Does the first graph of the function pass through the origin 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

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

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

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

Which expressions represent linear functions and parallel lines?

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

Which of the following describes linear functions and parallel lines?

$$ y=-4(x+1) $$ $$ y=8x-12(x+1) $$

Which of the following represent linear functions and parallel lines?

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

Which statement best describes the graph below? x y

The graph represents an ascending function.

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 $$ (5,3) $$.

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!

At what point does the graph intersect the x axis?

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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 what point does the graph intersect the yaxis?

Question 2

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

Question 3

Choose the correct answer

$$

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

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 #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

Choose the correct answer

Step-by-Step Solution

The blue line is a straight line, therefore it remains constant.

Let's note that the red line is rising because it starts in the negative part (negative values) and rises to the positive part (positive values).

Therefore, the correct answer is D.

Answer

Answers B and C are correct

Exercise #4

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

Video Solution

Step-by-Step Solution

Let's remember that the origin of the coordinate system is: $(0,0)$

We'll highlight the point on the graph, and note that it doesn't lie on any of the plotted graphs.

Therefore, the answer is C. If we plot the point $(3,1)$ , we'll see that it lies on the first graph (the blue one)

Answer

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

Exercise #5

Which expression 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.

Let's look at answer B and see that the formula indeed holds:

$y=mx+b$

Answer

$y=4x+1$

Do you know what the answer is?

Question 1

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

Question 2

What representations describe a linear function?

Question 3

Which expression describes a linear function?

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

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

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