# 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 $y=ax+b$.

The graphical representation of this function is a straight line that is ascending, descending, or parallel to the $X$ axis but never parallel to the $Y$ 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 $Y$ is isolated on one side and its coefficient is $1$)

## Test yourself on graphical representation!

Which statement best describes the graph below?

## A -> the slope of the line

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

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

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

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

$b$ the y-intercept $Y$

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

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

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

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

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 $b$.

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## Examples of Graphical Representation of a Linear Function

### Example 1 (use of the graph)

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

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

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 $X$ and $Y$ and find out the points that form the line.

### Example 2 (using the table)

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

We will replace three different values and obtain:

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

## 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 $(6,41)$.

Which function corresponds to the line described?

### 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)

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

### Exercise #2

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

### Video Solution

$(4,2)$

### Exercise #3

At what point does the graph intersect the yaxis?

### Video Solution

$(0,2)$

### Exercise #4

At what point does the graph intersect the x axis?

### Video Solution

Does not cut the axis x

### Exercise #5

What representations describe a linear function?