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## How to Graph an Equation

Graphing equations is a much simpler process that most people realize. You don’t have to be a math genius or straight-A student to learn the basics of graphing without using a calculator. Learn a few of these methods for graphing linear, quadratic, inequality, and absolute value equations.

## Graphing Linear Equations

**Use the y=mx+b formula.** To graph a linear equation, all you have to do it substitute in the variables in this formula.^{[1]}

- In the formula, you will be solving for (x,y).
- The variable m= slope. The slope is also noted as rise over run, or the number of points you travel up and over.
- In the formula, b= y-intercept. This is the place on your graph where the line will cross over the y-axis.

2 **Draw your graph.** Graphing a linear equation is the most simple, as you don’t have to calculate any numbers prior to graphing. Simply draw your Cartesian coordinate plane.

3 **Find the y-intercept (b) on your graph.** If we use the example of y=2x-1, we can see that ‘-1’ is in the point on the equation where you would find ‘b.’ This makes ‘-1’ the y-intercept.

- The y-intercept is always graphed with x=0. Therefore, the y-intercept coordinates are (0,-1).
- Place a point on your graph where the y-intercept should be.

4 **Find the slope.** In the example of y=2x-1, the slope is the number where ‘m’ would be found. That means that according to our example, the slope is ‘2.’ Slope, however, is the rise over run, so we need the slope to be a fraction. Because ‘2’ is a whole number and a fraction, it is simply ‘2/1.’

- To graph the slope, begin at the y-intercept. The rise (number of spaces up) is the numerator of the fraction, while the run (number of spaces to the side) is the denominator of the fraction.
- In our example, we would graph the slope by beginning at -1, and then moving up 2 and to the right 1.
- A positive rise means that you will move up the y-axis, while a negative rise means you will move down. A positive run means you will move to the right of the x-axis, while a negative run means you will move to the left of the x-axis.
- You can mark as many coordinates using the slope as you would like, but you must mark at least one.

5 **Draw your line.** Once you have marked at least one other coordinate using the slope, you can connect it with your y-intercept coordinate to form a line. Extend the line to the edges of the graph, and add arrow points to the ends to show that it continues infinitely.

### Method 2 Graphing Single-Variable Inequalities

1 **Draw a number line.** Because single-variable inequalities only occur on one axis, you don’t have to use Cartesian coordinates. Instead, draw a simple number line.

2 **Graph your inequality.** These are pretty simple, because they only have one coordinate. You will be given an inequality such as x<1 to graph. To do this, first find ‘1’ on your number line.

- If you are given a “greater than” symbol, which is either > or <, then draw an open circle around the number.
- If you are given a “greater than or equal to” symbol, either > or <, then fill in the circle around your point.

3 **Draw your line.** Using the point you just made, follow the inequality symbol to draw a line representing the inequality. If it is ‘greater than’ the point, then the line will go to the right. If it is ‘less than’ the point, then the line will be drawn to the left. Add an arrow to the end to show that the line continues and is not a segment.^{}

4 **Check your answer.** Substitute in any number to equal ‘x’ and mark it on your number line. If this number lies on the line you have drawn, your graph is accurate.

### Method 3 Graphing Linear Inequalities

1 **Use the slope intercept form.** This is the same formula used to graph regular linear equations, but instead of an ‘=’ sign being used, you will be given an inequality sign. The inequality sign will either be <, >, <, or >.

- Slope intercept form is y=mx+b, where m=slope and b=y-intercept.
- Having an inequality present means that there are multiple solutions.

2 **Graph the inequality.** Find the y-intercept and the slope to mark your coordinates. If we use the example of y>1/2x+2, then the y-intercept is ‘2’. The slope is ½, meaning you move up one point and to the right two points.

3 **Draw your line.** Before you draw it though, check the inequality symbol that is being used. If it is a “greater than” symbol, your line should be dashed. If it is a “greater than or equal to” symbol, your line should be solid.

4 **Shade your graph.** Because there are multiple solutions to an inequality, you must show all possible solutions on your graph. This means you will shade all of your graph above or below your line.

