HAND SIDE! DO NOT MOVE STUFF FROM THE LEFT SIDE TO THE RIGHT SIDE! DO NOT MOVE STUFF FROM THE RIGHT SIDE TO THE LEFT SIDE! DO NOT PASS GO! DO NOT COLLECT $200! JUST LEAVE THE OTHER SIDE ALONE!

Now let's try asample identity:

Rational Functions!

DEFINITION: a rational function is a polynomial divided by a polynomial

Examples:

AND
(corresponding to graphs seen below)

* Rational functions often have guidelines called ASYMPTOTES (shown above in red)

• The first graph (see above) has a vertical asymptote of x=0 and a horizontal asymptote of y=0.
• The second graph has two VAs and one HA.
  
  

To FIND vertical asymptotes: set the denominator equal to zero.

Ex: set x=0 ------ VA (x=0) ---- Domain is All Reals except 0

Ex: set to 0 ------ VA (x= +/- 2) ---- Domain: Reals except +/- 2

Ex: set to 0 ------ NO VA ---- Domain: All Reals

Algebraic Rules for horizontal asymptotes (HA) for the rational function:

1. If n is less than m, then the HA is y=0 (x-axis)
2. If n=m, then the HA is y= a/b
3. If n>m, then there is no HA

Examples:

1. ------ 1<2 y="0

2. ------ 2=2 (n=m) ---- HA: y= 2/3

3. ------ 3>2 (n>m) ---- NO HA

REVIEW: Roots, Y-intercept, x/- Intervals, Symmetry

1. Roots: set numerator equal to zero ---- ---- ---- x= +/- 1

2. Y-intercept: sub in zero ---- = = ------ (0, 1/4)

3. +/- Intervals: breakpoints (roots and discontinuities) {+/-1, +/-2}

VA: x= +/- 2

Then...
Test f(x) in each interval with a number line numbered from (in this case) -2 through 2

4. Symmetry: f(-x) = f(x) ---- Even (about y-axis)

5. VA: x= +/-2 HA: y=1

6. Sketch:

The EFFECT Of Multiplicity of Roots on Graph at VA's

is the same as

VA: ------ x=0, 4 (mult 2) *****

HA: 3=3 ---- 1/1 ---- y=1

Roots: ---- x= -2 and x=2 (mult 2)

Number line...

AND

The Actual Graph:

Common Factors in Numerator and Denominator ---- Hole in the Graph

Graph will be the same shape as except undefined at x=2

On calculator table: for x=0 and 2 --- y will show "error"

** Cancel common factor -- shape of graph -- factors that cancel are the hole -- factors that don't cancel are the VA **

Algebraic Rules For Slant Asymptotes

A function f(x) has a slant asymptote iff ("if and only if") the degree of the numerator is exactly one more than the degree of the denominator.

and n-m=1

To sum things up:

• if n is less than m --- y=0
• n=m --- y=a/b
• n-m=1 --- slant
  

The equation of the slant asymptote is the quotient of the numerator divided by the denominator (disregard any remainder)

EXAMPLE:

You do long division just as you would with normal numbers.

• This comes out with a remainder of 27x-26
• Completely disregard the remainder
• The equation on top is the equation of the slant asymptote!

In the end, you get a slant asymptote of y=2x-7

Lastly...

GIVEN GRAPH, Write The Equation

1.

Roots: (1,0)

• Numerator: (x-1)
• VA: x=2
• Denominator: (x-2)

Solution:

2.

Roots: (-1,0) (1,0)

• Num: (x+1)(x-1)=
• VA: x=2, x=-2
• Denom: (x-2)(x+2)=

Solution:

3.

Roots: (0,0) mult 3

• Num:
• VA: x=3, x= -3
• Denom: (x-3)(x+3)=

Solution:

** Sometimes you just have to fool around with it in your calculator- plug in a negative or an integer.

** Just for the record, when a graph "bounces" at a root- the muliplicity of that root is 2.

And now for...

