Graphical Representation of a Function

As we learned in an article on functions, the standard "correspondence rule" is a connection between a dependent variable $(Y)$ and an independent variable $(X)$ .

By means of a graph or drawing, which gives a visual aspect to the concept of the function. From the graph it is possible to understand whether it is a linear function (straight line), a quadratic function (parabola) and more.

Remember that when it comes to a graphical representation of a function, each point in the domain $X$ will always have only one point within the range $Y$ . Therefore, not every drawing is a graphical representation of a function. Here is an example.

A1 - Graphical representation of a function

Detailed explanation

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Is the given graph a function?

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

Graphical representation of a function

Collecting and Organizing Data - Statistical Research

Reading information from graphs

Graphing

Discrete graph

Continuous graph

Functions for seventh grade

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

Exercises on the graphical representation of a function

Exercise 1

Task

Given the function:

$x=y-4+2x$

Through which of the following points does the graph of the function pass?

Solution

$x=y-4+2x$

$y=x-2x+4$

$y=-x+4$

$-1\cdot\left(-1\right)+4=5$

Answer

$\left(-1,5\right)$

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Question 1

Is the given graph a function?

Question 2

Is the given graph a function?

Question 3

Is the given graph a function?

Exercise 2

Task

Given the line whose slope is $6$ and passes through the point $(6,41)$ .

Which of the algebraic representations corresponds to the given line?

Solution

$y=mx+b$

$m=6$

$\left(6,41\right)$

Replace accordingly

$41=6\cdot6+b$

$41-36=5=b$

$y=6x+5$

$y=2\left(3x+2\frac{1}{2}\right)$

Answer

$y=2\left(3x+2\frac{1}{2}\right)$

Exercise 3

Task

Given the line whose slope is $2$ and passes through the point $(3,7)$ .

Which of the algebraic representations corresponds to the given line?

Solution

$y=mx+b$

$m=2$

$\left(3,7\right)$

Replace accordingly

$7=2\cdot3+b$

$7=6+b$

$1=b$

$y=2x+1$

Answer

$y=2x+1$

Do you know what the answer is?

Question 1

Is the given graph a function?

Question 2

Determine whether the given graph is a function?

Question 3

Does the graph below represent a function?

Exercise 4

Task

Given the straight line parallel to another straight line

$y=2x+5$

passing through the point $(4,9)$

Which of the algebraic representations corresponds to the given line?

Solution

$y=2x+5$

Parallel to the line

$m=2$

$\left(4,9\right)$

Replace accordingly

$9=2\cdot4+b$

$9=8+b$

$1=b$

$y=2x+1$

Answer

$y=2x+1$

Exercise 5

Task

Choose the correct answer

a.The graph passes through $(3,5)$

b.The graph passes through $(3,7)$

c.The graph passes through $(5,4)$

d.The graph passes through $(-1,6)$

Answer

The correct answer is a. The graph passes through $(3,5)$

Check your understanding

Question 1

Determine whether the following table represents a constant function:

Question 2

Determine whether the following table represents a function

Question 3

Determine whether the following table represents a constant function

Review questions

What is a graphical representation of a function?

As we know a function can be represented in different ways: verbally, algebraically, table of values and graphically. The last representation means that a function can be graphed in the Cartesian plane, according to the values obtained in the table, this representation can be observed as a straight line, a parabola, among others, depending on the type of function we are working on.

How is a function represented graphically?

Let's see an example of how a function should be represented graphically.

Example:

Represent the following function in a graph

Task

Graph the following function $Y=X-1$

Solution

We are going to give values to $X$ , to find the value of $Y$ , according to the correspondence rule, and we are going to register these values in a table:

First, we are going to input the values of $X$ To get the values that correspond to $Y$ , the algebraic expression of this function is:

$Y=X-1$

Then,

When $X=-4$

$Y=-4-1=-5$

When $X=-3$

$Y=-3-1=-4$

When $X=-1$

$Y=-1-1=-2$

When When $X=0$

$Y=0-1=-1$

When $X=2$

$Y=2-1=1$

When $X=5$

$Y=5-1=4$

According to this data now we are going to input it in the table

Once we have the values in the table, we are going to look for these pairs of coordinates in the Cartesian plane, where we are going to find points and connect them as follows to obtain the final graph of the function

Graph:

According to these points that were located in the Cartesian plane, we can observe that a straight line was drawn, which means that the function is linear.

