# Slope in the Function y=mx

šPractice slope

The concept of slope in the function $y=mx$ expresses the angle between the line and the positive direction of the $X$ axis.
$M$ represents the slope of the function ā the rate of change of $Y$ relative to the rate of change of $X$.
When two points on a certain line are known, the slope of the line can be calculated from them.Ā

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

This calculation is done using the following formula:Ā

$Ā m=\frac {(Y2-Y1)}{(X2-X1)}$

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

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

Note:

The greater the slope ā the steeper the graph.
The smaller the slope ā the more moderate ā 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 ā it is very, very moderate ā flat.
Therefore, as it increases, the graph will be steeper.

## Test yourself on slope!

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

## 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 $\left(1,5\right)$ and $\left(2,8\right)$.

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.

Join Over 30,000 Students Excelling in Math!
Endless Practice, Expert Guidance - Elevate Your Math Skills Today

## Example 2:

Let's see an example of finding the slope:

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

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

The slope of the line is ā1.
We can sketch a graph, considering the two points it passes through and the fact that its slope is negative ā a descending line.

(Image 1)

Do you know what the answer is?

## Examples with solutions for Slope

### Exercise #1

Given the linear function:

$y=x-4$

What is the rate of change of the function?

### Step-by-Step Solution

Let's remember that the rate of change equals the slope.

In this case, the slope is:

$m=1$

$m=1$

### Exercise #2

Which best describes the function below?

$y=2-3x$

### Step-by-Step Solution

Remember that the rate of change equals the slope.

In this function:

$m=-3$

Therefore, the function is decreasing.

The function is decreasing.

### Exercise #3

Choose the correct answer for the function.

$y=-x+1$

### Step-by-Step Solution

In a linear function, to check if the functions are parallel, you must verify if their slope is the same.

y = ax+b

The slope is a

In the original formula:

Ā y = -x+1

The slope is 1

In option A there is no a at all, which means it equals 1, which means the slope is not the same and the option is incorrect.

Ā

Option B:

To check if the function passes through the points, we will try to place them in the function:

-1 = -(-2)+1

-1 = 2+1

-1 = 3

The points do not match, and therefore the function does not pass through this point.

Ā

Option C:

We rearrange the function, in a way that is more convenient:

y = -1-x

y = -x-1

You can see that the slope in the function is the same as we found for the original function (-1), so this is the solution!

Ā

Option D:

When the slope is negative, the function is decreasing, as the slope is -1, the function is negative and this answer is incorrect.

The graph is parallel to the graph of function

$y=-1-x$

### Exercise #4

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

Negative slope

### Exercise #5

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