REVIEW PROBLEMS

1. Give the equation of the horizontal asymptote of .

2. Give the equation of the slant asymptote of .

3. Find the equations of all vertical asymptotes of .

4. Write the equation of a rational function which has a horizontal asymptote of y=3/2 and y-intercept of -2.

5. Write the equation of a rational function which has roots 0 and -5 and whose vertical asymptotes are x=+/-2.

6. Given the graph, find the equation of the rational function.

7. Sketch the rational function .

8. Find the correct equation for the graph below that has a SA of y= 1/2x + 5/4 and a VA of x= 1/2

9. Given the equation, , find the horizontal and vertical asymptotes.

10. For the following equation, , find the roots, the y-intercept, vertical asymptotes, horizontal asymptote, +/- intervals, symmetry, and domain.



---------------------------------------------------- Answers below --------------------

Answers:

1. HA: y= -4

2. SA: y= 2x+4

3. VA: x= -3/2 and x=2

4.

5.

6.

7.

8.

9.

VA: x= 5/2 and x= -1

HA: y= 5

10.

Roots:+/-i (no real roots)

Y-int: (0,1)

VA: x= +/- 1

HA: y=1

Intervals: - (less than -1), + (-1-0), + (0-1), - (greater than 1)

Symmetry: y-axis (x=0)

D: (-infinity, -1)(-1,1)(1, infinity)

Conics

Parabolas

Definition: the set of all points (x,y) in a plane that are equidistant from a fixed line, also known as the directrix, and a fixed point, the focus, which is not on the line.

Vertex - the midpoint of the directrix and focus of the parabola

Axis - the line perpindicular to the directrix that passes through the vertex

The graph to the below is a parabola with the vertex at the origin with the equation:



That equation is simply the standard equation
when h and k are set to zero







If you switch x and y, you end up with a graph looking like this:

The Standard Equation of a Parabola:

(opens up/down)

(opens left/right)

p= distance from vertex to focus
focal width: 4p




*How to find "P" if you have a graph:

Find a point on the graph (not the origin) and sub in to the standard equation

Check to make sure you use the correct standard equation, based on the direction the graph opens

Solve for P

HINT: the parabola always opens towards the non-squared term

Ellipses

Definition: An ellipse is the set of all points in a plane, the SUM of whose distances from two foci is constant

Vertices: the two points which the line through the foci intersect

Standard Equation of an Ellipse
When the ellipse has a longer horizontal axis:

When the ellipse has a longer vertical axis:



Finding the Foci:





Eccentricity: measures the ovalness of an ellipse



When e is closer to zero, the ellipse is more circular
When e is closer to one, the ellipse is more elliptical

Hyperbolas

Definition: A hyperbola is the set of all points in a plane, the DIFFERENCE of whose distances from two foci is constant

In the hyperbola, there are two vertices instead of one, and the two curved lines open in opposite directions



The STANDARD EQUATION of a hyperbola is:

(when the hyperbola "branches" open left/right)



(when the hyperbola "branches" open up/down)





(slant asymptote)


Eccentricity: When e is larger, the branches are flatter

When e is smaller, (closer to 1) the branches are more pointed

Circles

Definition: A circle is the locus of all points equidistant from a point

Standard equation:


where (h,k) is the center
and r is the radius

PRACTICE PROBLEMS:

1. Write the equation of the locus of points in a plane tangent to x=3 and equidistant from the point (1, -4)


2. Determine the coordinates of the foci for





3. Write the equation of the hyperbola whose foci are (-5, -2) and (-5, 10) and whose vertices are(-5, 0) and (-5, 8)


A point of the parabola is (6, -1). Find:


a) P
b) vertex
c) focus
d) equation of directrix



5. If the eccentricity of
is 3/4, find b.




6. Find the coordinates of the center and radius of a circle whose equation is





7. Write the equation of the locus of points in a plane equidistant from x = 8 and (2, 4)





8. Find the coordinates of the vertices of the hyperbola







9. Find the eccentricity of an ellipse with c = 4.5 and a = 5.
Does it closely resemble a circle?



10. Write the equation of a hyperbola with vertices (0, ±3)
and asymptotes of y = ±3x

ANSWERS:

