# 8: Graphing

• 8.1: Graphing
In this chapter we will introduce readers to the Cartesian coordinate system and explain the correspondence between points in the plane and ordered pairs of numbers. Once an understanding of the coordinate system is sufficiently developed, we will develop the concept of the graph of an equation. In particular, we will address the graphs of a class of equations called linear equations.
• 8.2: The Cartesian Coordinate System
Let’s begin with the concept of an ordered pair of whole numbers.
• 8.3: Graphing Linear Equations

## 1st Grade Magic of Math Unit 8: Graphing Are you looking to hit those Common Core Standards or TEKS during your math block? We have 20 days worth of graphing activities that will have your students thinking critically while graphing away! These activities are hands-on AND engaging for your little learners! Here's what is included:

The Magic Of Math Unit 8 for FIRST GRADE focuses on:

Week One: Pictographs

Week Two: Bar Graphs

Week Three: Tally Charts with Collecting Data

Week Four: Graphing Wrap-Up

Want to know what activities are included? Check out below!:

Week One: Pictographs:

Labeling a Pictograph with Templates

Class Graph: Favorite Holiday

Pictograph Question Stems

Interactive Notebook: Shoe Graph

Class Graph: Favorite Sport

Class Graph: Favorite Instruments

Class Graph: Our Favorite Pizza

Interactive Notebook: Favorite Dinosaur

Class Graph: Favorite Transportation

Week Two: Bar Graphs:

Label a Bar Graph Templates

Class Graph: Favorite Fruit

Pet Shop Bar Graph and Pictograph

Class Graph: Do You have a pet?

Class Graph: Would You Rather

Activity: Build an Alien and Graph

Interactive Notebooks: Spin a Monster

Activity: Greater or Less Than

Week Three: Tally Charts:

Labeling a Tally Chart with Templates

Tally Chart Question Stems

Would You Rather Visit the Ocean or Mountains Class Chart

Tally Chart Activity: Target Toss

Interactive Notebook: Have you been to the circus

Have you ever gone fishing? Class Chart

Add Some Sprinkles Tally Chart Activity

Spin an Animal Interactive Notebook

Which Do You Like Better? Cats or Dog Class Chart

Scoop a Shape Tally Chart Activity

Have You Ever Been To Another Country Interactive Notebook

Which Do You Like Better: Cereal or Oatmeal Class Chart

The Spotted Snake Tally Chart Activity

The Great Food Spin Interactive Notebook

Would You Rather Visit the Farm or the Zoo Class Chart

I Spy Scoot: Tally Chart Activity

Week Word Problem Wrap-Up:

Solving Word Problems with Graphs Class Chart- (2)

Interactive Notebook: Graphing Word Problems- (2)

Graphing Project: Choosing a Topic, Creating a Survey, Collecting Data, Creating Graphs, and Generating Questions

Class Graph Talk: Generating Information Learned from Graphs - (2)

Interactive Notebook: Creating Graphs from Tally Charts and Generating Information Learned (2)

But that's not all, we also include:

- Overview with TEKS and CC

For Other 1st Grade Units Click Below:

What is the Magic of Math?

The Magic of Math is a series of math lesson plans and activities that can be used as your math curriculum or as a supplement to the program that you are already using. We provide the daily lesson plans, word problem, mini-lesson, activity, and interactive notebook entries for four weeks at a time. If you don't have time for it all, that's okay too! You can just pick and choose the pieces that you want to incorporate into your math block!

Are these activities repeats from other units you have?

Absolutely NOT! We create all new activities for each Magic of Math unit. They are all unique to this purchase and are not copy/pasted from things we have previously made.

What is the difference between Magic of Math and other units you have?

Magic of Math is organized as daily lesson plans with everything you need rather than just activities.

WHAT IT IS NOT:

It is NOT a collection of worksheets. Instead we use games, hands-on activities, engaging mini-lessons, and interactive notebooks to build our curriculum. We do try to keep in mind the amount of prep that is involved with the lessons, but some things will need to be printed, cut, and put together to use in your classroom.

Want to know more about Magic of Math? You can find a blog post about it HERE!

## Quick Link for All Graph Paper

Click the image to be taken to that Graph Paper.

##### StandardGraph Paper ##### Single Quadrant 1 Per PageGraph Paper ##### Single Quadrant 4 Per PageGraph Paper ##### Four Quadrant 1 Per PageGraph Paper ##### Four Quadrant 4 Per PageGraph Paper ##### Four Quadrant 12 Per PageGraph Paper ##### Polar CoordinateGraph Paper ##### Logarithmic Graph Paper ##### Horizontal Number LinesGraph Paper ##### Vertical Number LinesGraph Paper ##### Cornell Notes Template ##### Writing Paper ##### Notebook Paper ##### Dot Graph Paper ##### Trigonometric Graph PaperZero to 2 Pi ##### Trigonometric Graph PaperMinus 2 Pi to Plus 2 Pi ## 1/8 Inch Graph Paper This 1/8 inch graph paper is covered with a grid of small squares which are light blue colored. It’s suitable for the classroom, home school or other academic settings where math is taught. This printable 1/8 graph paper PDF has eight squares per inch. Paper size: US Letter. Document width and height: 8.5 by 11 inches.

### More Printable Small Graph Paper Templates ### How To Print This Small Graph Paper Template

After you click the "Download" button, save the archived PDF file on your PC. Once it's finished downloading, unpack the archived PDF and open it in the PDF viewer program or application of your choice. From there on, you can probably find the "Print" option listed under the "File" menu.

For more information, here are the instructions on how to print PDFs using Adobe Reader on Windows and how to print a PDF with Preview on a Mac.

