Types of graphs

Here you will learn about types of graphs, including points on the coordinate plane, linear graphs, plotting linear equations, and interpreting linear graphs.

Students first learn about graphs in the early grades when gathering and plotting data on line graphs and circle graphs with their work in measurement and data.

In the 6th grade, students begin to work with the coordinate plane in order to graph ordered pairs with their work in expressions and equations.

What are types of graphs?

Types of graphs are visual representations of mathematical relationships. Types of graphs can include points, straight lines, curves, polygons, and circles, depending on the type of graph that is needed. 

Graphing is a major component to the understanding of mathematics because it provides a visual representation of a mathematical relationship. Let’s look at different types of charts and graphs.

Name of graph	Sample of graph
Pie graph: ● Also known as a circle graph or pie chart ● A graphical representation or visual representation of data that is displayed in a circle. ● The data is typically in percent form.
Pictograph: ● Also known as pictogram ● Displays data in the form of pictures or symbols. ● The pictures or symbols represent the frequency of the data.
Bar Graph: ● Also known as bar chart or column chart ● Displays data points and frequency distributions. ● A stacked bar chart takes the set of data and divides it into different categories.
Line Graph: ● Also known as a line chart or line plot ● Displays continues data over a period of time ● Shows how the data increases and decreases.
Box and whisker plot: ● Also known as a box plot ● Represents numerical data on a number line. ● Displays median, interquartile range, minimum value and maximum value and any outliers.
Coordinate plane: ● Used to represent ordered pairs in the form of (x, y) ● Has horizontal axis called the y -axis and a vertical axis called the x -axis. ● Used to plot algebraic equations such as linear equations, quadratic equations, exponential equations, polynomial equations, etc…
Scatter plot: ● Also known as a scatter graph ● Displays data in the form of ordered pairs on the coordinate plane.
Venn diagram: ● Shows the relationship between data sets.

Coordinate plane and plotting points

Let’s focus on the coordinate plane and how it is used.

\textbf{x} axis and \textbf{y} axis: The x axis and y axis are straight lines that make up the coordinate plane, where each point on the plane can be represented as a coordinate of the form (x,y). ● The x -axis is a horizontal line (left to right). ● The y -axis is a vertical line (up and down). ● The x -axis and y -axis intersect at the origin (0,0).

The x axis and y axis divides the plane into 4 parts, or 4 quadrants.	Points on the coordinate plane are made up of an x -coordinate and a y -coordinate and is written as (x, y) Plotting the point (2, 3) on the coordinate plane

Each quadrant has unique characteristics.

Step by step guide: Coordinate plane

The coordinate plane can be used to find the distance between points.

For example, plot the coordinates A \, (-2, 3) and B \, (6, 3) on the coordinate plane and then find the distance between the points.

Plot the points and then count the units between the points.

Count the number of units from point A to point B. It is 8 units. You can also find the absolute value of the difference to find the distance.

|-2-6|=|-8|=8

Step by step guide: Plot points on a graph.

Linear graphs

Linear graphs are graphs that form straight lines on a coordinate plane. 

For example, the graphs below show linear graphs because they are straight lines.

Linear graphs can represent proportional relationships if they contain the point (0, 0).

If a linear graph is proportional, the points have a common ratio in the form of \, \cfrac{y}{x} \, .

Graph 1 is a proportional relationship because the line passes through the origin, (0, 0) and the points have a common ratio of \, \cfrac{-1}{1} \, .

Graph 2 is not a proportional relationship because the line does not pass through the origin and the points do not have a common ratio.

Step-by-step guide: Linear graphs

Equations can represent linear relationships and be graphed on the coordinate plane.

For example, graph the equation y=\cfrac{1}{2} \, x on the coordinate graph by making a table of values.

The x -coordinate is the independent variable, so you can choose any value you want for x. Substitute the values of x into the equation to find the y -coordinates.

The y -coordinate is the dependent variable because it’s value depends on the x value.

The equation y=\cfrac{1}{2} \, x is a linear graph, and it passes through the origin.

Also, the ratio of the points in the form of \, \cfrac{y}{x} \, is \, \cfrac{1}{2} \, .

\begin{aligned} &(2,1) \, \rightarrow \, \cfrac{1}{2}\\\\ &(-2,-1) \, \rightarrow \, \cfrac{-1}{-2}=\cfrac{1}{2} \end{aligned}

Direct variation

Linear equations in the form of y=kx represent proportional relationships where k is the common ratio, which also represents the constant of proportionality.