2. (-5, 2) and (-5, 6)

4. a) 2; b) (2, -3); c) (2, -1); d) y= -5



6. center: (1, -3)




8. vertices: (2, 2) (2, -10)

9. 9/10; No

MATRIX MADNESS

First off, don't mess this up:

Singular: matrix
Plural: matrices

Fun facts about matrices:)
A Matrix is a rectangular array of real numbers called entries (spreadsheet)

They have dimensions!
rows x columns (order matters)

Ex: Write dimensions of this matrix.

ANS: 2x3

You can add and subtract them!
But be careful kids... they must have the same dimensions

Ex:

ANS: They're incompatible. Gotcha!

Ex:

You can even multiply them together!
-AxB is possible if and only if the number of columns in A equals the number of rows in B

Ex: If A(2x3) and B(3x1), the answer will be (2x1)

Who are you matrix? Show us your identity

Matrix identities are defined, but only for square matrices (nxn)

for a 2x2:

for a 3x3:

Determinants!
ewww... what's that?

Definition:A DETERMINANT is any real number associated with any square matrix
-vertical bars on both sides of the matrix can also denote a matrix

Determinant of a 2x2 Matrix:

Coefficient Matrix Determinant

determinant (A) = IAI= ad-bc


NOTE: The determinant is the difference of the products of the 2 diagonals of the matrix

Ex: Find the determinant of the matrix


Ans: det(A)= 2(2)- 1(-3) = 4+3=7


USING A CALCULATOR!
Knowing this will make your life ten times easier, maybe even 100...

1. Hit 2nd matrix

2. Enter matrix as [A]

3. Arrow over to math and choose determinant feature(choice 1)

Determinant of a 3x3 matrix:
NOTE: its supposed to be a plus sign before a13 it wouldn't let me fix it!

Ex: Find IAI if

Solution:

IAI=0(-1)+2(5)+1(4)=14

Now from this we can tell that determinants are so much fun...but what's even more fun is using them to find a 2X2 inverse!!

Given:
and det A= ad-bc


*switch negate


Guess what? The best part about matrices is... nothing! Haha just kidding! You can use them to solve systems of equations!


Ways:

1) Reduced Row Echelon Form
2) Cramer's Rule
CRAMER'S RULE! because he's just that good!

Let's try an example to show how it works!

2x+3y=17
4x+y= 9


1st: Set up coefficient matrix


x y
IAI= 2-12= -10

2nd: Set up replacement matrices...we know its tedious but keep going, it's worth it in the end!


Replace x column: Replace y column:




3rd- Use Cramer's rule:


Isn't that cool!!!

Applications of Determinants!

You might be thinking where and when will I ever use this information in life?
Well, you will be suprised that there are many real life examples that you can amaze your friends with everyday!

1.)Have you ever randomly felt the urge to know the area of an object?

- well now you can easily find this answer on the spot with the matrix knowledge that you have just obtained!

Here's how:

All you need to do is plug the x values of the points into the first column, the y values into the second column and 1 into the third column of the matrix!

2.) You can also use this same method if three points are collinear if the area=0 and the equation of the line passing through a point!

Just look at all of the amazing things you can do with determinants!

PRACTICE PROBLEMS!

Now that you are matrix masters yourselves, you should be anxious to try some problems on your own!!!

2.) Let A=Find A^2

3.) Find the inverse of: [4 -1]

[-2 1]

Solve for x and y:

8.) Solve using matrices:

A-4B-3C=-1

2A+3B+7C=5

3A-2B+4C=1

9.) Find the area of the triangle with the vertices (-3,1), (3,0), and (5,4)

10.) Show, using determinants, that (-2,-5), (1,-2), and (5,2) are collinear.

ANSWERS!