- Choose a coordinate – the origin at (0,0) is often the easiest. Make sure that you note if this coordinate is above or below the line you’ve drawn.
- Substitute these coordinates into your inequality. Following our example, it would be 0>1/2(0)+1. Solve this inequality.
- If the coordinate pair is a point above your line and the answer is true, then you would shade above the line. If the answer to the inequality is false, then you would shade below the line. If the coordinate lies below your line and the answer is true, then you shade below your line. If your answer is false, then shade above our line.
- In our example, (0,0) is below our line and creates a false solution when substituted into the inequality. That means that we shade the remainder of the graph above the line.

### Method 4 Graphing Quadratic Equations

1 **Examine your formula.** A quadratic equation means that you have at least one variable that is squared. It will typically be written in the formula y=ax(squared)+bx+c.

- Graphing a quadratic equation will give you a parabola, which is a ‘U’ shaped curve.
- You will need to find at least three point to graph it, beginning with the vertex which is the centermost point.

2 **Find ‘a,’ ‘b,’ and ‘c’.** If we use the example y=x(squared)+2x+1, then a=1, b=2, and c=1. Each letter corresponds to the number directly before the variable it sits next to in the equation. If there is no number before ‘x’ in the equation, then the variable is equal to ‘1’ because it is assumed that there is 1x.

3 **Find the vertex.** To find the vertex, the point in the middle of the parabola, use the formula -b/2a. In our example, this equation would change to -2/2(1), which equals to -1.

4 **Make a table.** You now know the vertex, -1, which is a point on the x-axis. However, this is only one point of the vertex coordinate. To find the corresponding y-coordinate as well as two other points on your parabola, you will need to make a table.

5 **Make a table that has three rows and two columns.**

- Place the x-coordinate for the vertex in the top center column.
- Choose two more x-coordinates an equal number in each direction (positive and negative) from the vertex point. For example, we could go up two and down two, making the two numbers we fill in the other blank table spaces ‘-3’ and ‘1’.
- You can choose any numbers you want to fill in the top row of the table, as long as they are whole numbers and the same distance from the vertex.
- If you want to have a clearer graph, you can find five coordinates instead of three. Doing this is the same process as above, but give your table five columns instead of three.

6 **Use your table and formula to solve for the y-coordinates.** One at a time, take the numbers you have selected to represent the x-coordinates from your table and insert them into the original equation. Solve for ‘y’.

- Following our example, we could use our chosen coordinate of ‘-3’ to substitute into the original formula of y=x(squared)+2x+1. This would change to y= -3(squared)+2(3)+1, giving an answer of y=4.
- Place the new y-coordinate underneath the x-coordinate that you used into your table.
- Solve for all three (or five, if you want more) coordinates in this fashion.

7 **Graph the coordinates.** Now that you have at least three complete coordinate pairs, mark them on your graph. Draw a connecting them all into a parabola, and you’re finished!

### Method 5 Graphing a Quadratic Inequality

1 **Solve the quadratic formula.** A quadratic inequality uses the same formula as the quadratic formula but will use an inequality symbol instead. For example, it will look like y<ax(squared)+bx+c. Using the complete steps from above in “Graphing a Quadratic Equation,” find three coordinates to graph your parabola.

2 **Mark the coordinates on your graph.** Although you have enough points to make your complete parabola, don’t draw the shape yet.

3 **Connect the points on your graph.** Because you are graphing a quadratic inequality, the line you draw will be a bit different.

- If your inequality symbol was “greater than” or “less than” (> or <), then you will draw a dashed line between the coordinates.
- If your inequality symbol was “greater than or equal to” or “less than or equal to” (> or <), then the line you draw will be solid.
- End your lines with arrow points to show that the solutions extend beyond the range of your graph.

4 **Shade the graph.** In order to show multiple solutions, shade the portion of the graph in which the solution could be found. To find out which part of the graph should be shaded, test a pair of coordinates in your formula. An easy set to use is (0,0). Note whether or not these coordinates lie within or outside of your parabola.

