Which statement best describes the graph below? x y

The graph represents a decreasing function.

The graph represents a constant function.

Which statement is true according to the graph below? 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 0 0 0

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

The graph intersects the x axis at the point $$ (4,0) $$.

The graph intersects the y axis at the point $$ (0,4) $$.

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

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

Does line I pass through the origin point of the axes? 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 –5 –5 –5 –4 –4 –4 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 0 0 0 x y II I

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

Yes, it goes through $$ (0,0) $$.

No, it passes through $$ (3,2) $$.

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

No, the second line is cut off at point $$ (1,2) $$.

Which expressions represent linear functions and parallel lines?

$$ y=2(x^2+1) $$ $$ y=2x^2+2 $$

Graphs of Direct Proportionality Functions

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 $1$ )

A - Graphs of Direct Proportionality Functions

Detailed explanation

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Test yourself on graphical representation!

Which statement best describes the graph below?

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A -> the slope of the line

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

1- When a is positive the line is ascending

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

2 -When a 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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Test your knowledge

Question 1

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

Question 2

At what point does the graph intersect the yaxis?

Question 3

At what point does the graph intersect the x axis?

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:

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 $y=2X$ .

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

Exercise #1

Which statement best describes the graph below?

Step-by-Step Solution

To solve this problem, let's analyze the given graph:

The graph shows a straight line plotted on a coordinate grid.
The line is drawn from the lower-left corner to the upper-right corner. This indicates that for increasing values of $x$ , the corresponding $y$ values also increase.

According to the properties of linear graphs:

A line with a positive slope rises from left to right, which indicates an ascending function.
A line with a negative slope falls from left to right, which indicates a descending function.
A line with a zero slope is horizontal, indicating a constant function.

Given that the line in the graph rises as $x$ increases, it has a positive slope. Therefore, it represents an ascending function.

Thus, the correct statement about the graph is: The graph represents an ascending function.

Answer

The graph represents an ascending function.

Exercise #2

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

Video Solution

Step-by-Step Solution

Let's pay attention to the point where the lines intersect. We'll mark it.

We'll find that:

$X=4,Y=2$

Therefore, the point is:

$(4,2)$

Answer

$(4,2)$

Exercise #3

At what point does the graph intersect the x axis?

Video Solution

Step-by-Step Solution

Note that the line intersects only the Y-axis. In other words, it does not go through the X-axis at all.

Therefore, the answer is (d).

Answer

It does not intersect the x axis.

Exercise #4

What representations describe a linear function?

Video Solution

Step-by-Step Solution

To solve this problem, we need to determine which representations describe a linear function by analyzing each given choice:

Choice 1: $y = 1 - 2x$
- This is in the form $y = mx + b$ , where $m = -2$ and $b = 1$ , making it a linear function.
Choice 2: $y = -2x^2 + x$
- The term $x^2$ indicates a quadratic polynomial, which is not linear due to the power of 2 on $x$ .
Choice 3: $y = x$
- In the form $y = mx + b$ , it is $y = 1x + 0$ , with $m = 1$ and $b = 0$ , thus a linear function.
Choice 4: Asserts that both Choice 1 and Choice 3 are correct, which aligns with our analysis.

Based on this examination, choices forming linear functions are ones where the equation stays in the standard linear form $y = mx + b$ with no additional exponents or variable products. Thus, the correct answer is:

Answers A + C are correct

Answer

Answers A + C are correct

Exercise #5

Determine which of the following expressions describes a linear function?

Video Solution

Step-by-Step Solution

Note that in answer A there is an exponent, therefore the answer is incorrect.

Note that in answer C, if we multiply X by X we get X to a power, therefore the answer is incorrect.

Note that in answer D there is an exponent, therefore the answer is incorrect.

In answer B the following formula can be observed.

$y=mx+b$

Answer

$y=4x+1$

Do you know what the answer is?

Question 1

What representations describe a linear function?

Question 2

Determine which of the following expressions describes a linear function?

Question 3

Which of the following represent linear functions and parallel lines?

Start practice

Graphs of Direct Proportionality Functions

Test yourself on graphical representation!

A -> the slope of the line

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

Examples of Graphical Representation of a Linear Function

Example 1 (use of the graph)

Example 2 (using the table)

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

Exercise #1

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

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

Answer

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Graphical Representation