1.)[9 -7]

[-9 4]

2.)[7 -18]

[9 34 ]

3.) [31 12]

[15 11 ]

4.)[2 -4 -6]

[-1 2 3]

[1 -2 -3]

5.) y=2

x=0

6.)x=6 x=-1

7.) 120, 240, and 0 degrees

8.)A=5 B=3 C=-2

9.) 13=area

10.) yes, collinear because area=0

Some Pointers

Those who have published their (unfinished, I realize) posts so far are coming along nicely. It looks like many of you were able to insert equations and graphs successfully.

Just a few pointers for everyone:

1. Many of you need to delete some whitespace from your posts. Just delete blank lines so there are not so many skipped lines within the text.

2. Posts will be easier to read if they are divided into sections with titles. If the titles are larger and bold, even better.

3. Try to make your review sheet readable to a classmate who hasn't seen the material in a while. This may mean adding some narrative text to your lists of "need to know" formulas and definitions.

Looking good so far! I can't wait to see how this turns out.

Transformations and Functions

Function: A set of ordered pairs in which each input (x) has at most one output (y).
Domain: set of usuable x's
Range: set of usuable y's


Vertical line test: in graph form, if any vertical line intersects the graph twice, it is not a function Odd Function: A function is odd if f (x) = - f (- x).
Even Function: A function is even if f (x) = f (- x).

Domain of a function
1. Polynomial Domain: [,] all reals


2.) restriction: stuff greater than or equal to 0


3.) restriction: stuff does not equal zero

Finding Domain and Range:
ex.
Domian:
set the y (f(x)) equal to 0 and then solve for x


0=

0=4-

= 4

x=2, x=-2

D: -2 x 2

D: [-2,2]

Range:
set the x equal to zero and solve for f(x)

f(x)=

f(x)= 2

R: 0 y 2

R: [0,2]

Graphs of a Function
*Increase as x value increases and y value increases

*Decrease as x value increases and y value decreases

*Constant as x value increases and y value remains the same

Relative Minimum and Maximum Values

-A function increases to its relative maximum value then decreases to its relative minimum then increases again. It also works the other way.

-On TI, maxima and minima can be found with 2nd Trace

ex.




Def: an odd function has the property f(-x)= -f(x). An odd function is symmetric to the origin. (x,y) is on the graph as well as (-x,-y)

Def: an even function has the property f(-x)= f(x). An even function is symmetric to the y-axis. (x,y) is on the graph as well as (-x,y)

Examples: Odd, Even or Neither
1.)

even

2.)

odd


3.)


f(-x)= x^2-x neither

Piecewise Functions: a domain-restricted function

ex. f(x)= x

f(x)= x ("v" shaped graph. Vertex at origin. You should know what this graph looks like)

f(x)= x -x less than 0 (picture graph with -x and x values)
x greater than or equal to 0


Transformations of function graphs
ex. f(x)=x+3

-f(x)= -(x+3)
-f(x)= -x-3 reflection over x-axis

f(-x)= -x+3 reflection over y-axis

3f(x)= 3(x+3) y-value times 3. x value stays the same
3f(x)= 3x+9


f(x-3)+1=... shift graph 3 units to the right and 1 unit up

f(x+1)-4 shift graph 1 unit to the left and 4 units down

Combination of Functions
Let f(x)= x g(x)=x^2

a.) (f+g)(x)
x+x^2
b.) (f x g)(x)
(x)(x^2)
x^3
etc.

Composition of functions

(f (sign looks like little o) g) NOT MULTIPLY!!!

f(x)= 5x-3 g(x)= 2x+5

ex. f(g(5))
g(5)= x-1
g(5)= 5-1
g(5)= 4
f(g(5))= f(4)
f(4)= 2(4)+1
f(4)=9

Decomposition of Functions

f(g(x))= (4x-12)^3

g(x)= 4x-12
f(x)= x^3


PRACTICE PROBLEMS

1. What is the domain and range of the relation y = 2x-3
2. Is the graph of (x-4)^2 - (y-1)^2 = 16 a function?
3. Determine the domain of f(x)= 2x/(x-3)
4. If f(x)= 4^x and g(x)= 2x-5 and f(g(x))=16, find x
5. Reflect the graph of y=x over the y-axis
6. What is the domain and range of y=x^2+2
7. Which of the following isnt a function? (a) y=x+1 (b) y=x^2+3x+6 (c) y=4 (d) x=0
8. Reflect y=3x-6 over the x-axis
9. If f(x)= 5x and g(x)= x^2 and f(g(x))=16, find x
10. What is the domain of f(x) = ?

