Reading Graphs

One of the most important skills in mathematics is knowing how to read and interpret a graph, particularly when studying the topic of functions. Often, functions are represented by a diagram or some graph, hence the importance of interpreting the data in front of you in order to analyze it and draw conclusions. Indeed, reading information from a graph or table is not "rocket science". This is an acquired skill, which requires understanding a series of basic rules and practices. In this article, you will find a series of tools that will allow you to "dive" into the topic on the right foot.

What is a graph?

A graph is basically any representation in the coordinate system that allows data to be transmitted in a visual and accessible way (for a more detailed explanation, see the dedicated article "graph")

Types of Graphs

There are several types of graphs:

Discrete graph (for a more detailed explanation, see the dedicated article " Discrete graph") whose most common form is a bar chart.

Discrete graphs are made up of distinct points that are not connected. Each point represents a specific value or data point. To read a discrete graph:

Identify each point on the graph.
Match the $x$ -coordinate (horizontal axis) with the $y$ -coordinate (vertical axis) for each point.
Analyze the pattern or relationship between the points.

Continuous graphs (for a more detailed explanation, see the dedicated article " Continuous graphs"), which as its name implies is a continuous line of values on the Cartesian plane.

Continuous graphs have connected lines or curves, representing data that changes smoothly over a range. To read a continuous graph:

Observe the overall trend or shape of the graph (e.g., increasing, decreasing, or constant).
Locate specific points by matching values on the $x$ - and $y$ -axes.
Understand intervals or areas where the function’s behavior changes.

What is a value table?

A value table (for a more detailed explanation, see the dedicated article "Value Table") generally aims to summarize the data on which the graph is based, whether it is a discrete or continuous graph.

Example of a value table for $X = Y+2$

A value table is a tabular representation of pairs of $x$ - and $y$ -values, showing the relationship between them. To read a value table:

Look at the $x$ -column to find the input values.
Match each $x$ -value to its corresponding $y$ -value in the adjacent column.
Use the table to analyze trends, plot points, or construct graphs.

If you are interested in more information about "graphs" you can find detailed information in the following articles:

Data Collection and Organization - Statistical Research

Graph

Discrete Graph

Continuous Graph

Graphical Representation of a Function

In the Tutorela blog you will find a variety of articles with interesting explanations about mathematics

Examples and practices of reading and interpreting information within a graph

Example No. 1

The following line graph shows the number of students at the university from the moment its doors open.

A5 - graph of number of students and number of hours

Based on the graph, answer the following questions

How many students were at the university after one hour?
How many students were at the university after three hours?
After how many hours were there 500 students at the university?
After how many hours were there 200 students at the university?

Solution:

Before addressing the solution to each of the questions, it is necessary to understand well what each of the axes in front of us represents:

The $X$ axis represents the time in hours from the moment the university opens, meanwhile the $Y$ axis represents the number of students at the university as a function of time in hours.

To solve this task, we will observe the graph and find on the $X$ axis the value $1$ which represents one hour. Now we will draw a dashed line from the value towards the direction of the graph. In the next step, we will draw a dashed line from the point on the graph towards the $Y$ axis and we will see that the value we obtain is $100$ . That is, it means that after one hour there were $100$ students at the university.
We will repeat the action we performed in the previous section. Let's observe the graph and find on the $X$ axis the value $3$ which represents three hours. Now we will draw a dashed line from the value towards the direction of the graph. Next, we will draw a dashed line from the point on the graph towards the $Y$ axis and we will see that the value we obtain is $500$ . That is, after three hours, the university had $500$ students.
In this section, we will have to act in reverse. We will observe the graph and find on the $Y$ axis the value $500$ which represents the number of students. Now we will draw a dashed line from the value towards the direction of the graph. We will find two coinciding points on the graph. In the next step, we will draw a dashed line from each of the points on the graph towards the $X$ axis and it will be seen that the values we obtain are $3$ and $6.5$ . That is, the university had $500$ students at $3$ hours and at $6.5$ hours.
We will repeat the action we performed in the previous section. Let's observe the graph and find on the $Y$ axis the value $200$ which represents the number of students. Now we will draw a dashed line from the value towards the direction of the graph. We will find two coinciding points on the graph. Next, we will draw a dashed line from each of the points on the graph towards the $X$ axis and therefore the values we obtain are $1.5$ and $5$ . That is, it means that the university had $500$ students after $1.5$ hours and after $5$ hours.

