Graphical Representation - Examples, Exercises and Solutions

Question Types:
Graphical Representation: Using additional geometric shapesGraphical Representation: Determining the slope of a function with a pointGraphical Representation: Finding points on the graph of a functionGraphical Representation: Identifying the algebraic expression of a linear function

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

Detailed explanation

Suggested Topics to Practice in Advance

  1. Function
  2. Linear Function
  3. The Linear Function y=mx+b
  4. Slope in the Function y=mx
  5. Positive and Negativity of a Linear Function
  6. Finding a Linear Equation

Practice Graphical Representation

Question 1

Which statement is true according to the graph below?

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Question 2

Choose the correct answer

\( \)xy

Question 3

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

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Question 4

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

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Question 5

At what point does the graph intersect the x axis?

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Examples with solutions for Graphical Representation

Exercise #1

Which statement is true according to the graph below?

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Video Solution

Step-by-Step Solution

If we plot all the points, we'll notice that point (3,5) (3,5) is the correct one, because:

x=3,y=5 x=3,y=5

And they intersect exactly on the line where the graph passes.

Answer

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

Exercise #2

Choose the correct answer

xy

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 #3

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

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Video Solution

Step-by-Step Solution

Let's remember that the origin of the coordinate system is: (0,0) (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) (3,1) , we'll see that it lies on the first graph (the blue one)

Answer

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

Exercise #4

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

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 X=4,Y=2

Therefore, the point is:

(4,2) (4,2)

Answer

(4,2) (4,2)

Exercise #5

At what point does the graph intersect the x axis?

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Video Solution

Step-by-Step Solution

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

Therefore, the answer is D.

Answer

Does not cut the axis x

Question 1

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

Which function corresponds to the line described?

Question 2

At what point does the graph intersect the yaxis?

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Question 3

Which expression describes a linear function?

Question 4

Which expressions represent linear functions and parallel lines?

Question 5

What representations describe a linear function?

Exercise #6

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 inserting the available data 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 #7

At what point does the graph intersect the yaxis?

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Video Solution

Answer

(0,2) (0,2)

Exercise #8

Which expression describes a linear function?

Video Solution

Answer

y=4x+1 y=4x+1

Exercise #9

Which expressions represent linear functions and parallel lines?

Video Solution

Answer

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

y=3+2x y=3+2x

Exercise #10

What representations describe a linear function?

Video Solution

Answer

Answers A + C are correct

Question 1

Which of the following describes linear functions and parallel lines?

Question 2

Which of the following represent linear functions and parallel lines?

Question 3

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

Which equation corresponds to the line?

Question 4

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

Which of the following equations corresponds to the line?

Question 5

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?

Exercise #11

Which of the following describes linear functions and parallel lines?

Video Solution

Answer

y=−4(x+1) y=-4(x+1)

y=8x−12(x+1) y=8x-12(x+1)

Exercise #12

Which of the following represent linear functions and parallel lines?

Video Solution

Answer

y=12x+10 y=\frac{1}{2}x+10

y=12(x+2) y=\frac{1}{2}(x+2)

Exercise #13

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

Which equation corresponds to the line?

Video Solution

Answer

y=2x+1 y=2x+1

Exercise #14

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

Which of the following equations corresponds to the line?

Video Solution

Answer

y=9x+37 y=9x+37

Exercise #15

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

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

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

Video Solution

Answer

y=2 y=2

Topics learned in later sections

  1. Representation of Phenomena Using Linear Functions