ANSWERS:
1. All reals, all reals
2. no
3. all reals but x=3
4. x= 3.5
5. y=-x
6. all reals, y> (or equal to) 2
7. d
8. -3x-6
9. 4/5
10. x ≥ 4, x ≠ 8

Polynomials

NOTE: This is Chapter 2, Section 2. Reviewing Chapter 2, Section 1 before coming here might not be a bad idea. For really quick review, go to the bottom section, labeled..."Polynomial Review and Theorems."

A polynomial function of nth degree has at most n roots.

has 2 roots.

has 3 roots.

has 4 roots.

Descartes' Rule of Signs:
Let f(x) be a polynomial function with real coefficients.


The max number of positive real roots of f(x) equals the number of sign changes between consecutive coefficients or less that number by an even integer.

Ex. 4 -> 2 -> 0


The max number of negative real roots equals the number of sign changes between consecutive coefficients of f(-x) or less that number by an even integer.

Ex. 3 -> 1

A variation in sign means that consecutive (nonzero) coefficients have opposite signs.


Ex.
at most 2 positive roots. (sign change between 1x and -4x, and -4x,1x)

at most 1 negative roots. (sign change between -1x and +12)

Rational Roots of Polynomials:
Rational Zero Test:
If


The possible rational roots of f(x) must be in form:

Ex. Find the rational zeros of

Solution: The leading coefficient is 2 and the constant term is 3.

Possible rational zeros:

By synthetic division, you can determine that is a rational zero. So f(x) factors as and you can calculate the rational zeros of f to be

To Determine the Number of Zeros of a Polynomial Function:
The Fundamental Theorem of Algebra and The Linear Factorization Theorem are EXISTENCE theorems; they tell you how many zeros a polynomial has, but not HOW to find them.

The Fundamental Theorem of Algebra: If f(x) is a polynomial of degree n, where n is greater than zero, then f has at least one zero in the complex number system (that would be the realm of imaginary numbers).

Linear Factorization Theorem:

If f(x) is a polynomial of degree n where n is greater than zero, then f has precisely n linear factors where are complex numbers. N zeros may be real or complex, and may be repeated.

Real Zeros of a Polynomial Function:
Don't forget multiplicity:
has two zeros

Real and Complex Zeros of a Polynomial Function:

has the zeros because when factored completely as ,



-->
-->

Though Descartes' Analysis will not immediately identify imaginary roots, it is very helpful in reducing the equation; imaginary roots are generally easily solved for once the real rationals are found using synthetic division (i.e., use the quadratic formula).

Finding the Zeros of a Polynomial Function
To write

as a product of linear factors and list the zeros:

The possible rational zeros are (if you don't understand where this came from, go back and re-read "the rational zero test"). Using synthetic division, you can determine that -2 is a zero and 1 is a repeated zero. SO:


can be factored as
which finally comes out to

as the factored equation and

as the zeros.

Polynomial Review and Important Theorems:

Intermediate Value Theorem: If 2 consecutive x values yield y values of opposite sign (+ -) or (- +), then there is a real root between the x-values.

Remainder Theorem: When f(x) is divided by (x-k), the remainder is the value f(k)

Factor Theorem: If (x-k) is a factor, then k is a root.


Rational Zero Test: The possible rational roots of a polynomial function are


Descartes’ Rule of Signs: For a polynomial function , the number of :
Positive roots = the number of sign changes in f(x) or less that number by an even integer.
Negative roots = the number of sign changes in f(-x) or less that number by an even integer.
An nth degree polynomial equation has n roots.