Answer:

$100$ students
$500$ students
After $3$ hours and after $6.5$ hours
After $1.5$ hours and after $5$ hours

Exercise No. 2 on reading information from a graph:
The bar chart presented below describes the number of city residents who participated in activities organized by the municipality of Ibarra, Ecuador, by each type of activity performed.

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Exercises on Reading Information from Graphs:

Exercise 1:

Statement

In the graph of the linear function that passes through the points $A(0,7)$ and $B(-4,-9)$

Find the slope of the graph.

Solution

$m=\frac{y_2-y_1}{x_2-x_1}$

We replace according to the existing data

$\frac{-9-7}{-4-0}=$

$\frac{-16}{-4}=4$

Answer

$4$

Exercise 2:

Statement

A graph of a linear function passes through the points $A\left(\frac{1}{2},5\right),B(5\frac{1}{2},10)$

Solution

We use the slope formula

$m=\frac{y_2-y_1}{x_2-x_1}$

We replace the data accordingly

$\frac{10-5}{5\frac{1}{2}-\frac{1}{2}}=\frac{5}{5}$

If the slope is equal to $1$ , then the slope is positive and the function is increasing

Answer

Increasing function

Exercise 3:

Statement

The graph of the linear function passes through the points $A(6,5), B(0,5),$

Is the function: increasing, decreasing or constant?

Solution

Answer

The function is a constant function.

Exercise 4:

Statement

The graph of the linear function passes through the points $A(2,0), B(0,0)$

Solution

Answer

Constant function

Exercise 5:

Statement

At what point does the graph of the first function intersect with the graph of the second function II?

Solution

At what point does the graph of the first function I intersect with the graph of the second function II

Answer

$4,2$

review questions:

How to interpret the results of statistical graphs?

A statistical graph will show us quantitative data that was collected, therefore it is very important to identify what type of graph we are working with, these can be: a bar diagram, percentage graph, statistical graphs, histogram graph and hence the importance of knowing how to read and interpret the information that the graph is giving us, whether on the $X$ axis and on the $Y$ axis, in the case of a bar diagram or a histogram, in order to analyze and interpret them and thus reach a conclusion.

How to interpret a bar graph?

First of all, we must locate the $X$ axis and the $Y$ axis, then we will observe what data these indicate, and in this way be able to read the data and relate it to the data of each axis and reach a conclusion.

Bar graph example

Assignment

In the following bar diagram, the heights of $20$ students are represented

Solution:

We can observe in the graph that on the $X$ axis the obtained heights are represented and on the $Y$ axis we can observe the number of students who recorded those heights

For example, we can observe that the height of $1.48$ m if we draw a line upwards we can see that it reaches on the $Y$ at the number $5$ . This can be interpreted as: $5$ students recorded that height.

On the $X$ the height of $1.55$ m can be related to the number $8$ on the $Y$ axis, this means that $8$ students have the height of $1.55$ m. and in this way, we can continue interpreting the data provided by the graph.

How is a graph read?

First, we must determine what type of graph is being presented to us, once this is identified, for example, if it is a histogram, we should observe what data we have on the coordinate axes and analyze them to interpret said data.

How is a line graph interpreted?

Normally a line graph will indicate changes that are usually occurring over time and distance, so first, we should observe what data is presented to us on the coordinate axes in order to interpret the data that will be studied.

What should we take into account in bar graphs?

To be able to analyze the data in a bar graph, it is important to take into account the data that is being provided on the $X$ axis and on the $Y$ axis, and thus interpret the information provided to us in a better way.