Graphs of Direct Proportionality Functions

🏆Practice graphical representation

The graphical representation of a function that represents direct proportionality is actually the ability to express an algebraic expression through a graph.

As it is a direct proportionality, the graph will be of a straight line.

A function that represents direct proportionality is a linear function of the family y=ax+b y=ax+b .

The graphical representation of this function is a straight line that is ascending, descending, or parallel to the X X axis but never parallel to the Y Y axis.

Note: we observe the line from left to right.

We can now recognize in the equation of the line what the graphical representation of each function looks like:

(only when the equation is explicit Y Y is isolated on one side and its coefficient is 1 1 )

A - Graphs of Direct Proportionality Functions

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Test yourself on graphical representation!

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Which statement best describes the graph below?

xy

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A -> the slope of the line

When a>0 a > 0 is positive: the line is ascending

1- When a is positive the line is ascending


When a<0 a < 0 is negative: the line is descending

2 -When a is negative the line is descending


When a=0 a = 0 : the line is parallel to the X X axis

When a = 0 the line is parallel to the X axis


B -> the point of intersection with the Y-axis

b b the y-intercept Y Y

b b indicates at which point the line crosses the Y Y axis.

If b b has a positive coefficient, the line will intersect the positive part of the Y Y axis at the point b b .

If b has a negative coefficient, the line will intersect the negative part of the Y Y axis at the point b b .

If b=0 b=0 , the line will cross the Y Y axis at the origin where Y=0 Y=0 .

To know exactly what the graph of the line's equation looks like, we will have to examine both parameters at the same time, both a and b b .


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Examples of Graphical Representation of a Linear Function

Example 1 (use of the graph)

y=5x4 y=5x-4
We will examine the linear equation.

a=5 a=5 The slope is positive, the line ascends
b=4 b=-4 The line crosses the Y Y axis at the point where Y=4 Y=-4

We will plot the graph based on the data:

We will plot the graph based on the data

Keep in mind that this is just a sketch.

If you want to draw the graph accurately, you can construct a table of values for X X and Y Y and find out the points that form the line.


Example 2 (using the table)

The function y=2X y=2X represents a direct proportionality between the values of X X and Y Y . That is, for each value of X X that we input, the value of Y Y will be double.

We will replace three different values and obtain:

for each value of X that we input, the value of Y will be double

Now let's plot the three points on the coordinate system and connect them. This is actually the graph of the function for y=2X y=2X .


Examples and Exercises with Solutions on Graphical Representation of a Function Representing Direct Proportionality

Exercise #1

A straight line has a slope of 6y and passes through the points (6,41) (6,41) .

Which function corresponds to the line described?

Video Solution

Step-by-Step Solution

To solve the exercise, we will start by placing the data we have into the equation of the line:
y = mx + b
41 = 6*6 + b
41 = 36 +b
41-36 = b
5 = b
 
Now we have the data for the equation of the straight line:
 
y = 6x + 5
But it still does not match any of the given options.

Keep in mind that a common factor can be excluded:
y = 2(3x + 2.5)

Answer

y=2(3x+212) y=2(3x+2\frac{1}{2})

Exercise #2

At which point does the graph of the first function (I) intersect the graph of the second function (II)?

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Video Solution

Answer

(4,2) (4,2)

Exercise #3

At what point does the graph intersect the yaxis?

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Video Solution

Answer

(0,2) (0,2)

Exercise #4

At what point does the graph intersect the x axis?

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Video Solution

Answer

Does not cut the axis x

Exercise #5

What representations describe a linear function?

Video Solution

Answer

Answers A + C are correct

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