Math 4 The Final Project

A collaborative review of topics studied

submitted by Math 4 students

Project requirements:

• Each students will post at least two but no more

than 4 pages of individual work • Topics can only be used once and will be posted

on a first come basis on wall wisher at http://www.wallwisher.com/wall/math4project

• Topics chosen must be useful and mathematically

correct (that is part of the assessment) • Topics may be a chapter review or a specific topic

such as writing the equation of a line given two coordinates

• Each page must be submitted as a hard copy and in

electronic form (email or flash drive) • Pages should be colorful and engaging

• All students must use Microsoft Word with 0.5 as

the margin setting • All students must include a ‘tag cloud

WALL WISHER

• wall wisher is a site where you can place a post it note for everyone to see

• http://www.wallwisher.com/wall/math4project

• Your wall wisher post must include your name and

topic • Check wall wisher before you begin your project

Invitation:

Hi Guys n Gals

I have created a Wallwisher wall. It is a simple webpage where we all can post our messages easily.

So simply go to http://www.wallwisher.com/wall/math4project and post your message there

Happy Posting!

Mrs. Muse

TAG CLOUD CREATORS

• These are tools that create clouds of words for text analysis

• Text is copied into the creator and transformed

into a ‘cloud’

• Words used most often are the largest, words used less often are the smallest

• Many creators allow the deletion of specific

words or a maximum number of words

• Many creators allow you to alter the shape and/color of your cloud

• Some Tag Cloud creators that you can use:

(Google these)

• Wordle • Tag Crowd • Word It Out • ABC-Ya • Tagxedo

Math 4 final project notes: • Wall Wisher posting is due May 18, 2010

• Your work must be done in ms word both

electronically and a hard copy. • Your work will be compiled with the work of

others. Be creative. • This is half of your Final Exam Grade – the other

half is an in – school exam. • This project must be completed by June 1st. For

every day late, there will be a 2 point reduction in your grade.

• Your work should include at least one major

topic, mathematical examples, definitions and theorems covered in the course.

• Although your project will be added to an online

magazine, you may present the project as a review guide, story, review questions (include answers!) or a text on the topic chosen. Include graphs when appropriate.

END BEHAVIORS

By: Kevin Brown Definition: A function's values for extreme values of its variable; the value a function, f(x) approaches when x is extremely large or when x is extremely small; the function value, f(x) or y, approximated by the value of another function defined by the curve which is approached but never "reached" by a graph.y =

( x + 6)(x – 7)

Example Problems

EX. 1: When x is extremely large, the values of f(x) are close to the values of y = 0, so, f(x) is positive and approximately 0. When x is extremely small, the values of f(x) are close to the values of y = 0, so, f(x) negative and approximately 0 Which of the following could be the graph of a polynomial whose leading term is "–3x4"?

By

The Twelve Basic Functions

The 12 basic functions consist of identity, squaring, cubing, reciprocal, square root, exponential, natural logarithm, sine, cosine, absolute value, greatest integer, and logistic functions.

The basic functions each are divided up by their characteristics such as domain, range, continuity, behavior in terms of increasing or decreasing, symmetry, bounded ness, local extrema, horizontal and vertical asymptotes, and end behavior.

The first characteristics of all the twelve functions are domain and range.

The domain and range of the functions are as followed:

f(x)= x | domain = (-∞,∞) / range = (-∞,∞)

f(x)= x^3 | domain = (-∞,∞) / range = (-∞,∞)

f(x)= √x | domain = (0,∞) / range = (0,∞)

f(x)= x^2 | domain = (-∞,∞) / range = (0,∞)

f(x)= 1/x | domain = (-∞,0)v(0,∞) / range = (-∞,0)v(0,∞)

f(x)= e^x | domain = (-∞,∞) / range = (0,∞)

f(x)= ln x | domain = (0,∞) / range = (-∞,∞)

f(x)= cos x | domain = (-∞,∞) / range = [-1,1]

f(x)= int(x) | domain = (?,?) / range = (?,?)

f(x)= sin x | domain = (-∞,∞) / range = [-1,1]

f(x)= |x| | domain = (-∞,∞) / range = [0,∞)

f(x)= 1/(1+e^x) | domain = (-∞,∞) / range = (0,1)

Math 4 project Functions and their Properties

By:

Dylan Bajer

Quadratic Functions

The general form of a quadratic is "y = ax2 + bx + c". For graphing, the leading coefficient "a" indicates how "fat" or how "skinny" the parabola will be.

