Slope in the Function y=mx

Slope - Examples, Exercises and Solutions
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Slope
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The concept of slope in the function y=mx y=mx expresses the angle between the line and the positive direction of the X X axis.
M M represents the slope of the function – the rate of change of Y Y relative to the rate of change of X X .
When two points on a certain line are known, the slope of the line can be calculated from them. 

If M>0 M>0 is positive - the line rises
If M<0 M<0 is negative - the line falls
If M=0 M=0 the line is parallel to the X X axis. (In a graph like this, where b=0 b=0 the line coincides with the X X axis.)

This calculation is done using the following formula: 

 m=(Y2−Y1)(X2−X1)  m=\frac {(Y2-Y1)}{(X2-X1)}

where the two points (X1,Y1) \left(X1,Y1\right) and (X2,Y2) \left(X2,Y2\right) are on the mentioned line. 

It is important to emphasize that the slope is constant for any line. 

Note:

The greater the slope – the steeper the graph.
The smaller the slope – the more moderate – flatter the graph.
How will you remember this?
Remember that when the slope is equal to 0, the graph is parallel to the X-axis – it is very, very moderate – flat.
Therefore, as it increases, the graph will be steeper.

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Test yourself on slope!

einstein

For the function in front of you, the slope is?

XY

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Examples of questions on the topic of the slope of a function

Which of the graphs has a greater slope?
We see that the orange graph is "flatter" than the purple graph, so the slope of the purple graph is greater.

Example 1:

Given two points (1,5) \left(1,5\right) and (2,8) \left(2,8\right) .

We know that the two points lie on a certain line.
We are asked to find the slope of the line.
We will use the formula mentioned earlier and substitute the values:

( m=\frac{(8-5)}{(2-1)}= \frac{3}{1}=3 )

In other words, the result we obtained is actually the slope of the desired line.

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Test your knowledge
Question 1

For the function in front of you, the slope is?

XY

Question 2

For the function in front of you, the slope is?

XY

Question 3

For the function in front of you, the slope is?

XY

Example 2:

Let's see an example of finding the slope:

Given two points that the line passes through: (2,4),(5,1) \left(2,4\right),\left(5,1\right)
We will calculate the slope using the formula:

m=(1−4)(5−2)=−33=−1 m=\frac{(1-4)}{(5-2)}= \frac{-3}{3}=-1

The slope of the line is −1.
We can sketch a graph, considering the two points it passes through and the fact that its slope is negative – a descending line.

(Image 1)

Do you know what the answer is?
Question 1

For the function in front of you, the slope is?

XY

Question 2

For the function in front of you, the slope is?

XY

Question 3

For the function in front of you, the slope is?

XY

Examples with solutions for Slope

Exercise #1

For the function in front of you, the slope is?

XY

Video Solution

Step-by-Step Solution

To solve this problem, we need to determine the slope of the line depicted on the graph.

First, understand that the slope of a line on a coordinate plane indicates how steep the line is and the direction it is heading. Specifically:

  • A positive slope means the line rises as it goes from left to right.
  • A negative slope means the line falls as it goes from left to right.

Let's examine the graph given:

  • We see that the line starts at a higher point on the left and descends to a lower point on the right side.
  • As we move from the left side of the graph towards the right, the line goes downwards.

This downward trajectory clearly indicates a negative slope because the line is declining as we move horizontally left to right.

Therefore, the slope of this function is Negative.

The correct answer is, therefore, Negative slope.

Answer

Negative slope

Exercise #2

For the function in front of you, the slope is?

XY

Video Solution

Step-by-Step Solution

To determine the slope of the line shown on the graph, we perform a visual analysis:

  • We examine the orientation of the line from left to right.
  • The red line starts at a higher point on the left and descends to a lower point on the right.
  • This indicates a downward movement, which corresponds to a negative slope.

Therefore, by observing the direction of the line, we conclude that the slope of the function is negative. This positional evaluation confirms that the correct answer is negative slope.

Answer

Negative slope

Exercise #3

For the function in front of you, the slope is?

XY

Video Solution

Step-by-Step Solution

To solve this problem, follow these steps:

  • Step 1: Observe the given graph and the plotted line.
  • Step 2: Determine the direction of the line as it moves from left to right across the graph.
  • Step 3: Understand that a line moving downwards from left to right represents a negative slope.

Now, let's work through these steps:

Step 1: The graph shows a straight line that starts higher on the left side and descends towards the right side.

Step 2: As the line moves from left to right, it descends. This is a key indicator of the slope type.

Step 3: A line that moves downward from the left side to the right side of the graph (decreasing in height as it proceeds to the right) is characteristic of a negative slope. Conversely, a positive slope would show a line ascending as it moves rightward.

Therefore, the solution to the problem is the line has a negative slope.

Answer

Negative slope

Exercise #4

For the function in front of you, the slope is?

XY

Video Solution

Step-by-Step Solution

To solve this problem, let's analyze the given graph of the function to determine the slope's sign.

The slope of a line on a graph indicates the line's direction. A line with a positive slope rises as it moves from left to right, indicating that for every step taken to the right (along the x-axis), we move upward. Conversely, a line with a negative slope falls as it moves from left to right, meaning each step to the right results in moving downward.

Examining the graph provided, the red line starts higher on the left and goes downward towards the right visually. This indicates that the line is rising as it goes from left to right, which confirms it has a positive slope.

Therefore, the solution to the problem, regarding the slope of the line, is that it is a Positive slope.

Answer

Positive slope

Exercise #5

For the function in front of you, the slope is?

XY

Video Solution

Step-by-Step Solution

To determine the slope of the line segment shown in the graph, follow these steps:

  • Identify the line segment on the graph; it's shown as a red line from one point to another.
  • Examine the direction the line segment travels from the leftmost point to the rightmost point.
  • Visually analyze whether the line segment is rising or falling as it moves from left to right.

Here is the detailed analysis:
- The red line segment starts lower on the left side and ends higher on the right side.
- This suggests that as we move from left to right, the line is rising.
- In terms of slope, a line that rises as it moves from left to right has a positive slope.

Therefore, the slope of the line segment is positive.

Thus, the correct answer is Positive slope.

Answer

Positive slope

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