This Instructable will show you how to Graph, from Bar Graphs to Exponential functions, I'm hoping to win in Burning Questions 6.5, so if you think this is good, please vote!

First, We'll go through the Basics of Graphing, then build upon that, and go through the different types of graphing, All Types of Graphs besides Histograms, and Circle Graphs are included in this Instructable.

Materials Needed:

Paper (Graphing Paper Ideally)

Pencil

Not Shown, for Obvious Reasons:

Brain

I Apologize in advance for my Sloppy Handwriting.

First, We'll go through the Basics of Graphing, then build upon that, and go through the different types of graphing, All Types of Graphs besides Histograms, and Circle Graphs are included in this Instructable.

Materials Needed:

Paper (Graphing Paper Ideally)

Pencil

Not Shown, for Obvious Reasons:

Brain

I Apologize in advance for my Sloppy Handwriting.

## Step 1: Very Basics

The Very Basics of Graphing are somewhere around here...

First, we need to know our Axis. The Y-Axis goes up and down, the vertical Axis, and the X-Axis goes side to side, the Horizontal Axis. The Axis are commonly referred to by just the Letter, the Y-Axis is often called just "The Y", and the X-Axis is called just "The X"

First, we need to know our Axis. The Y-Axis goes up and down, the vertical Axis, and the X-Axis goes side to side, the Horizontal Axis. The Axis are commonly referred to by just the Letter, the Y-Axis is often called just "The Y", and the X-Axis is called just "The X"

## Step 2: The Basics

The X and Y Axis are lines, so they go on indefinitely, and most higher level Math looks at them as more of a "Plus" Sign. The Center Point, (0,0) is called the Origin. Not (0,0.01), not (0,0.00000001), exactly (0,0), were the two Axis cross.

There are four Quadrants in a Graph, and they start at the top right one, and move Counter/Anti-Clockwise. As you can see, they're named Quadrant I, Quadrant II, Quadrant III, and Quadrant IV. (Roman Numerals for 1,2,3,4). In Quadrant I, the X and Y are both positive numbers. In Quadrant II, the X is negative, and the Y is Positive. In Quadrant III, the X and the Y are negative. And in Quadrant IV, the X is positive, and the Y is not.

There are four Quadrants in a Graph, and they start at the top right one, and move Counter/Anti-Clockwise. As you can see, they're named Quadrant I, Quadrant II, Quadrant III, and Quadrant IV. (Roman Numerals for 1,2,3,4). In Quadrant I, the X and Y are both positive numbers. In Quadrant II, the X is negative, and the Y is Positive. In Quadrant III, the X and the Y are negative. And in Quadrant IV, the X is positive, and the Y is not.

## Step 3: The Less Basic Basics

On the X and Y Axis, put numbers every block, you can go one block is one, or one block is two, However many you like. Technically speaking, is you're going one by one, you don't have to number each line, but I did for ease of explanation.

If you have really high numbers, put a breaking sign directly after zero, and at the top continue counting. These are most commonly used on the Y axis, but it could be used on both Axis.

Points are essential to learn. They're in the order of the X Coordinate, and the Y Coordinate, in alphabetical order. For Example, if you had the Point (3,4), you go over 3 on the X Axis, and up for on the Y Axis. On the other hand, if you have the Point (-3,-4), you go back 3 on the X and down 4 on the Y.

If you have really high numbers, put a breaking sign directly after zero, and at the top continue counting. These are most commonly used on the Y axis, but it could be used on both Axis.

Points are essential to learn. They're in the order of the X Coordinate, and the Y Coordinate, in alphabetical order. For Example, if you had the Point (3,4), you go over 3 on the X Axis, and up for on the Y Axis. On the other hand, if you have the Point (-3,-4), you go back 3 on the X and down 4 on the Y.

## Step 4: Putting It All Together!

Let's Put everything we learned so far Together (Wow, this Instructable is Reminiscent of Beginning French Class...)

And the Picture Below is a Brief Summary of Everything up to this Point.

And the Picture Below is a Brief Summary of Everything up to this Point.

## Step 5: Section 2, Bars and Lines

Now we're going to Review how to Graph Bar Graphs and Line Graphs.

First, the Bar Graph.

Bar Graphs are used for Directly comparing things.

The Setup is like a view of Quadrant I, in most cases, but instead of Numbers on the X Axis, there Are Words. Then, the Y axis has the Numbers. For Example, If I'm Graphing Popular Foods, The Types of Foods would be across the Bottom, and the Number of People who chose the Food would be across the Y. You need to label the X and Y Axies, and give the Graph a Title.