What are the types of graphical representation of a function?

There are many graphs of functions, this is according to the type of function that is being graphed, among the most common are:

Graph of a constant function

Graph of a linear function

Graph of a quadratic function

Graph of a cubic function

Graph of an exponential function

Do you think you will be able to solve it?

Question 1

Determine whether the following table represents a function

Question 2

Determine whether the data in the following table represent a constant function

Question 3

Determine whether the following table represents a linear function

Examples with solutions for Graphical Representation of a Function

Exercise #1

Is the given graph a function?

Video Solution

Step-by-Step Solution

To determine if the graph is a function, we will use the Vertical Line Test.

The Vertical Line Test states that a graph represents a function if and only if no vertical line intersects the graph at more than one point.

Let's apply this test to the given graph, where a horizontal line is drawn. This line represents the function the graph should be verified against.

Step 1: Conceptualize vertical lines passing through different x-values across the domain of the graph.
Step 2: Observe if any of these vertical lines intersect the graph at more than one point.

Upon inspection of the graph, we see that every vertical line intersects the graph at exactly one point.

This indicates that for every input (x-value), there is a unique output (y-value), fulfilling the criteria for the definition of a function.

Therefore, according to the Vertical Line Test, the given graph is indeed a function.

The correct choice is: Yes

Answer

Yes

Exercise #2

Is the given graph a function?

Video Solution

Step-by-Step Solution

To determine whether the graph represents a function, we apply the Vertical Line Test. Here are the steps we follow:

Step 1: Visualize placing a vertical line across various parts of the graph.
Step 2: Check if the vertical line intersects the graph at more than one point at any given position.

Step 1: On evaluating the given graph carefully, there is a notable presence of a vertical line passing through multiple y-values. Specifically, the vertical line goes from $y = -3$ to $y = 3$ at $x = 3$ .

Step 2: Since this vertical line at $x = 3$ intersects the graph at an infinite number of points, it fails the Vertical Line Test.

Therefore, the graph does not represent a function. According to our analysis and the Vertical Line Test, the correct answer is No.

Answer

Exercise #3

Is the given graph a function?

Video Solution

Step-by-Step Solution

To determine if the graph in question represents a function, we'll employ the Vertical Line Test. This test helps to ascertain whether each input value from the domain (x-values) is connected to a unique output value (y-values).

According to the Vertical Line Test, a graph represents a function if no vertical line can intersect the graph at more than one point.
In the provided diagram, the graph is a straight line.
Visual inspection shows that any vertical line drawn at any point along the x-axis intersects the line exactly once.
This indicates that for each x-value, there is a unique corresponding y-value. Therefore, the relationship depicted by the graph meets the criteria for a function.

Thus, the given graph correctly characterizes a function.
Therefore, the solution to the problem is Yes.

Answer

Yes

Exercise #4

Is the given graph a function?

Video Solution

Step-by-Step Solution

To determine if the given graph represents a function, we use the vertical line test: if any vertical line intersects the graph at more than one point, the graph is not a function.

Let's apply this test to the graph:

Examine different sections of the graph by drawing imaginary vertical lines.
Look for intersections where more than one point exists on the vertical line.

Upon examining the graph, we observe that there are several vertical lines that intersect the graph at multiple points, particularly in areas with loops or overlapping curves. This indicates that at those $x$ -values, there are multiple $y$ -values corresponding to them.

Since there exist such vertical lines, according to the vertical line test, the graph does not represent a function.

Thus, the solution to this problem is that the given graph is not a function.

Answer

Exercise #5

Is the given graph a function?

Video Solution

Step-by-Step Solution

It is important to remember that a function is an equation that assigns to each element in domain X one and only one element in range Y

Let's note that in the graph:

$f(0)=2,f(0)=-2$

In other words, there are two values for the same number.

Therefore, the graph is not a function.

Answer

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Graphical Representation of a Function

Test yourself on representations of functions!

Exercises on the graphical representation of a function

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Review questions

Examples with solutions for Graphical Representation of a Function

Exercise #1

Video Solution

Step-by-Step Solution

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Exercise #2

Video Solution

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Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

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Representations of Functions