# Graphical representation - Examples, Exercises and Solutions

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 $1$)

## examples with solutions for graphical representation

### 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?

### Video Solution

Answers A + C are correct

## examples with solutions for graphical representation

### Exercise #1

Which expression describes a linear function?

### Video Solution

$y=4x+1$

### Exercise #2

Which expressions represent linear functions and parallel lines?

### Video Solution

$y=2(x+1)$

$y=3+2x$

### Exercise #3

Which of the following represent linear functions and parallel lines?

### Video Solution

$y=\frac{1}{2}x+10$

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

### Exercise #4

Which of the following describes linear functions and parallel lines?

### Video Solution

$y=-4(x+1)$

$y=8x-12(x+1)$

### Exercise #5

Which statement is true according to the graph below?

### Video Solution

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

## examples with solutions for graphical representation

### Exercise #1

Does the first graph of the function pass through the origin of the axes?

### Video Solution

No, it passes through $(3,1)$

### Exercise #2

Given the line parallel to the line $y=4$

and passes through the point $(1,2)$.

Which of the algebraic representations is the corresponding one for the given line?

### Video Solution

$y=2$

### Exercise #3

A straight line with a slope of 2y passes through the point $(3,7)$.

Which equation corresponds to the line?

### Video Solution

$y=2x+1$

### Exercise #4

Given the line parallel to the line $y=2x+5$

and passes through the point $(4,9)$.

Which of the algebraic representations is the corresponding one for the given line?

### Video Solution

$y=2x+1$

### Exercise #5

A straight line with the slope 9y passes through the point $(-5,-8)$.

Which of the following equations corresponds to the line?

### Video Solution

$y=9x+37$