The Bars should all be the Same width, say. 2 squares, but they can/should be different hieghts, depending on the data. In Plain Bar Graphs, the bars shouldn't touch.

To find the Hieght of the Bar, look at your data (I apologize for mine's Inabillity to be read...), and put a line at the number of that specifec piece of Data, the Pizza had 23 Votes, put a line at 23 above the Pizza collum, and turn that line into a Collum, and Shade.

First, the Bar Graph.

Bar Graphs are used for Directly comparing things.

The Setup is like a view of Quadrant I, in most cases, but instead of Numbers on the X Axis, there Are Words. Then, the Y axis has the Numbers. For Example, If I'm Graphing Popular Foods, The Types of Foods would be across the Bottom, and the Number of People who chose the Food would be across the Y. You need to label the X and Y Axies, and give the Graph a Title.

The Bars should all be the Same width, say. 2 squares, but they can/should be different hieghts, depending on the data. In Plain Bar Graphs, the bars shouldn't touch.

To find the Hieght of the Bar, look at your data (I apologize for mine's Inabillity to be read...), and put a line at the number of that specifec piece of Data, the Pizza had 23 Votes, put a line at 23 above the Pizza collum, and turn that line into a Collum, and Shade.

## Step 6: The Line Graph

The Line Graph shows progression of something over time, say, Plant Growth. It shows how one or more things change over time (Multiple Lined Line Graphs).

The Setup is similar to the Line Graph, but on the X Axis is the Measurements of time (Days, Weeks, Quarters, Years), and on the Y Axis is what you're measuring (In, Cm., Revenue, Jobs, etc.).

Using your data, you graph the points, and connect the dots from left to right.

The Setup is similar to the Line Graph, but on the X Axis is the Measurements of time (Days, Weeks, Quarters, Years), and on the Y Axis is what you're measuring (In, Cm., Revenue, Jobs, etc.).

Using your data, you graph the points, and connect the dots from left to right.

## Step 7: Section 3 Y=MX+B

Now we're going into Section 3, Algebraic Graphs, we're going into higher level Graphing. We'll be graphing Equations, inequalities, and Exponential Functions.

The First and Most Important/ Mucho Importante/ Tres Importante Equation is Y=MX+B.

It means that a Coordinate for Y, equals the Slope time X plus the Y-Intercept, An Equation in this form is in Point-Intercept Form.

First, we need to find the Slope. and the symbol for this is M.

The Equation to find the Slope is [Y(Sub-2) Minus Y (Sub-1)] divided by [X (Sub-2) minus X (Sub-1)] This basically means, that out of the Two Points, the second Y coordinate, minus the first y coordinate, divided by the second X coordinate, minus the first X coordinate. Then, we can move on.

The First and Most Important/ Mucho Importante/ Tres Importante Equation is Y=MX+B.

It means that a Coordinate for Y, equals the Slope time X plus the Y-Intercept, An Equation in this form is in Point-Intercept Form.

First, we need to find the Slope. and the symbol for this is M.

The Equation to find the Slope is [Y(Sub-2) Minus Y (Sub-1)] divided by [X (Sub-2) minus X (Sub-1)] This basically means, that out of the Two Points, the second Y coordinate, minus the first y coordinate, divided by the second X coordinate, minus the first X coordinate. Then, we can move on.

## Step 8: The Rest of Y=mx+b

Then, after you find the slope, you solve the rest of the Equation.

But before we go into that, we need to figure out how to get there from Standard Form.

Standard Form is "Ax+By=C" which is not "y=mx+b". To get to this, the first thing you need to do, is subtract the X from both sides, So, if the Equation was 6x+3y=9, you subtract the 6x from both sides, so you have 3y=-6x+9, but it's still not in Point-Intercept Form! In Point-Intercept, the "Y" doesn't have a Coefficient (Number that goes with it), so we have to divide both sides by three, in the example equation. So now, it's y=-2+3, and that's in Point-Intercept form.

Now, Putting it all together!

Ex. Question:

Write, in point-intercept form, and equation for a line that passes through the points (2,2) and (3,5), graph.

Find the slope. 5-2=3 3-2=1 3/1=3 Slope=3

Find the Y-Intercept (B): y=mx+b, 2=3(2)+b, 2=6+b 6-4=2 B=-4

Graph y=3x-4

First, put one point on the y-intercept, where the line will cross the Y-Axis, it will be on the Y-axis, and however high up B is, so for the example equation, ,B=-4 so down 4 on the Y-Axis, to the point (0,-4), and put a point there. Then, using the slope, turn it into a fraction, the slope is 3, so turn it into 3/1. Now, do "Rise/Run" Rise 3 on the Y axis, and go over one on the X-Axis. Connect the two points with a line, and you're done!!