__________________________________________________________________





Practice Problems

1) Write an equation given the roots

2) Write the equation of a polynomial function whose possible rational roots are:

3) Identify the maximum number of positive real and negative real roots in the equation:

4) Write the equation for the following graph:

5) One root of is -3. Find the other roots.

6) Find the rational and irrational roots of

7) Question: Find the equation of the polynomial graph:

8) Find all roots of

9) Give the equation for this graph:

10) Given equation , find the zeros.





11) Find all roots of

_________________________________________________________________

SOLUTIONS:

1)

2)

3) Max +real roots: 2
Max -real roots: 2

4)

5)

6)

7)

8)

9)

10) Zeros:




11)

Sequences & Series

A sequence is a function whose domain is only natural numbers.

In the general formula, it is common to use n instead of x.

• Sequences can be infinite, so
  Sometimes, an extra term is needed, so precedes
  

Summation Notation:
n=> upper limit

i=> index of summation

1=>lower limit

Geometric Progressions or Geometric Sequences:

• consecutive terms have a common ratio (r)
  

Methods of Representation:

1. Sequence: (list of numbers)

2. n-th term: (formulas)

3. Recursion: (using past terms)

Practice Word Problem:

A man wishes to save money by setting aside one cent the 1st, two cents the 2nd day, four cents the 3rd day and so on, doubling the amount each day. Assuming he does not run out of money, what is the total amount of money he would have saved at the end of 30 days?

• First, you would recognize this as a geometric sequence. Using the equation, , you would plug in to solve for r.


• Then with the given r value, which should be calculated as 2, you would then use the Geometric Sum formula, to find what the sum of the money would be.
• With correct substitution of values, the answer comes out to be $3.00
  

How to find sequence/sum of series on calculators:
LIST MENU (2nd STAT) =>OPS=>seq(formula, variable, lower, upper)=>MATH sum(seq(...))

FORMULAS:
General Sequence:

General Series:

Arithemic Sequence:

nth term of an arithemic sequence:

Finding d:

Arithmetic Sum:

Geometric Sequence: Recursive Definition:

nth term of a geometric sequence:

Finding r:

Geometric Sum:

Infinite Series:

Practice Problems:

1. Find a formula for the nth term of the sequence below.

2. Find the 15th term of the geometric progression below.
2, 6, 18, 54,...

3. How many terms are in the sequence below?
4,12,36,108,...708588

4. The sum of the first n positive odd integers is 1521. Find n.

5. Insert 3 arithmetic means between 19 and 32.

6. The 7th and 10th terms of a geometric sequence are 821 and 6568. Find the 11th term.

7. The seats in a lecture room increase in each row from front to back. The first row has 10 seats. The second row has 11 seats, etc. Find the number of seats in the twelfth row.

8. Using the given information in problem 7, find the number of seats in the room if the twelfth row is the last row. 9. How many terms are in the sequence

10. For the sequence below, determine the next two terms:

1,5,13,29,61,...

Solutions to practice problems

1.

2. 9,565,938

3. 12 terms

4. n=39

5. 22.25, 25.5, 28.75

6. 13,136

7. 21 seats

8. 186 seats

9. 24 terms

10. 125, 253

Admin Priveleges

Hi Everyone -

I may live to regret this, but in order for multiple people to be able to edit the same post, I had to give everyone admin priveleges. Please don't mess with the settings, or edit anyone's post except your own. If you have any questions or comments about this, you know what to do!