- Solve the inequality with the coordinates you’ve chosen. If we use an example of y>x(squared)-4x-1 and substitute the coordinates (0,0), then it will change to 0>0(squared)-4(0)-1.
- If the solution to this is true and the coordinates are inside the parabola, shade inside the parabola. If the solution is false, shade outside of the parabola.
- If the solution to this is true and the coordinates are outside the parabola, shade the outside of the parabola. If the solution is false, shade inside the parabola.

### Method 6 Graphing an Absolute Value Equation

1 **Examine your equation.** The most basic absolute value equation will appear as y=|x|. Other numbers or variables may be involved though.

2 **Make the absolute value equal to 0.** To do this, make everything in the absolute value lines | | =0. If we use the example y=|x-2|+1, then we get the absolute value by making |x-2|=0. Then the absolute value becomes 2.

- The absolute value is the number of points from |x| to ‘0’ on a number line. So the absolute value of |2| is 2, and the absolute value of |-2| is also two. This is because in both cases, ‘2’ and ‘-2’ are 2 steps away from zero on the number line.
- You may have an absolute value equation where ‘x’ is alone. In that case, the absolute value is ‘0’. For example, y=|x|+3 changes to y=|0|+3, which equals to ‘3’.

3 **Make a table.** You want it to have three rows and two columns.

- Put the first absolute value coordinate in the into the top center column for ‘X’.
- Choose two other numbers an equal distance from your x-coordinate in each direction (positive and negative). If |x|=0, then move up and down an equal number of spaces from ‘0’.
- You can choose any numbers, although ones that are near the x-coordinate are most helpful. They must also be whole numbers.

4 **Solve the inequality.** You need to find the y-coordinate that pairs with the three x-coordinates you have. To do this, substitute the x-coordinate values into the inequality and solve for ‘y’. Fill these answers in on your table.

5 **Graph the points.** You only need three points to graph an absolute value equation, but you can use more if you would like. An absolute value equation will always form a “V” shape on your graph. Add arrows to the ends to show that the line extends further than the edge of your graph

## Equations and Graphs

**Overview:** In this module we review the essentials of interpreting and preparing graphical data and revisit the graphical addition of vectors.

### Skills:

- Determining slopes and y-intercepts from graphs
- Estimating specific data values from graphs

In the sciences, many times it is necessary to be able to interpret graphs as well as be able to graph certain equations. Often data is available in a graphical format and you must be able to extract the necessary information. Other times, it may be helpful to plot an equation in order to fully understand a problem. However, graphing can be difficult for some students. The format in this section is a little different. The first part will be simply showing what some special equations look like in graphical form and the second part will be a series of questions to help you understand graphs better.

This graph shows a straight line and the corresponding equation is y = mx + b. Y is the y value (distance along the y-axis, which is the vertical axis) and x is the x value (the distance along the x-axis, which is the horizontal axis). The slope of the line is m, which is also the rise/run or the Dx/Dy. The y-intercept of the line is b. This is the y value where the line crosses the y-axis (when the x = 0). In the graph here, the slope is 1 and b is +2. Notice that a line crosses each axis only once (at most).

This graph represents a quadratic function, which is y = ax^{2} + bx + c. The parameters a, b and c are constants. In this graph the actual equation is y = 2x^{2} + 3x – 2. Notice that in this graph the function crosses the y-axis once and the x-axis twice. This has to do with the fact that x is squared in the equation and y is not. The function crosses the axis (up to) the same number of times as the power of that value. In this graph x is squared so the line crosses the x-axis up to two times.

This graph shows a cubic equation, which is expressed mathematically as y = ax^{3} + bx^{2} + cx + d. For this particular function, y = x^{3} + 2 x^{2} – 2 x – 3. Because x is raised to the third power, the function crosses the x-axis 3 times. It crosses the y-axis only once however.

This is a log plot. Notice that the x-axis is quite different than in the other graphs. In this graph, if we look at the y-axis we see that the distance from 1 to 10 is the same as that from 10 to 100. But we know that the span from 10 to 100 represents a much larger range of x-values than that from 1 to 10. This is a property of a log plot. Be sure to look at the axis on a log plot (as well as all graphs) in order to understand exactly what the graph is trying to show.

In order to answer the following questions, you need to have a good understanding of the preceding subjects in the tutorial.