• Lined Paper
• Lined Paper PDF
• Lined Paper Template
• Printable Lined Paper
• College Ruled Paper
• Polar Graph Paper
• Isometric Graph Paper
• Dot Grid Paper
• Hex Paper
• 1 Inch Graph Paper
• Isometric Dot Paper
• Wide Ruled Paper
• Printable Graph Paper With Axis
• Coordinate Graph Paper
• Hexagonal Graph Paper
• Polar Coordinate Graph Paper
• 1/4 Inch Graph Paper
• Centimeter Graph Paper
• 1/2 Inch Graph Paper
• College Ruled Lined Paper Template
• Narrow Ruled Paper
• 3D Graph Paper
• College Ruled Lined Paper
• Wide Ruled Lined Paper
• Handwriting Paper
• Printable Handwriting Paper
• Writing Paper With Picture Box
• 1/8 Inch Graph Paper
• To Do List Templates
• Handwriting Practice Paper
• Printable Battleship Game
• Printable Dot Game
• Blank Sheet Music

I used the 4 squares per inch graph paper template to crochet a bookmark for my adorable niece. It was just what I needed since it's not easy to find this type of paper in stores anymore.

Michelle Bogart
Bozeman, Montana, US

## 8: Graphing

• Digital Notebook
• Practice problems (one per topic)
• Create Study Groups
• Custom Settings
• Full access to solution steps
• Web & Mobile subscription
• Notebook (Unlimited storage)
• Personalized practice problems
• Quizzes
• Detailed progress report
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• Full access to solution steps
• Web & Mobile subscription
• Notebook (Unlimited storage)
• Personalized practice problems
• Quizzes
• Detailed progress report
• No ads

## Deadlock Detection (Cycle In Directed Graph)

In the wait-for graph above, our deadlock detection program will detect at least one cycle and return true.

For this algorithm, we’ll use a slightly different implementation of the directed graph to explore other data structures. We are still implementing it using the adjacency list but instead of an object (map), we’ll store the vertices in an array.

The processes will be modeled as vertices starting with the 0th process. The dependency between the processes will be modeled as edges between the vertices. The edges (adjacent vertices) will be stored in a Linked List, in turn stored at the index that corresponds to the process number. Wait-for graph (a) implementation

• Every vertex will be assigned 3 different colors: white, gray and black. Initially all vertices will be colored white. When a vertex is being processed, it will be colored gray and after processing black.
• Use Depth First Search to traverse the graph.
• If there is an edge from a gray vertex to another gray vertex, we’ve discovered a back edge (a self-loop or an edge that connects to one of its ancestors), hence a cycle is detected.
• Time Complexity: O(V+E)

## 8: Graphing

The following problems illustrate detailed graphing of functions of one variable using the first and second derivatives. Problems range in difficulty from average to challenging. If you are going to try these problems before looking at the solutions, you can avoid common mistakes by carefully labeling critical points, intercepts, and inflection points. In addition, it is important to label the distinct sign charts for the first and second derivatives in order to avoid unnecessary confusion of the following well-known facts and definitions.

Here are instruction for establishing sign charts (number line) for the first and second derivatives. To establish a sign chart (number lines) for f ' , first set f ' equal to zero and then solve for x . Mark these x -values underneath the sign chart, and write a zero above each of these x -values on the sign chart. In addition, mark x -values where the derivative does not exist (is not defined). For example, mark those x -values where division by zero occurs in f ' . Above these x -values and the sign chart draw a dotted vertical line to indicate that the value of f ' does not exist at this point. These designated x -values establish intervals along the sign chart. Next, pick points between these designated x -values and substitute them into the equation for f ' to determine the sign ( + or - ) for each of these intervals. Beneath each designated x -value, write the corresponding y -value which is found by using the original equation y = f ( x ) . These ordered pairs ( x , y ) will be a starting point for the graph of f . This completes the sign chart for f ' . Establish a sign chart (number line) for f '' in the exact same manner. To avoid overlooking zeroes in the denominators of f ' and f '' , it is helpful to rewrite all negative exponents as positive exponents and then carefully manipulate and simplify the resulting fractions.

1. If the first derivative f ' is positive (+) , then the function f is increasing ( ) .

2. If the first derivative f ' is negative (-) , then the function f is decreasing ( ) .

3. If the second derivative f '' is positive (+) , then the function f is concave up ( ) .

4. If the second derivative f '' is negative (-) , then the function f is concave down ( ) .

5. The point x = a determines a relative maximum for function f if f is continuous at x = a , and the first derivative f ' is positive (+) for x < a and negative (-) for x > a . The point x = a determines an absolute maximum for function f if it corresponds to the largest y -value in the range of f .

6. The point x = a determines a relative minimum for function f if f is continuous at x = a , and the first derivative f ' is negative (-) for x < a and positive (+) for x > a . The point x = a determines an absolute minimum for function f if it corresponds to the smallest y -value in the range of f .

7. The point x = a determines an inflection point for function f if f is continuous at x = a , and the second derivative f '' is negative (-) for x < a and positive (+) for x > a , or if f '' is positive (+) for x < a and negative (-) for x > a .

8. THE SECOND DERIVATIVE TEST FOR EXTREMA (This can be used in place of statements 5. and 6.) : Assume that y = f ( x ) is a twice-differentiable function with f '( c )=0 .

a.) If f ''( c )<0 then f has a relative maximum value at x = c .

PROBLEM 1 : Do detailed graphing for f ( x ) = x 3 - 3 x 2 .

## Activities

• 1.2 Equivalent Graphs
Students investigate graphs that are equivalent, in the sense that they represent the same data, though they look different because they employ different scales. (Addresses Graph Literacy Objective 1.2: Understand how zooming, panning, stretching, and shrinking do not change the data within a graph.)

This material is based upon work supported by the National Science Foundation under Grant No. DRL-1256490. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.