The equation y=kx can be called direct variation.

Direct variation is when x, the independent variable, varies directly with y, the dependent variable. This means that the ratio between the x -coordinate and the y -coordinate is constant.

For example, if x and y vary directly, find the equation of direct variation when x = 3 and y = 12.

The x -coordinate and y -coordinate vary directly, which means the ratio of the x and y coordinates is the same or constant.

\begin{aligned} & k=\frac{y}{x} \\\\ & k=\frac{12}{3}=4 \end{aligned}

Since k=4 , the equation of direct variation is y=4x .

Step by step guide: Direct Variation

Graphs can be used to represent real world situations where you may be asked to interpret the graph.

Make sure to look at the key information on the graph. You may be asked to read a value from the graph, interpret a point on the graph, or find a constant of variation (proportionality) from the graph.

For example, the graph represents a proportional, linear relationship where the distance you are from lightning varies directly with the time it takes for you to hear the thunder. Interpret the point (4, 12) from the graph.

Step-by-step guide: Interpreting graphs

What are types of graphs?

Common Core State Standards

How does this relate to 6th grade math, 7th grade math, and 8th grade math?

• Grade 6 – Number System (6.NS.C.6.b)
  Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

• Grade 6 – The Number System (6.NS.C.8)
  Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

• Grade 6 – Geometry (6.G.A.3)
  Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

• Grade 6 – Expressions and Equations (6.EE.C.9)
  Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

• Grade 7 – Ratios and Proportional Relationships (7.RP.A.2.b)
  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

• Grade 7 – Ratios and Proportional Relationships (7.RP.A.2.d)
  Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

• Grade 8 – Expressions and Equations (8.EE.B.5)
  Graph proportional relationships, interpreting the unit rate as the slope of the graph.

How to recognize types of graphs

In order to recognize types of graphs:

1. Identify the graph as a linear relationship or a non-linear relationship.
2. Identify if the linear relationship represents direct variation.

Recognize types of graphs examples

Example 1: recognize the types of graphs

Does the graph represent a linear relationship?

1. Identify the graph as a linear relationship or a non-linear relationship.

The graph is a non-linear relationship because it does not form a straight line.

2Identify if the linear relationship represents direct variation.

The graph is not linear.

Example 2: recognize the types of graphs

Does the graph represent a linear relationship?

Example 3: recognize the types of graphs

Classify each of the graphs as linear or non-linear.

How to plot types of graphs

In order to plot different types of graphs

1. Complete the table of values.
2. Plot the coordinates.
3. Draw a straight line through the points.

Example 4: plot types of graphs

Plot the graph represented by the equation y=x +3.

Complete the table of values.

Substitute the values of x into the equation, y=x +3. Write the values of y in the table.

Plot the coordinates.

Draw a straight line through the points.

Example 5: plot types of graphs

Plot the graph represented by the equation y=3.

Complete the table of values.

In this case the the y value will be -2 for each x value.

Plot the coordinates.

Draw a straight line through the points.

Example 6: plot types of graphs

Plot the graph represented by the equation y=-4x.

Complete the table of values.

Substitute the values of x into the equation, y=-4x. Write the values of y in the table.

Plot the coordinates.

Draw a straight line through the points.

Teaching tips for types of graphs

• Creating and interpreting graphs is an essential skill in mathematics. Graphs help to organize mathematical information while also providing a visual model. Be sure to emphasize these skills in the classroom as they lead to data literacy.

• Using discovery-based activities which provide students with the opportunity to compare and contrast graphs and connect linear equations to the graphs they represent helps students formulate their own understanding.

• Infuse graphs that represent real world situations so that students can connect the math concepts to the world around them.

Easy mistakes to make

• Making arithmetic errors when substituting \textbf{x} values into the equation which produces incorrect \textbf{y} -values

• Plotting a point incorrectly
  If there is a point that does not join up with the other points when you draw a smooth line, go back and check the coordinate. In the example below, there is a point that does not lie on the line so needs checking.
  
  

  

The next lessons are

• Number patterns
• Angles
• 2D shapes
• x and y axis
• Lines

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