Easiest way to remember if a quadratic is positive or negative. positive quadratic y = x2 negative quadratic y = –x2

The only other consideration regarding the vertex is the "axis of symmetry". If you look at a parabola, you'll notice that you could draw a vertical line right up through the middle, which would split the parabola into two mirrored halves. This vertical line, right through the vertex, is called the axis of symmetry. The vertical line test determines if the equation is a function or not. Every x value must have one y value. If one x value has two y values it’s not a function.

Linear Functions Linear equations are the simplest equations that you'll deal with. One example is 2x-2=6. All you have to so is solve for x. When solving the equation, whatever you do to one side you must do to the other side as well. The linear function formula is y=mx+b.

Polynomial Functions Polynomials are sums of these "variables and exponents" expressions. Each piece of the polynomial, each part that is being added, is called a "term". Polynomial terms have variables which are raised to whole-number exponents (or else the terms are just plain numbers); there are no square roots of variables, no fractional powers, and no variables in the denominator of any fractions. Terms are the number of x’s in an equation like, 4x2+3x-7. The coeffients are the numbers in the equation. When you are solving one of these functions, you are given an x value to plug in for x, and then which you have to solve.

Math 4 Project Vertical Stretch and Shrinks

By: Joshua Ramsey

A shrink is a transformation in which all distances on the coordinate plane are shortened by multiplying

either all x-coordinates (horizontal compression) or all y-coordinates (vertical compression) of a graph by a

common factor less than one. A stretch is a transformation in which all distances on the coordinate plane are

lengthened by multiplying either all x-coordinates (horizontal dilation) or all y-coordinates (vertical dilation) by

a common factor greater than 1. Stretches and shrinks effectively pull the base graph outward or compress the

base graph inward, changing the overall dimensions of the base graph without altering its shape. When a graph

is stretched or shrunk vertically, the x -intercepts act as anchors and do not change under the transformation. A

shrinks and stretches are done quite simply just by changing the x value in the formula: g(x) = k f (x). For a

shrink you use a factor that is greater than zero but less than one and for a stretch you use a factor that is greater

than one. When you shrink a linear equation, you put in the factor at which you want to shrink into the Y1= in

the Y= function and then you then hit graph. From there the calculator will do it for you. It will then either

vertically or horizontally shrink your equation set on the standards at which you set. Then for a stretch it is the

exact same thing. The only thing you need to worry about is if you want to change the y-intercept or the x-

intercept. From there you're done. Included is an example of what happens when you vertically stretch an

equation.

Word Cloud

Andrew Gilliam

Definition

In mathematics real numbers are rational and irrational. Also real numbers are all the numbers which can be expressed as

decimals. A real number can also be called a floating point number.

Interval Description Picture

Closed: [a,b] When the real number line has end points on each side.

Open: (a,b) When the real numbers line does not end points on either side.

Half Open: (a,b] When only one side of the interval has an end point and the other side does not.

Infinite: [a, � ) When one side has and endpoint and

the other side is infinite. Which of the following is real number? A.)5.9 B.)2 Squared C.) All of These D.) 1000

For The Following Draw The Graph “Number Line” and Label The Type Of Interval:

[3,8) - Graph - ____________________ Type:

(8,-3) - Graph - ____________________ Type:

[-1, � ) Graph - ____________________ Type:

Looking At The Graphs Determine The Type And Show The Notation:

° °

<----|----------------|--> Type: Notation:

-2 5 • °

<---|------------------|-> Type: Notation:

-9 -1 •

<-|----------------|-----> Type: Notation:

3 �

By: Jonathan Tirado Definition: Polynomial functions are functions with x as an input variable, made up of several terms, each term is made up of two factors, the first being a real number coefficient, and the second being x raised to some non-negative integer power. Definition of terms and symbols when dividing polynomials: Dividend: f(x) Divisor: h(x) Quotient: q(x) Remainder: r(x) If any of these are constant then variable, rather than function, notation may be used for that value, as in r(x)=5 is also equal to r=5. Example 1: Graph f ( x)= x4 − 10 x2 + 9. f ( x)= x4 − 10 x2 + 9 can be factored

The zeros of this function are −1, 1, −3, and 3. That is, −1, 1, −3, and 3 are the x-intercepts of this function. When x < −3, say, x=−4, then

So for x < −3, f ( x) > 0. When −1 < x < 1, say, x=0, then

So for −1 < x < 1, f ( x) > 0. In a similar way, it can be seen that

Zeros of a function.

The y-intercept of this function is found by finding f (0).

so (0, 9) is a point on the graph. To complete the graph, find and plot several points. Evaluate f ( x) for several integer replacements; then connect these points to form a smooth curve (see Figure 3 ).

