A function is a connection between an independent variable (X) and a dependent variable (Y). The relationship between the variables is called a "correspondence rule".
An algebraic representation of a function is actually a description of the relationship between the dependent variable(Y) and the independent variable(X) by means of an equation.
The following is the typical structure of a graphical representation:
Y=X+3, Y=2X−5
For example, if the data is that every month, Daniel earns20.000 dollars.
The algebraic representation will be X for the number of months Yf(X) for the amount earned. f(x)=20000X
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Exercises on algebraic representation of a function
Exercise 1
Task
On the graph of the linear function that passes through the points A(2,10) and B(−5,−4)
Find the slope of the graph.
Solution
m=x2−x1y2−y1
Replace accordingly
x1=−5,y1=−4
x2=2,y2=10
2−(−5)10−(−4)=
714=2
Answer
2
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Test your knowledge
Question 1
Is the given graph a function?
Incorrect
Correct Answer:
No
Question 2
Does the graph below represent a function?
Incorrect
Correct Answer:
Yes
Question 3
Determine whether the following table represents a function
Incorrect
Correct Answer:
Yes
Exercise 2
Task
From the following function, find the slope of the line:
y=−x+1
Solution
In order to find the slope of the line we must remember how is the algebraic representation of a line:
y=mx+b
Where m is the slope, that is, the coefficient of the independent variable is the slope or the slope of the function, therefore in the equation
y=−x+1
m=−1
And from here we deduce that the slope is −1
Answer
m=−1
Exercise 3
Assignment
Find the slope of the straight line through the following points (0,4),(−5,6)
Solution
m=x2−x1y2−y1
Replace accordingly according to the data
x1=0,y1=4
x2=−5,y2=6
−5−06−4=
−52=−52
Answer
−52
Do you know what the answer is?
Question 1
Determine whether the following table represents a constant function:
Incorrect
Correct Answer:
Yes, it does
Question 2
Determine whether the data in the following table represent a constant function
Incorrect
Correct Answer:
No
Question 3
Determine whether the following table represents a function
Incorrect
Correct Answer:
No
Exercise 4
Task
Given the linear function whose slope of the graph is −3 and passes through the point (−6,−3).
Find the algebraic representation of the function
Solution
y=m⋅x+b
m=−3
Replace accordingly
(−6,−3)
−3=(−3)⋅(−6)+b
−3=18+b
−21=b
y=−3⋅x−21
Answer
y=−3⋅x−21
Exercise 5
Task
On the graph of the linear function passing through the points A(0,7) and B(8,−3)
Find the slope of the graph
Solution
m=x2−x1y2−y1
Replace accordingly using the data
x1=0,y1=7
x2=8,y2=−3
8−0−3−7=
8−10=−45
Answer
−45
Check your understanding
Question 1
Determine whether the following table represents a function
Incorrect
Correct Answer:
Yes
Question 2
Is the given graph a function?
Incorrect
Correct Answer:
No
Question 3
Is the given graph a function?
Incorrect
Correct Answer:
Yes
Review questions
How is an algebraic function represented?
In some related articles, it had been mentioned that a function can be represented in different ways, verbally, algebraically, with tables and graphically. In the case of how to represent a function algebraically, in a few words we can say that it will be represented in an equation, which will indicate the rule of correspondence between the dependent variable Y and the independent variable. X
What is the algebraic representation of a linear function?
The representation of a linear function is the one where we are going to visualize an equation where it represents a straight line, that is to say the independent variable X this with an exponent one, that is to say, of first degree.
Examples
Some algebraic representations of a linear function are the following:
y=x+4
y=−x+1
y=x−5
y=−x−1
We can observe that the variable X has as exponent one, and if we graph these functions we will always get a straight line, so they represent a linear function.
How is the algebraic representation of a quadratic function?
A quadratic function can be seen as an equation where the variable X will have the exponent 2, which will represent a parabola if it is graphed. Some examples are the following:
y=x2+4
y=x2−3
y=−x2+5
How to get the algebraic representation of a table?
A function can be represented verbally, algebraically, in a table of values and graphically, therefore, we can go from one representation to another, in this case we are going to study how to go from a table to an algebraic representation, for such a case let's see the following example:
Example
Assignment
From the following table find its algebraic representation.
Solution:
From the previous table we are going to take any two points, in order to get its slope:
Let the points be
A=(−3,−2),B=(−1,0)
From where:
X1=−3,Y1=−2
X2=−1,Y2=0
Then we substitute in the formula of the slope
m=x2−x1y2−y1
m=−1−(−3)0−(−2)
m=−1−(−3)0−(−2)=−1+30+2
m=−1+30+2=22=1
Then m=1
Now taking the first point
A=(−3,−2)
We substitute in the equation of the line:
y=mx+b
−2=1⋅−3+b
−2=−3+b
−2+3=b
1=b
Knowing m and b
Substitute again in the equation of the line
y=mx+b
y=1⋅x+1
y=x+1
Result
The algebraic expression is y=x+1.
Do you think you will be able to solve it?
Question 1
Is the given graph a function?
Incorrect
Correct Answer:
No
Question 2
Is the given graph a function?
Incorrect
Correct Answer:
Yes
Question 3
Which of the following equations corresponds to the function represented in the graph?
Incorrect
Correct Answer:
\( y=-x-2 \)
Examples with solutions for Algebraic Representation of a Function
Exercise #1
Determine whether the given graph is 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
We should note that for every X value found on the graph, there is one and only one corresponding Y value.
Therefore, the graph is indeed a function.
Answer
Yes
Exercise #2
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
No
Exercise #3
Does the graph below represent a function?
Video Solution
Step-by-Step Solution
It is important to remember that a function is an equation that assigns to each value in domain x only one value in range y.
Since we can see that for every x value found on the graph there is only one correspondingy value, the graph is indeed a function.
Answer
Yes
Exercise #4
Determine whether the following table represents a function
Video Solution
Step-by-Step Solution
It is important to remember that a constant function describes a situation where as the X value increases, the function value (Y) remains constant.
In the table, we can observe that there is a constant change in X values, meaning an increase of 1, and a constant change in Y values, meaning an increase of 3
Therefore, according to the rule, the table describes a function.
Answer
Yes
Exercise #5
Determine whether the following table represents a constant function:
Video Solution
Step-by-Step Solution
It is important to remember that a constant function describes a situation where, as the X value increases, the Y value remains constant.
In the table, we can see that there is a constant change in the X values, specifically an increase of 2, while the Y value remains constant.
Therefore, the table does indeed describe a constant function.