3D Graphing

Vocab You Need to Know

3D graph- used using three variables instead of two: X,Y and Z

Trace- the cross-intersection of a 3D graph, and they help graph two variable equations in the 3D plane

Lateral Surface Area- area of just sides (not bases)

Total surface area-sum of area of all sides



General formulas for 3D graphs

Basic Formulas To Keep In Mind

Area of Trapezoid:
Area Regular Polygon:

Area of Circle:

Midpoint:


Volume of a Sphere:

Surface Area of a Sphere:

Volume of a prism:

Graphing a 2 variable equation in 3D

• use the trace to sketch the graph of the 2 variable equation



• stretch the trace in the direction of the missing variable
  

Graphing a 3 variable equation in 3d

• find each intercept (x,0,0) (0,y,0) (0,0,z)



• find the equations for all 3 traces

Finding Intersections

• can be solved algebraically by substitution:
  

ex: and

the intersection is a circle

Review Questions:

1. find and label the shape of the xy trace for the equation:

2. write a possible equation for a graph with the following traces:

xy: no intersection

yz: hyberbola

xz: hyperbola

3. describe the intersections between the graphs of the following equations:

Given: a right prism of height 12 with an equilateral triangle base of side length 4.

4. find Lateral Surface Area (LSA)

5. find Total Surface Area (TSA)

6. find the Volume of the prism

7. Find the coordinates of the midpoint of segment AB if A(-5, 8,-3) and B(3, 4, -3)

8. Write the equation of a cylinder with diameter 10 whose xz trace is a circle

9. Write the equation of a plane that passes through the points: A(0,6,0) B(2,0,0) C(0,0,-7)

10. A plane intersects a sphere at a distance of 5 from the centre, the radius of the circle formed is 12. Find:

a. the radius of the sphere

b. the surface area of the sphere

c. the volume of the sphere

Answers:

1. ellipse

2. xy: no intersection DNE

yz: hyperbola

xz: hyperbola

final: (2cups)

3. cirlce

4. LSA:

5. TSA:

6.Volume:

7. (-1, 6, -3)

8.diameter= 10, so radius=5

9. given x, y, z intercepts for plane

10. a. radius:

special right triangle 5,12, 13

r=13

b. surface area:

=

c.volume:

Exponents and Logs

Exponents

Definition: If n is a positive integer an x is a numerical base, then

(n times)

Exponential Rules




Radicals as Exponents

Exponential Functions

1. Standard form of an exponential function is f(x)= a , where b>0, b1 and a is the y intercept.
   

Red - When x is a negative exponent

Blue - When x is a positive exponent

2. Shifting the Graph:

g(x) = f(x-2) shift 2 right

h(x) = f(x) - 2 shift 2 down

j(x) = -f(x) reflection in the x axis

Radioactive Decay

t = time, m = initial mass, h = half life

Natural Base e

• e is a irrational number and is derived from computation:
  
  
• e can be approximated using the following expression:

• To imput into your calcuter, hit 2nd LN

Compounding Countinuously

Logarithm- An exponent

(log to the base a of x)

Properties of logs




Log base e (ln)

a=ln(a)

Change of Base Formula

Exponential Function
The inverse of y= is y=x

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Example Questions:
Unless other directions are given, solve for x:

1. 


2. 
   
3. 


4. 


5. Expand as a function of individual logs.


6. 


7. Graph


8. 


9. Given, for what value of t will P be greater than 200,000? (round to the thousandth.)
   
10. 
    
    

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Answers
1)
2) x = 119.89
3) x = 2.77
4) x = 1.54
5) 3 ln(x) - (2 ln(y) +5 ln(z))
6) x = 7
7)

8) x = -1, x = 5
9) t = 24.26
10) x = 9

Quadratics

Quadratics

There are two different types of forms.

Form One: f(x) = ax² + bx + c

1. Opens up if a > 0, opens down if a <>

2. Axis of symmetry:

3. Vertex: to find the x value, and then plug that value into the equation to find the y value

4. Y-intercept: (0, c)

5. Roots: Factor ax² + bx + c or use the quadratic formula

Form Two: f(x) = a(x-h)² + k , where a does not equal 0

1. Vertex (h,k)

2. Axis of symmetry x = h

3. Opens up if a > 0, opens down if a < style="font-weight: bold;">

4. Shift: h units horizontally, k units vertically


Finding Min/Max on calculator:

1. Enter equation into Y1

2. Graph

3. 2nd CALC -> 4: Maximum -> Left, Right, Guess

Monomial and Polynomial Functions:

Degree tells how many roots

If the degree is odd and the leading coefficient is positive -> falls to left, rises to right

If the degree is odd and the leading coefficient is negative -> rises to left, falls to right

If the degree is even and the leading coefficient is positive -> rises in both directions

If the degree is even and the leading coefficient is negative -> falls in both directions

*Any factor ( )² is said to have a multiplicity of two

A factor with a multiplicity of two is a "player root". Instead of going through the x-axis like a normal root, "player roots" are tangent to the x-axis.