## Function Graph

## How to Draw a Function Graph

First, start with a blank graph like this. It has x-values going left-to-right, and y-values going bottom-to-top:

## Plotting Points

A simple (but not perfect) approach is to calculate the function at **some points** and then plot them.

A function graph is the set of points of the values taken by the function.

**Example: y = x ^{2} − 5**

Let us calculate **some points**:

Not very helpful yet. Let us add some **more points**:

Looking better!

We can now guess that plotting **all the points** will look like this:

A nice parabola.

We should try to plot enough points to be confident in what is going on!

Example: y = x^{3} − 5x

With these calculated points:

So “plotting some points” is useful, but **can lead to mistakes**.

## Complete Graph

For a graph to be “complete” we need to show all the important features:

- Crossing points
- Peaks
- Valleys
- Flat areas
- Asymptotes
- Any other special features

This often means thinking carefully about the function.

Example: (x−1)/(x^{2}−9)

## Using Calculus

We can also find Maxima and Minima using derivatives :

## Tools to Help You

- The Function Grapher can help you. Enter the equation as “y=(some function of x)”. You can use zoom to find important points.
- If you can’t write the equation as “y=(some function of x)”, you can try the Equation Grapher, where you enter equations like “x^2+y^2=9” (meaning
**x**).^{2}+y^{2}=9

But remember they are just a help! They are only computer programs, and could easily miss some important thing on the graph, or not plot something correctly.

Note: you may hear the phrase “satisfy the equation”, which means where the equation is **true**.

## Description

It can plot an equation where x and y are **related somehow** (not just y=…), like these:

Examples:

**x^2+y^2=9**(an equation of a circle with a radius of 3)**sin(x)+cos(y)=0.5****2x−3y=1****cos(x^2)=y****(x−3)(x+3)=y^2****y=x^2**

If you don’t include an equals sign, it will assume you mean “**=0**“

It has not been well tested, so **have fun** with it, but **don’t trust it**.

If it gives you problems, let me know.

Note: it may take a few seconds to finish, because it has to do lots of calculations.

If you just want to graph a function in “y=…” style you may prefer Function Grapher and Calculator

## Zooming

Use the zoom slider (to the left zooms in, to the right zooms out).

To reset the zoom to the original bounds click on the **Reset** button.

## Dragging

Click-and-drag to move the graph around. If you just click-and-release (without dragging), then the spot you clicked on will be the new center

Note: the plots use **computer calculations**. Round-off can cause errors or values can be missed completely.

## All Functions

### Operators

+ | Addition operator | |
---|---|---|

– | Subtraction operator | |

* | Multiplication operator | |

/ | Division operator | |

^ | Exponent (Power) operator |

### Functions

sqrt | Square Root of a value or expression. | |
---|---|---|

sin | sine of a value or expression | |

cos | cosine of a value or expression | |

tan | tangent of a value or expression | |

asin | inverse sine (arcsine) of a value or expression | |

acos | inverse cosine (arccos) of a value or expression | |

atan | inverse tangent (arctangent) of a value or expression | |

sinh | Hyperbolic sine (sinh) of a value or expression | |

cosh | Hyperbolic cosine (cosh) of a value or expression | |

tanh | Hyperbolic tangent (tanh) of a value or expression | |

exp | e (the Euler Constant) raised to the power of a value or expression | |

ln | The natural logarithm of a value or expression | |

log | The base−10 logarithm of a value or expression | |

floor | Returns the largest (closest to positive infinity) value that is not greater than the argument and is equal to a mathematical integer. | |

ceil | Returns the smallest (closest to negative infinity) value that is not less than the argument and is equal to a mathematical integer. | |

round | Round to the nearest integer. Examples: round(−2.5) = −2, round(-0.1) = 0, round(0.1) = 0, round(2.5) = 3 | |

abs | Absolute value (distance from zero) of a value or expression | |

sign | Sign (+1 or −1) of a value or expression |

### Constants

pi | The constant π (3.141592654…) | |
---|---|---|

e | Euler’s number (2.71828…), the base for the natural logarithm |

## Equations and their Graphs

✅ Math Formulas ⭐️⭐️⭐️⭐️⭐

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