But before we go into that, we need to figure out how to get there from Standard Form.

Standard Form is "Ax+By=C" which is not "y=mx+b". To get to this, the first thing you need to do, is subtract the X from both sides, So, if the Equation was 6x+3y=9, you subtract the 6x from both sides, so you have 3y=-6x+9, but it's still not in Point-Intercept Form! In Point-Intercept, the "Y" doesn't have a Coefficient (Number that goes with it), so we have to divide both sides by three, in the example equation. So now, it's y=-2+3, and that's in Point-Intercept form.

Now, Putting it all together!

Ex. Question:

Write, in point-intercept form, and equation for a line that passes through the points (2,2) and (3,5), graph.

Find the slope. 5-2=3 3-2=1 3/1=3 Slope=3

Find the Y-Intercept (B): y=mx+b, 2=3(2)+b, 2=6+b 6-4=2 B=-4

Graph y=3x-4

First, put one point on the y-intercept, where the line will cross the Y-Axis, it will be on the Y-axis, and however high up B is, so for the example equation, ,B=-4 so down 4 on the Y-Axis, to the point (0,-4), and put a point there. Then, using the slope, turn it into a fraction, the slope is 3, so turn it into 3/1. Now, do "Rise/Run" Rise 3 on the Y axis, and go over one on the X-Axis. Connect the two points with a line, and you're done!!

## Step 9: Inequalities

Inequalities are like equations, but instead of an equal sign, they have a Less Then, Greater Then, Less Then or Equal to, or Greater Then or Equal To, signs (<,>,d,e).

You solve them basically like an Equation, using y=mx+b, or solving from Standard Form.

For Example, 2x+y>1.

Subtract 2x from both sides, so y>-2x+1, and Graph that line like normal, except the line should be dashed, because it's not "...or equal to", if the inequality uses < or >, use a dashed line. Then, pick a point, (0,0) or (1,1) and test it in the equation, 0>-2(0)+1, 0>1 X, so shade the other side from the point you picked, but if the point worked, shade that side.

Drat..... The Less/Greater Then or Equal to Signs Come out as D and E....

You solve them basically like an Equation, using y=mx+b, or solving from Standard Form.

For Example, 2x+y>1.

Subtract 2x from both sides, so y>-2x+1, and Graph that line like normal, except the line should be dashed, because it's not "...or equal to", if the inequality uses < or >, use a dashed line. Then, pick a point, (0,0) or (1,1) and test it in the equation, 0>-2(0)+1, 0>1 X, so shade the other side from the point you picked, but if the point worked, shade that side.

Drat..... The Less/Greater Then or Equal to Signs Come out as D and E....

## Step 10: Exponential Funtions

Exponential functions are when a numbers exponent is a variable, for example, 2

To graph, you need to make a table of the Exponent and it's possible number solutions. For example, if 3

^{x.}To graph, you need to make a table of the Exponent and it's possible number solutions. For example, if 3

^{x, and x=1, the Answer is 3, but if X=-1, the Answer is 1/3, anything to the zero power is one. So, /on the X axis, graph what X equals, -1,0,1,2, etc. Then, on the Y axis, get what the number raised to what X equals equals, and graph the points, eg. (-1,1/3) (0,1) (1,3) (2,9) (3,27). Then connect these dots, and it forms a curve. Exponential Functions Increase or Decrease Rapidly.}## Step 11: Thanks!

Thanks for Reading my Instructable, now you know how to Make a Graph, Graph a Bar Graph, to Graphing Exponential Functions, or, you just needed a brush up, you know now!

Thanks For Reading!!

Thanks For Reading!!

I like your explanations. I think that for demonstration purposes, an ink pen (fine point black marker) and a ruler would be great to use so your examples are clean and clear. Other than that, i think you did a great job.

I completely agree about the choice of pens or why not a softer and darker pencil, anyway you did a pretty good work

Now all you have to do is not skip the steps so that you won't get lost (to all those people who are only beginners in algebra)

Not bad, keep going though (graphing in 3 dimensions, graphing in polar, cylindrical, spherical coordinates)

It's hard to draw a 3-dimensional graph on a piece of paper (unless you roll it into a cone and say it's (x²+y²) = z²). In my experience graphing 3D involves computer programs or a good imagination.<br/>