Polynomial Long Division:

1.Put the polynomial being divided under the division sign and the polynomial being divided by to the left of the division sign.

2.Find how many times the second polynomial fits into the first one by multiplying it and then subtracting that from the first polynomial.

3. If there is no remainder:(a-d are true)
a. The polynomial is a factor.
b. The root(s) are roots of the polynomial.
c. The quotient is also a factor, and the factors of the quotient are as well.
d. The roots of the quotient are roots of the polynomial.

4. If there is a remainder: (e-g are true)
e. The polynomial is not a factor.
f. The root(s) are not roots of the polynomial.
g. f(the root of the dividing polynomial) = the remainder.

5. When dividing a fourth degree polynomial or greater repeat the first two steps with the quotient being the divided polynomial until an answer is reached that cannot be divided further.

example:


Synthetic Division:

Usable when a number is suspected to be a root of an equation to simplify division.

1. The root is put to the left of the upside-down division sign and the coefficients are put to the right and over the division sign. note: if a coefficient is 0, it is included as a placeholder.

2. Begin dividing by taking the first coefficient and bringing it down and then multiplying the root by that coefficient and subtracting that from the second coefficient.

3. This is done until no more coefficients remain.

4. The coefficients are then put back into a polynomial of the degree below it(A polynomial with a variable to the third would become one with a variable to the second and so on)

5. If there is a remainder, the number is not a root of the polynomial and the f(the number divided by) is equal to the remainder.

example:


The Remainder Theorem:

If a polynomial is divided by (x+k) then f(-k) is equal to the remainder.

If a polynomial is divided by (x-k) then f(k) is equal to the remainder.

The Factor Theorem:

A polynomial only has a factor of (x+k) if -k is a root of the equation.

A polynomial only has a factor of (x-k) if k is a root of the equation.

Sample Questions:

1. Find the equation of the parabola whose roots are 3 and -2 and whose y – intercept is 1.

2. Find the equation of the parabola with a vertex is (1,2) and passes through the point (3, - 6).

3. Find the other two roots.

4.
a. What is x³+6x²+15x+6 divided by x+3 equal to?
b. What are all the roots of x³+6x²+15x+6?

5. f(x) = x³+5x²+3x-11, find f(4) using polynomial division or synthetic division.

6. x+3 is a factor of x⁴ -x³-11x²+9x+18, find all roots of this equation.

7. The cost of a product is modeled by the equation: C = 800 - 10x + .25x². Find the value for x and c that will minimize cost (Hint: Find Vertex).

8. Describe the end behavior of 2x³+2 and make a sketch.

9. Roots {-1, 0, 0, 3, 5} Write an equation given these roots.

10. Give the degree of:



Answers:

1. y = (1/6)(x²) - (1/6) x - 1

2. y = -2 (x-1)² + 2

3. -2, 1

4. a.) x²+3x+2 b.) -3, -2, -1

5. 145

6. -1, 2, 3, -3

7. x = 20, c = 700

8. Down to left, Up to Right



9. (x+1)(x²)(x-3)(x-5)

10. 5

Selected Answers Explained:

1. Given roots: find factors and multiply

y = (x-3) (x+2)

y = a(x² - x - 6)

Other information is needed to know which equation, such as a y - intercept (parabola could open up or down)

However, since we know the y - intercept is 1. Just plug in the y - intercept information to find full equation.

y = a(x² - x - 6)

1 = a (0² - 0 - 6)

1 = -6a

a = (-1/6)