Figure 3 Graph of f( x)

Notice that f ( x)= x4 − 10 x2 + 9 has a leading term with an even exponent. The far right and far left sides of the graph will go in the same direction. Because the leading coefficient is positive, the two sides will go up. If the leading coefficient were negative, the two sides would go down. Example 2: Graph f ( x)= x3 − 19 x + 30. f ( x)= x3 − 19 x + 30 can be factored using the rational zero theorem:

f( x) can now be written in factored form and further factored.

The zeros of this function are 2, 3, and −5 (see Figure 4 ).

Transformations By: Jordan Morrison

Translation Shift Formulas of y=f(x) : Shift Vertically Up: y=f(x)+c Shift Vertically Down: y=f(x)-c Shift Horizontally to the Left: y=f(x+c) Shift Horizontally to the Right: y=f(x-c) Dilation Stretch or Shrink Formulas of y=f(x) Vertical Stretch: y=cf(x) Vertical Shrink: y=1/2 f(x) Horizontal Shrink: y=f(cx) Horizontal Stretch: y=f(1/2 x) Reflection Formulas of y=f(x) Vertical Reflection over the x-axis: y= -f(x) Horizontal Reflection over the y-axis: y=f(-x) Examples:

1.) If the formula of the original equation is y=|x| which answer is the equation shifted to the right by 3?

a) The green line: y=abs(x+3) b) The blue line y=abs(x-3)

2.) If the formula of the original equation is y=|x| which answer is the equation vertically stretched by 2?

a) The green line y=2(abs(x)) b) The blue line y=abs((1/2)x)

Data and Statistics Jessica Golombek Data- this term means groups of information that represent the qualitative or quantitative attributes of a variable or set of variables.

-Data can be organized in charts, images and graphs including bar graphs, pie charts, histograms and line graphs.

Bar Graph Pie Chart Histogram Line Graph Scatter Plot- The data is displayed as a collection of points, each having the value of one variable determining the position on the horizontal axis and the value of the other variable determining the position on the vertical axis. Enter all information into STAT in calculator set STAT PLOT and adjust window to display data and graph correctly. ZOOM 9. Line Best Fit: Y=, VARS, 5, enter, zoom 9. Negative Positive

Measures of Central Tendency: -Mean- a set of n numbers is the sum of the numbers divided by n. Also called the average (add up all numbers then divide by all how many numbers were added). -Median- the middle value. EXAMPLE: Kurts grades on his report card are 88, 81,91, 83, and 86. 1.)Arrange the grades in numerical order. 2.)Mark off equall numbers from highest and lowest (81,81,91,88) 3.) Median is 86, which was the middle number. -Mode- the value that appears the most often in a set of data.

EXAMPLE: Sarah’s grades were 90, 88, 88, 85, 84, and 83… the mode is 88 because it appears twice and the other numbers only appear once.

-Range- the difference between the highest and lowest value in a set of data. Standard Deviation- a statistical population, a data set, or a probability distribution the square root of its variance. Can be found on most graphing calculators.

1.) GotoSTATandclearlists.2.) Enternewdataintocalculator.3.) Gotocatalogandscrolldownto“stdDev”4.) Enter.

Standard Deviation Bell Curve- normal distribution, if mean and standard deviation are known then there is access available to every point in the data set. Bell Curve is always symmetrical.

-Definition: A complex number is a number consisting of a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = � 1.

- Two complex numbers are equal if and only if1thier real and imagenary parts are equal

- Complex numbers can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram, representing the complex plane. Examples:

-6, 5i, 5/2i + 2/3i, -2 + 3i, 5 – 3i Sum: ( a+bi ) + ( c+di ) = ( a+c ) + ( b+di ) Difference: ( a+bi ) - ( c+di ) = ( a-c ) + ( b-di ) Distributive: (2 + 3i) • (4 + 5i) = 2(4 + 5i) + 3i(4 + 5i) Division: (4+2i)/(3-i)

SINCERE THANKS TO ALL WHO PARICIPATED AND FOLLOWED THROUGH ON THE

PROJECT SPECIFICATIONS!

YOU HAVE NOW PUBLISHED YOUR WORK ON-LINE AND CAN VIEW IT ON THE ISSY

SITE!

TO THOSE WHO DID ORIGINAL WORK, MAY THE EXPERIENCE OF COMPOSING AND

FORMATTING YOUR PROJECT BE OF VALUE TO YOU!

GOOD LUCK IN YOUR FUTURE ENDEAVORS!